Research ArticleIntegration in Orlicz-Bochner Spaces

Marian Nowak

Faculty of Mathematics Computer Science and Econometrics University of Zielona Gora Ul Szafrana 4A65ndash516 Zielona Gora Poland

Correspondence should be addressed to Marian Nowak mnowakwmieuzzgorapl

Received 25 January 2018 Accepted 31 March 2018 Published 14 May 2018

Academic Editor Alberto Fiorenza

Copyright copy 2018 Marian Nowak This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Let (Ω Σ 120583) be a complete 120590-finite measure space 120593 be a Young function and 119883 and 119884 be Banach spaces Let 119871120593(119883) denote theOrlicz-Bochner space andTand

120593 denote the finest Lebesgue topology on 119871120593(119883) We study the problem of integral representation of(Tand120593 sdot119884)-continuous linear operators119879 119871120593(119883) rarr 119884with respect to the representing operator-valuedmeasuresThe relationships

between (Tand120593 sdot 119884)-continuous linear operators 119879 119871120593(119883) rarr 119884 and the topological properties of their representing operator

measures are established

1 Introduction and Preliminaries

Throughout the paper (119883 sdot 119883) and (119884 sdot 119884) denote realBanach spaces and 119883lowast and 119884lowast denote their Banach dualsrespectively By 119861119883 and 119861119884lowast we denote the closed unit ball in119883 and in 119884lowast LetL(119883 119884) stand for the space of all boundedoperators from119883 and119884 equipped with the uniform operatornorm sdot

We assume that (Ω Σ 120583) is a complete 120590-finite measurespace Denote by Σ119891(120583) the 120575-ring of sets 119860 isin Σ with 120583(119860) ltinfin By 1198710(119883) we denote the linear space of 120583-equivalenceclasses of all strongly Σ-measurable functions 119891 Ω rarr 119883equipped with the topology T0 of convergence in measureon sets of finite measure

Nowwe recall the basic concepts and properties ofOrlicz-Bochner spaces (see [1ndash6] for more details)

By a Young function we mean here a continuous convexmapping 120593 [0infin) rarr [0infin) that vanishes only at 0 and120593(119905)119905 rarr 0 as 119905 rarr 0 and 120593(119905)119905 rarr infin as 119905 rarr infin Let 120593lowast standfor the complementary Young function of 120593 in the sense ofYoung

Let 119871120593(119883) (resp 119871120593) denote the Orlicz-Bochner space(resp Orlicz space) defined by a Young function 120593 thatis

119871120593 (119883) fl 119891 isin 1198710 (119883) intΩ120593 (120582 1003817100381710038171003817119891 (120596)1003817100381710038171003817119883) 119889120583

lt infin for some 120582 gt 0 = 119891 isin 1198710 (119883) 1003817100381710038171003817119891 (sdot)1003817100381710038171003817119883isin 119871120593

(1)

Then 119871120593(119883) equipped with the topologyT120593 of the norm

10038171003817100381710038171198911003817100381710038171003817120593 fl inf 120582 gt 0 intΩ120593(1003817100381710038171003817119891 (120596)1003817100381710038171003817119883120582 )119889120583 le 1 (2)

is a Banach space For a sequence (119891119899) in 119871120593(119883) 119891119899120593 rarr 0 ifand only if int

Ω120593(120582119891119899(120596)119883)119889120583 rarr 0 for all 120582 gt 0 Let119861119871120593(119883) fl 119891 isin 119871120593 (119883) 10038171003817100381710038171198911003817100381710038171003817120593 le 1 (3)

Let

119864120593 (119883)= 119891 isin 1198710 (119883) int

Ω120593 (120582 1003817100381710038171003817119891 (120596)1003817100381710038171003817119883) 119889120583 lt infin forall120582 gt 0 (4)

Then 119864120593(119883) is a sdot 120593-closed subspace of 119871120593(119883)

HindawiJournal of Function SpacesVolume 2018 Article ID 9380350 9 pageshttpsdoiorg10115520189380350

2 Journal of Function Spaces

Recall that a subset119867 of119871120593(119883) is said to be solidwhenever1198911(120596)119883 le 1198912(120596)119883 120583-ae and 1198911 isin 119871120593(119883) 1198912 isin 119867 imply1198911 isin 119867 A linear topology 120585 on 119871120593(119883) is said to be locally solidif it has a local basis at 0 consisting of solid sets (see [4])

According to [7 Definition 22] and [6] we have thefollowing definition

Definition 1 A locally solid topology 120585 on 119871120593(119883) is said to bea Lebesgue topology if for a net (119891120572) in 119871120593(119883) 119891120572(sdot)119883 (o)997888997888rarr 0in the Banach lattice 119871120593 implies 119891120572 rarr 0 in 120585

In view of the super Dedekind completeness of 119871120593 onecan restrict in the above definition to usual sequences (119891119899) in119871120593(119883) (see [7 Definition 22 p 173])

Note that for a sequence (119891119899) in 119871120593(119883) 119891119899(sdot)119883 (o)997888997888rarr 0in 119871120593 if and only if 119891119899(120596)119883 rarr 0 120583-ae and 119891119899(120596)119883 le119906(120596) 120583-ae for some 0 le 119906 isin 119871120593

For 120576 gt 0 let 119880120593(120576) = 119891 isin 119871120593(119883) intΩ120593(119891(120596)119883)119889120583 le120576 Then the family of all sets of the form

infin⋃119899=1

( 119899sum119894=1

119880120593 (120576119894)) (lowast)where (120576119899) is a sequence of positive numbers and is a localbasis at 0 for a linear topology Tand

120593 on 119871120593(119883) (see [4 6] formore details) Using [4 Lemma 11] one can show that thesets of the form (lowast) are convex and solid so Tand

120593 is a locallyconvex-solid topology

We now recall terminology and basic facts concerning thespaces of weaklowast-measurable functions 119892 Ω rarr 119883lowast (see[8 9]) Given a function 119892 Ω rarr 119883lowast and 119909 isin 119883 let119892119909(120596) = 119892(120596)(119909) for 120596 isin Ω By 1198710(119883lowast 119883) we denote thelinear space of the weaklowast-equivalence classes of all weaklowast-measurable functions 119892 Ω rarr 119883lowast In view of the superDedekind completeness of 1198710 the set |119892119909| 119909 isin 119861119883 is orderbounded in 1198710 for each 119892 isin 1198710(119883lowast 119883) Thus one can definethe so-called abstract norm120599 1198710(119883lowast 119883) rarr 1198710 by

120599 (119892) fl sup 10038161003816100381610038161198921199091003816100381610038161003816 119909 isin 119861119883 in 1198710 (5)

One can easy check that the following properties of 120599 hold120599(119892) = 0 if and only if 119892 = 0 and 119892 isin 1198710(119883lowast 119883)120599(120582119892) = |120582|120593(119892) for 120582 isin R and 119892 isin 1198710(119883lowast 119883)120599(1198921 + 1198922) le 120599(1198921) + 120599(1198922) if 1198921 1198922 isin 1198710(119883lowast 119883)120599(1119860119892) = 1119860120599(119892) for 119860 isin Σ and 119892 isin 1198710(119883lowast 119883)

It is known that for 119891 isin 1198710(119883) 119892 isin 1198710(119883lowast 119883) the function⟨119891 119892⟩ Ω rarr R defined by ⟨119891 119892⟩(120596) = ⟨119891(120596) 119892(120596)⟩ ismeasurable and

1003816100381610038161003816⟨119891 (120596) 119892 (120596)⟩1003816100381610038161003816 le 1003817100381710038171003817119891 (120596)1003817100381710038171003817119883 120599 (119892) (120596) 120583-ae (6)

Moreover 120599(119892) = 119892(sdot)119883lowast for 119892 isin 1198710(119883lowast) Let119871120593lowast (119883lowast 119883) fl 119892 isin 1198710 (119883lowast 119883) 120599 (119892) isin 119871120593lowast (7)

Clearly 119871120593lowast(119883lowast) sub 119871120593lowast(119883lowast 119883) If in particular 119883lowast has theRadon-Nikodymproperty (ie119883 is anAsplund space see [10p 213]) then 119871120593lowast(119883lowast 119883) = 119871120593lowast(119883lowast)

Let 119871120593(119883)lowast stand for the Banach dual of 119871120593(119883) equippedwith the conjugate norm sdot lowast120593

Recall that a Young function 120593 satisfies the Δ 2-conditionif 120593(2119905) le 119889120593(119905) for some 119889 gt 1 and all 119905 ge 0 We shall saythat a Young function 120595 is completely weaker than another120593 (in symbols 120595 ⊲ 120593) if for an arbitrary 119888 gt 1 there exists119889 gt 1 such that 120595(119888119905) le 119889120593(119905) for all 119905 ge 0 Note that a Youngfunction 120593 satisfies the Δ 2-condition if and only if 120593 ⊲ 120593 If120595 ⊲ 120593 then 119871120593 sub 119864120595 and it follows that 119871120593(119883) sub 119864120595(119883)

Now we present basic properties of the topology Tand120593 on119871120593(119883)

Theorem 2 Let 120593 be a Young function Then the followingstatements hold

(i) Tand120593 sub T120593 andTand

120593 = T120593 if120593 satisfies theΔ 2-condition(ii) Tand

120593 is the finest Lebesgue topology on 119871120593(119883)(iii) Tand

120593 is generated by the family of norms sdot 120595|119871120593(119883) 120595 ⊲ 120593(iv) (119871120593(119883)Tand

120593)lowast = 119865119892 119892 isin 119871120593lowast(119883lowast 119883) where for119892 isin 119871120593lowast(119883lowast 119883)

119865119892 (119891) = intΩ⟨119891 (120596) 119892 (120596)⟩ 119889120583 for 119891 isin 119871120593 (119883)

1003817100381710038171003817100381711986511989210038171003817100381710038171003817lowast120593 = sup intΩ

1003817100381710038171003817119891 (120596)1003817100381710038171003817119883 120599 (119892) (120596) 119889120583 119891 isin 119861119871120593(119883)= 1003817100381710038171003817120599 (119892)1003817100381710038171003817120593lowast

(8)

(v) (119871120593(119883)Tand120593) is a closed subset of the Banach space(119871120593(119883) sdot lowast120593)

(vi) If119883lowast has the Radon-Nikodym property then the space(119871120593(119883)Tand120593) is strongly Mackey hence Tand

120593 coincideswith the Mackey topology 120591(119871120593(119883) 119871120593lowast(119883lowast))

Proof (i)ndash(iii) See [4 Theorems 61 63 and 65](iv) In view of [6 Corollary 44 andTheorem 12] we get(119871120593(119883)Tand

120593)lowast = 119871120593(119883)sim119899 where 119871120593(119883)sim119899 stands for the ordercontinuous dual of 119871120593(119883) (see [7 8 11] for more details)According to [8 Theorem 41] 119871120593(119883)sim119899 = 119865119892 119892 isin119871120593lowast(119883lowast 119883)

Using [11 Theorem 13] for 119892 isin 119871120593lowast(119883lowast 119883) we have1003817100381710038171003817100381711986511989210038171003817100381710038171003817lowast120593 fl sup 1003816100381610038161003816100381610038161003816intΩ ⟨119891 (120596) 119892 (120596)⟩ 119889120583

1003816100381610038161003816100381610038161003816 119891 isin 119861119871120593(119883)= sup int

Ω

1003817100381710038171003817119891 (120596)1003817100381710038171003817119883 120599 (119892) (120596) 119889120583 119891 isin 119861119871120593(119883)= 1003817100381710038171003817120599 (119892)1003817100381710038171003817120593lowast

(9)

(v) See [12 sect 3 Theorem 2](vi) See [6 Theorem 45]

Journal of Function Spaces 3

Let 120574120593[T120593T0] (briefly 120574120593) denote the natural mixedtopology on 119871120593(119883) that is 120574120593 is the finest linear topologythat agrees with T0 on sdot 120593-bounded sets in 119871120593(119883) (see[5 13 14] for more details) Then 120574120593 is a locally convex-solidHausdorff topology (see [14 Theorem 32]) and 120574120593 and T120593

have the same bounded sets This means that (119871120593(119883) 120574120593)is a generalized DF-space (see [15]) and its follows that(119871120593(119883) 120574120593) is quasinormable (see [15 p 422]) Moreover fora sequence (119891119899) in 119871120593(119883) 119891119899 rarr 0 in 120574120593 if and only if 119891119899 rarr 0inT0 and sup119899119891119899120593 lt infin (see [14 Theorem 31])

We say that a Young function 120593 increases essentially morerapidly than another 120595 (in symbols 120595 ≪ 120593) if for arbitrary119888 gt 0 120595(119888119905)120593(119905) rarr 0 as 119905 rarr 0 and 119905 rarr infin

Theorem 3 Let 120593 be a Young function Then the mixedtopology 120574120593 on 119871120593(119883) is generated by the family of norms sdot 120595|119871120593(119883) 120595 ≪ 120593Proof It is known that the mixed topology 120574120593 on 119871120593 isgenerated by the family of norms sdot 120595|119871120593 120595 ≪ 120593 (see [16Theorem 21]) Since 119891120595 = 119891(sdot)119883120595 for 119891 isin 119871120593(119883) by[14 (54) p 97] the mixed topology 120574120593 on 119871120593(119883) is generatedby the family of norms sdot 120595|119871120593(119883) 120595 ≪ 120593

Since 120595 ≪ 120593 implies 120595 ⊲ 120593 in view of Theorems 2 and 3we get

120574120593 sub Tand120593 (10)

The problem of integral representation of bounded lin-ear operators on Banach function spaces of vector-valuedfunctions to Banach spaces in terms of the correspondingoperator-valued measures has been the object of much study(see [5 17ndash24]) In particular Dinculeanu (see [19 sect 13 Sect3] [20] [21 sect 8 Sect B]) studied the problem of integralrepresentation of bounded linear operators from 119871119901(119883) to aBanach space 119884 It is known that if 1 le 119901 lt infin 120583(Ω) lt infinand an operator measure 119898 Σ rarr L(119883 119884) vanishes on120583-null sets and has the finite 119902-semivariation 119902(Ω) (1 lt119902 le infin 1119901 + 1119902 = 1) then one can define the integralintΩ119891119889119898 for all 119891 isin 119871119901(119883) Moreover if 119879 119871119901(119883) rarr 119884

is a bounded linear operator then the associated operatormeasure 119898 Σ rarr L(119883 119884) has the finite 119902-semivariation119902(Ω) and 119879(119891) = intΩ 119891119889119898 for all 119891 isin 119871119901(119883) (see [19 sect 13Theorem 1 p 259] [20 Theorem 4]) The relationships of the119902-semivariation 119902 to the properties of operators from 119871119901(119883)to 119884 were studied in [22] Diestel [23] found the integralrepresentation of bounded linear operators from an Orlicz-Bochner space 119871120593(119883) to a Banach spaces if 120583(Ω) lt infin and aYoung 120593 satisfies the Δ 2-condition

The present paper is a continuation of [5] where weestablish integral representation of (120574120593 sdot 119884)-continuouslinear operators 119879 119871120593(119883) rarr 119884 We study the problemof integration of functions in 119871120593(119883) with respect to therepresenting operator measures of (Tand

120593 sdot 119884)-continuouslinear operators 119879 119871120593(119883) rarr 119884 An integral representationtheorem for (Tand

120593 sdot 119884)-continuous linear operators 119879 119871120593(119883) rarr 119884 is established (see Theorem 9 below) We study

the relationships between (Tand120593 sdot 119884)-continuous operators119879 119871120593(119883) rarr 119884 and the properties of their representing

measures119898 Σ119891(120583) rarrL(119883 119884)2 120593lowast-Semivariation of Operator Measures

Assume that 119898 Σ119891(120583) rarr L(119883 119884) is an additive measuresuch that119898 ≪ 120583 that is119898(119860) = 0 if 120583(119860) = 0

Let S(Σ119891(120583) 119883) denote the space of all 119883-valued Σ119891(120583)-simple functions on Ω Then 119904 isin S(Σ119891(120583) 119883) if 119904 = sum(1119860119894 otimes119909119894) where (119860 119894) is a finite pairwise disjoint sequence in Σ119891(120583)and 119909119894 isin 119883 For 119904 = sum119899119894=1(1119860119894 otimes 119909119894) isin S(Σ119891(120583) 119883) and 119860 isin Σwe can define the integralint

119860119904 119889119898 by

int119860119904 119889119898 fl

119899sum119894=1

119898(119860 119894 cap 119860) (119909119894) (11)

Note that

int119860119904 119889119898 = int

Ω1119860119904 119889119898 (12)

For 119910lowast isin 119884lowast we define a measure 119898119910lowast Σ119891(120583) rarr 119883lowast by theequality

119898119910lowast (119860) (119909) fl 119910lowast (119898 (119860) (119909))for 119860 isin Σ119891 (120583) 119909 isin 119883 (13)

For 119904 = sum119899119894=1(1119860119894 otimes 119909119894) isin S(Σ119891(120583) 119883) and 119860 isin Σ wedefine the integral int

119860119904 119889119898119910lowast by the equality

int119860119904 119889119898119910lowast fl 119899sum

119894=1

119898119910lowast (119860 119894 cap 119860) (119909119894) (14)

Then

119910lowast (int119860119904 119889119898) = int

119860119904 119889119898119910lowast (15)

Following [23] [19 sect 13] one can define the 120593lowast-semivariation120593lowast(119860) of119898 on 119860 isin Σ by

120593lowast (119860) fl sup1003817100381710038171003817100381710038171003817100381710038171003817119899sum119894=1

119898(119860 cap 119860 119894) (119909119894)1003817100381710038171003817100381710038171003817100381710038171003817119884 (16)

where the supremum is taken over all finite pairwise disjointsets 1198601 119860119899 in Σ119891(120583) and 119909119894 isin 119883 for 119894 = 1 119899 suchthat sum119899119894=1(1119860119894 otimes 119909119894)120593 le 1

One can observe that

120593lowast (119860)= sup1003817100381710038171003817100381710038171003817int119860 119904 119889119898

1003817100381710038171003817100381710038171003817119884 119904 isin S (Σ119891 (120583) 119883) 119904120593 le 1 (17)

Note that

120593lowast (119860) le 120593lowast (119861) if 119860 119861 isin Σ with 119860 sub 119861120593lowast (119860 cup 119861) le 120593lowast (119860) + 120593lowast (119861) for 119860 119861 isin Σ (18)

4 Journal of Function Spaces

Let (119898119910lowast)120593lowast(119860) stand for the 120593lowast-semivariation of119898119910lowast on 119860 isinΣ that is(119898119910lowast)120593lowast (119860)= sup 1003816100381610038161003816100381610038161003816int119860 119904 119889119898119910lowast

1003816100381610038161003816100381610038161003816 119904 isin S (Σ119891 (120583) 119883) 119904120593 le 1 (19)

The following lemma will be useful

Lemma 4 Let 120593 be a Young function and 119898 Σ119891(120583) rarrL(119883 119884) be a measure with 119898 ≪ 120583 and 120593lowast(Ω) lt infin Thenthe following statements hold

(i) If119891 isin 119864120593(119883) then there exists a sdot120593-Cauchy sequence(119904119899) inS(Σ119891(120583) 119883) such that 119904119899(120596)minus119891(120596)119883 rarr 0 120583-ae

(ii) If (119904119899) is a sdot 120593-Cauchy sequence inS(Σ119891(120583) 119883) thenfor119860 isin Σ (int

119860119904119899119889119898) is a Cauchy sequence in a Banach

space119884 and for every119910lowast isin 119884lowast (int119860119904119899119889119898119910lowast) is aCauchy

sequence in R(iii) If 119891 isin 119864120593(119883) and (1199041015840119899) and (11990410158401015840119899 ) are sdot 120593-Cauchy

sequence inS(Σ119891(120583) 119883) such that 1199041015840119899(120596)minus119891(120596)119883 rarr0 120583-ae and 11990410158401015840119899 (120596) minus 119891(120596)119883 rarr 0 120583-ae then for119860 isin Σ one haslimint

1198601199041015840119899119889119898 = limint

11986011990410158401015840119899 119889119898 (20)

and for every 119910lowast isin 119884lowast one haslimint

1198601199041015840119899119889119898119910lowast = limint

11986011990410158401015840119899 119889119898119910lowast (21)

Proof (i) Let 119891 isin 119864120593(119883) Then there exists a sequence (119904119899)in S(Σ119891(120583) 119883) such that 119904119899(120596) minus 119891(120596)119883 rarr 0 120583-ae and119904119899(120596)119883 le 119891(120596)119883 120583-ae for all 119899 isin N (see [21 Theorem 6p 4]) Using the Lebesgue dominated convergence theoremwe obtain that int

Ω120593(120582(119904119899(120596) minus 119891(120596)119883)119889120583 rarr 0 for all 120582 gt 0

so 119904119899 minus 119891120593 rarr 0 Hence (119904119899) is a sdot 120593-Cauchy sequence(ii) Assume that (119904119899) is a sdot 120593-Cauchy sequence in

S(Σ119891(120583) 119883) Hence for 119899 119896 isin N we have

1003817100381710038171003817100381710038171003817int119860 119904119899119889119898 minus int1198601199041198961198891198981003817100381710038171003817100381710038171003817119884 =

1003817100381710038171003817100381710038171003817int119860 (119904119899 minus 119904119896) 1198891198981003817100381710038171003817100381710038171003817119884

le 1003817100381710038171003817119904119899 minus 1199041198961003817100381710038171003817120593 120593lowast (119860) le 1003817100381710038171003817119904119899 minus 1199041198961003817100381710038171003817120593 120593lowast (Ω) (22)

It follows that (int119860119904119899119889119898) is a Cauchy sequence in 119884 Hence in

view of (15) for 119910lowast isin 119884lowast (int119860119904119899119889119898119910lowast) is a Cauchy sequence

in R(iii) Note that (1199041015840119899 minus 11990410158401015840119899 ) is a sdot 120593-Cauchy sequence and1199041015840119899(120596) minus 11990410158401015840119899 (120596)119883 rarr 0 120583-ae Hence there exists ℎ isin 119864120593(119883)

such that (1199041015840119899minus11990410158401015840119899 )minusℎ120593 rarr 0 Note thatT0|119864120593(119883) sub T120593|119864120593(119883)Hence (1199041015840119899minus11990410158401015840119899 )minusℎ rarr 0 inT0 and it follows that there exists asubsequence (1199041015840119896119899minus11990410158401015840119896119899) of (1199041015840119899minus11990410158401015840119899 ) such that (1199041015840119896119899(120596)minus11990410158401015840119896119899(120596))minus

ℎ(120596)119883 rarr 0 120583-ae Then ℎ(120596) = 0 120583-ae so 1199041015840119899 minus 11990410158401015840119899 120593 rarr 0and for 119860 isin Σ we get

1003817100381710038171003817100381710038171003817int119860 1199041015840119899119889119898 minus int11986011990410158401015840119899 1198891198981003817100381710038171003817100381710038171003817119884 =

1003817100381710038171003817100381710038171003817int119860 (1199041015840119899 minus 11990410158401015840119899 ) 1198891198981003817100381710038171003817100381710038171003817119884

le 100381710038171003817100381710038171199041015840119899 minus 11990410158401015840119899 10038171003817100381710038171003817120593 120593lowast (119860) (23)

It follows that

limint1198601199041015840119899119889119898 = limint

11986011990410158401015840119899 119889119898 (24)

and hence in view of (15) for every 119910lowast isin 119884lowast we havelimint

1198601199041015840119899119889119898119910lowast = limint

11986011990410158401015840119899 119889119898119910lowast (25)

Following [21 sect 13 Definition 1 p 254] in view ofLemma 4 we have the following

Definition 5 Let 120593 be a Young function and 119898 Σ119891(120583) rarrL(119883 119884) be an additive measure such that 119898 ≪ 120583 and120593lowast(Ω) lt infin Then for every 119891 isin 119864120593(119883) and 119860 isin Σ wecan define the integralint

119860119891119889119898 by the equality

int119860119891119889119898 fl limint

119860119904119899119889119898 (26)

and for 119910lowast isin 119884lowast we can define the integralint119860119891119889119898119910lowast by the

equality

int119860119891119889119898119910lowast fl limint

119860119904119899119889119898119910lowast (27)

where (119904119899) is an arbitrary sdot 120593-Cauchy sequence inS(Σ119891(120583) 119883) such that 119904119899(120596) minus 119891(120596)119883 rarr 0 120583-ae3 Integral Representation of ContinuousOperators on Orlicz-Bochner Spaces

For a bounded linear operator 119879 119871120593(119883) rarr 119884 let

119879120593 fl sup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) (28)

Proposition 6 Let 119879 119871120593(119883) rarr 119884 be a bounded linearoperator and

119898(119860) (119909) fl 119879 (1119860 otimes 119909) for 119860 isin Σ119891 (120583) 119909 isin 119883 (29)

Then the following statements hold

(i) For 119860 isin Σ119891(120583)119898(119860) isin L(119883 119884) and 119898(119860) le 119879120593 sdot1119860120593(ii) 119898 ≪ 120583(iii) 119898(119860119899) rarr 0 if 119860119899 darr 0 with 119860119899 isin Σ119891(120583)(iv) 119898 Σ119891(120583) rarr L(119883 119884) is countably additive that is119898(⋃infin119899=1 119861119899) = suminfin119899=1119898(119861119899) if (119861119899) is a pairwise disjoint

sequence in Σ119891(120583) with⋃infin119899=1 119861119899 isin Σ119891(120583)(v) 120593lowast(Ω) le 119879120593

Journal of Function Spaces 5

Proof (i) Let 119860 isin Σ119891(120583) Then for 119909 isin 119861119883 we have 1119860 otimes119909120593 le 1119860120593 and hence

119898 (119860) (119909)119884 = 1003817100381710038171003817119879 (1119860 otimes 119909)1003817100381710038171003817119884 le 119879120593 sdot 10038171003817100381710038171119860 otimes 1199091003817100381710038171003817120593le 119879120593 100381710038171003817100381711198601003817100381710038171003817120593 (30)

so 119898(119860) le 119879120593 sdot 1119860120593(ii) This follows from (i) because 1119860120593 = 0 if 120583(119860) = 0(iii) Assume that 119860119899 darr 0 with 119860119899 isin Σ119891(120583) Then

11198601(120596) ge 1119860119899(120596) darr 0 for 120596 isin Ω By the Lebesgue dominatedconvergence theorem we obtain that int

Ω120593(1205821119860119899(120596))119889120583 rarr 0

for every 120582 gt 0 This means that 1119860119899120593 rarr 0 and by (i)119898(119860119899) rarr 0(iv) Assume that (119861119899) is a pairwise disjoint sequence inΣ119891(120583) with 119861 = ⋃infin119899=1 119861119899 isin Σ119891(120583) Let 119860119899 = 119861 ⋃119899119894=1 119861119894 for119899 isin N Then 119860119899 isin Σ119891(120583) and 119860119899 darr 0 Hence by (iii) 119898(119861) minussum119899119894=1119898(119861119894) = 119898(119861) minus 119898(⋃119899119894=1 119861119894) = 119898(119860119899) rarr 0Statement (v) is obvious

Definition 7 Let119879 119871120593(119883) rarr 119884 be a bounded linear operatorand

119898(119860) (119909) fl 119879 (1119860 otimes 119909) for 119860 isin Σ119891 (120583) 119909 isin 119883 (31)

Then the measure 119898 Σ119891(120583) rarr L(119883 119884) will be called arepresenting measure of 119879Proposition 8 Let 119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot 119884)-continuous linear operator and 119898 Σ119891(120583) rarr L(119883 119884) beits representing measure Then there exists a Young function 120595such that 120595 ⊲ 120593 and 120595lowast(Ω) lt infin

Proof According toTheorem 2 there exist a finite set 120595119894 119894 =1 119899 of Young functions with 120595119894 ⊲ 120593 for 119894 = 1 119899 and119886 gt 0 such that1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 le 119886max1le119894le119899

10038171003817100381710038171198911003817100381710038171003817120595119894 forall119891 isin 119871120593 (119883) (32)

Let120595(119905) = max1le119894le119899120595119894(119905) for 119905 ge 0Then120595 is a Young functionwith 120595 ⊲ 120593 and

1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 le 119886 10038171003817100381710038171198911003817100381710038171003817120595 forall119891 isin 119871120593 (119883) (33)

Hence120595lowast (Ω)= sup 119879 (119904)119884 119904 isin S (Σ119891 (120583) 119883) 119904120595 le 1le 119886 lt infin

(34)

For a linear operator 119879 119871120593(119883) rarr 119884 and 119860 isin Σ let119879119860 (119891) fl 119879 (1119860119891) for 119891 isin 119871120593 (119883) (35)

Nowwe can state ourmain result that extends the classicalresults concerning the integral representation of operators onLebesgue-Bochner spaces 119871119901(119883) (1 le 119901 lt infin) (see [19 sect13 Theorem 1 pp 259ndash261]) to operators on Orlicz-Bochnerspaces 119871120593(119883)

Theorem 9 Let 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be its representingmeasure Then for 119860 isin Σ the following statements hold

(i) 119879119860 119871120593(119883) rarr 119884 is a (Tand120593 sdot 119884)-continuous linear

operator(ii) For 119891 isin 119871120593(119883) one has

119879119860 (119891) = int119860119891119889119898 (36)

and for 119910lowast isin 119884lowast one has119910lowast (119879119860 (119891)) = int

119860119891119889119898119910lowast (37)

(iii) For 119891 isin 119871120593(119883) the measure 119898119891 Σ rarr 119884 defined bythe equality

119898119891 (119860) fl int119860119891119889119898 for 119860 isin Σ (38)

is countably additive(iv) 119879119860120593 = 120593lowast(119860)

and for 119910lowast isin 119884lowast 119910lowast ∘ 119879119860lowast120593 = (119910lowast ∘ 119879)119860lowast120593 =(119898119910lowast)120593lowast(119860)(v) 120593lowast(119860) = sup(119898119910lowast)120593lowast(119860) 119910lowast isin 119861119884lowast(vi) For 119891 isin 119871120593(119883) one has

1003817100381710038171003817100381710038171003817int119860 1198911198891198981003817100381710038171003817100381710038171003817119884 le 120593lowast (119860) 10038171003817100381710038171198911003817100381710038171003817120593 (39)

and for 119910lowast isin 119884lowast one has1003816100381610038161003816100381610038161003816int119860 119891119889119898119910lowast

1003816100381610038161003816100381610038161003816 le (119898119910lowast)120593lowast (119860) 10038171003817100381710038171198911003817100381710038171003817120593 (40)

Proof (i) Assume that (119891120572) is a net in 119871120593(119883) such that119891120572 rarr 0inTand

120593 SinceTand120593 is a locally solid topology on 119871120593(119883) we get

1119860119891120572 rarr 0 inTand120593 Hence

1003817100381710038171003817119879119860 (119891120572)1003817100381710038171003817119884 = 1003817100381710038171003817119879 (1119860119891120572)1003817100381710038171003817119884 997888rarr 0 (41)

(ii) In view of Proposition 8 there exists a Young function120595 such that 120595 ⊲ 120593 and 120595lowast(Ω) lt infin Then 119871120593(119883) sub119864120595(119883) Let 119891 isin 119871120593(119883) Then there exists a sequence (119904119899)in S(Σ119891(120583) 119883) such that 119904119899(120596) minus 119891(120596)119883 rarr 0 120583-ae and119904119899(120596)119883 le 119891(120596)119883 120583-ae for all 119899 isin N (see [21 Theorem 6p 4])Then 119904119899 rarr 119891 inTand

120593 becauseTand120593 is a Lebesgue topology

Hence 119904119899 minus 119891120595 rarr 0 In view of Lemma 4 we can define theintegral int

119860119891119889119898 by the equality

int119860119891119889119898 fl limint

119860119904119899119889119898 (42)

6 Journal of Function Spaces

Since 119879119860(119904119899) = int119860119904119899119889119898 and by (i) 119879119860 is (Tand

120593 sdot 119884)-continuous we get

119879119860 (119891) = limint119860119904119899119889119898 (43)

Hence

119879119860 (119891) = int119860119891119889119898 (44)

and for 119910lowast isin 119884lowast we have119910lowast (119879119860 (119891)) = lim119910lowast (int

119860119904119899119889119898) = limint

119860119904119899119889119898119910lowast

= int119860119891119889119898119910lowast

(45)

(iii) Let 119891 isin 119871120593(119883) and (119860119899) be a sequence in Σ suchthat 119860119899 darr 0 Then 1119860119899(120596) darr 0 for 120596 isin Ω andhence 1119860119899(120596)119891(120596)119883 rarr 0 120583-ae and 1119860119899(120596)119891(120596)119883 le119891(120596)119883 120583-ae Hence 1119860119899119891 rarr 0 in Tand

120593 because Tand120593 is a

Lebesgue topology and by (i) we get

10038171003817100381710038171003817119898119891 (119860119899)10038171003817100381710038171003817119884 =100381710038171003817100381710038171003817100381710038171003817int119860119899 119891119889119898

100381710038171003817100381710038171003817100381710038171003817119884 =10038171003817100381710038171003817119879 (1119860119899119891)10038171003817100381710038171003817119884 997888rarr 0 (46)

(iv) Note that 120593lowast(119860) le 119879119860120593 To show that 119879119860120593 le120593lowast(119860) assume that 119891 isin 119861119871120593(119883) Choose a sequence (119904119899)in S(Σ119891(120583) 119883) such that 119904119899(120596) minus 119891(120596)119883 rarr 0 120583-ae and119904119899(120596)119883 le 119891(120596)119883 120583-ae for all 119899 isin N Since Tand

120593 isa Lebesgue topology we have 119904119899 rarr 119891 in Tand

120593 and hence119879119860(119904119899) minus 119879119860(119891)119884 rarr 0 Note that 119879119860(119904119899) = int119860 119904119899119889119898

Let 120576 gt 0 be given Choose 1198990 isin N such that 119879119860(119891) minusint1198601199041198990119889119898119884 le 120576 Then

1003817100381710038171003817119879119860 (119891)1003817100381710038171003817119884 le 1003817100381710038171003817100381710038171003817119879119860 (119891) minus int119860 11990411989901198891198981003817100381710038171003817100381710038171003817119884 +

1003817100381710038171003817100381710038171003817int119860 11990411989901198891198981003817100381710038171003817100381710038171003817119884

le 120576 + 120593lowast (119860) (47)

It follows that 119879119860120593 le 120593lowast(119860) so 120593lowast(119860) = 119879119860120593 Hencefor 119910lowast isin 119884lowast we easily get

1003817100381710038171003817(119910lowast ∘ 119879)1198601003817100381710038171003817lowast120593 = 1003817100381710038171003817119910lowast ∘ 1198791198601003817100381710038171003817lowast120593 = (119898119910lowast)120593lowast (119860) (48)

(v) Using (iv) we have

120593lowast (119860) = 10038171003817100381710038171198791198601003817100381710038171003817120593= sup 1003817100381710038171003817119879119860 (119891)1003817100381710038171003817119884 119891 isin 119871120593 (119883) 10038171003817100381710038171198911003817100381710038171003817120593 le 1= sup119910lowastisin119861119884lowast

1003816100381610038161003816(119910lowast ∘ 119879119860) (119891)1003816100381610038161003816 119891 isin 119871120593 (119883) 10038171003817100381710038171198911003817100381710038171003817120593 le 1= sup119910lowastisin119861119884lowast

1003817100381710038171003817119910lowast ∘ 1198791198601003817100381710038171003817lowast120593 = sup119910lowastisin119861119884lowast

(119898119910lowast)120593lowast (119860) (49)

(vi) This follows from (ii) and (iv)

For a sequence (119860119899) in Σ we will write 1198601198991205830 if 119860119899 darrand 120583(119860119899 cap 119860) rarr 0 for every 119860 isin Σ119891(120583)Definition 10 A measure 119898 Σ119891(120583) rarr L(119883 119884) with 119898 ≪120583 and 120593lowast(Ω) lt infin is said to be 120593lowast-semivariationally 120583-continuous if 120593lowast(119860119899) rarr 0 whenever 1198601198991205830 (119860119899) sub Σ

Using a standard argument we can show the following

Proposition 11 Let119898 Σ rarrL(119883 119884) be an additive measuresuch that 119898 ≪ 120583 and 120593(Ω) lt infin Then the followingstatements are equivalent

(i) 119898 is 120593lowast-semivariationally 120583-continuous(ii) The following two conditions hold simultaneously

(a) For every 120576 gt 0 there exists 120575 gt 0 such that120593lowast(119860) le 120576 whenever 120583(119860) le 120575 119860 isin Σ(b) For every 120576 gt 0 there exists 1198600 isin Σ119891(120583) such that120593lowast(Ω 1198600) le 120576

The following theorem characterizes 120593lowast-semivariation-ally 120583-continuous representing measures

Theorem 12 Let 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be its representingmeasure Then the following statements are equivalent

(i) 119898 is 120593lowast-semivariationally 120583-continuous(ii) 119879 is (120574120593 sdot 119884)-continuous(iii) 119879(119891119899)119884 rarr 0 if 119891119899 rarr 0 inT0 and sup119899119891119899120593 lt infin(iv) 119879119860119899120593 rarr 0 if 1198601198991205830 (119860119899) sub Σ

Proof (i)hArr (ii)hArr (iii) See [5 Corollary 28 and Proposition11]

(i)hArr (iv) This follows fromTheorem 9

Now assume that Ω is a completely regular Hausdorffspace LetB119886 denote the 120590-algebra of Baire sets in Ω whichis the 120590-algebra generated by the class Z of all zero sets ofbounded continuous positive functions on120596 ByPwedenotethe family of all cozero (=positive) in Ω (see [25 p 108])

Let 120583 B119886 rarr [0infin) be a countably additive measureThen 120583 is zero-set regular that is for every119860 isinB119886 and 120576 gt 0there exists 119885 isin Z with 119885 sub 119860 such that 120583(119860 119885) le 120576 (see[25 p 118]) It follows that for every 119860 isin B119886 and 120576 gt 0 thereexist 119880 isin P 119880 sup 119860 such that 120583(119880 119860) le 120576

We can assume that 120583 to be complete (if necessary wecan take the completion (ΩB119886 120583) of the measure space(ΩB119886 120583))Proposition 13 Assume that Ω is a completely regular Haus-dorff space and (ΩB119886 120583) is a complete finite measure spaceLet119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operatorand119898 B119886 rarrL(119883 119884) be its representing measure Then thefollowing statements are equivalent

(i) 119898 is 120593lowast-semivariationally 120583-continuous

Journal of Function Spaces 7

(ii) For every sequence (119860119899) in B119886 such that 119860119899 darr and120583(119860119899) rarr 0 there exists a sequence (119880119899) inPwith119860119899 sub119880119899 darr such that 120593lowast(119880119899) rarr 0(iii) For every sequence (119860119899) in B119886 such that 119860119899 darr and120583(119860119899) rarr 0 there exists a sequence (119880119899) inPwith119860119899 sub119880119899 darr such thatsup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) supp119891 sub 119880119899 997888rarr 0 (50)

Proof (i)rArr (ii) Assume that (i) holds and (119860119899) is a sequencein B119886 such that 119860119899 darr and 120583(119860119899) rarr 0 Then there exists asequence (119880119899) inP such that119860119899 sub 119880119899 darr and120583(119880119899119860119899) le 1119899for 119899 isin N

Let 120576 gt 0 be given Then in view of Proposition 11 thereexists 120575 gt 0 such that 120593lowast(119860) le 1205762 if 120583(119860) le 120575with119860 isinB119886Choose 1198991 isin N such that 120583(119880119899 119860119899) le 120575 for 119899 ge 1198992 Then120593lowast(119880119899 119860119899) le 1205762 for 119899 ge 1198991 Since 120593lowast(119860119899) rarr 0 we canchoose 1198992 isin N such that 120593lowast(119860119899) le 1205762 for 119899 ge 1198992 Then for119899 ge 1198990 = max(1198991 1198992) we get

120593lowast (119880119899) le 120593lowast (119880119899 119860119899) + 120593lowast (119860119899) le 120576 (51)

that is (ii) holds(ii)rArr (iii) Assume that (ii) holds and (119860119899) is a sequence

in B119900 such that 119860119899 darr and 120583(119860119899) rarr 0 Then there exists asequence (119880119899) inP with 119860119899 sub 119880119899 darr such that 120593lowast(119880119899) rarr 0Note that for 119891 isin 119861119871120593(119883) with supp119891 sub 119880119899 for 119899 isin N byTheorem 9 we have

1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 = 1003817100381710038171003817100381710038171003817intΩ 1198911198891198981003817100381710038171003817100381710038171003817119884 =

100381710038171003817100381710038171003817100381710038171003817int119880119899 119891119889119898100381710038171003817100381710038171003817100381710038171003817119884 le 120593lowast (119880119899) (52)

It follows that (iii) holds(iii)rArr (i) Assume that (iii) holds and119860119899 darrwith 120583(119860119899) rarr0Then there exists a sequence (119880119899) inPwith119860119899 sub 119880119899 darr such

that

sup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) supp119891 sub 119880119899 997888rarr 0 (53)

Assume on the contrary that (i) fails to hold Then withoutloss of generality we can assume that

120593lowast (119860119899) gt 1205760 for some 1205760 gt 0 all 119899 isin N (54)

Choose 1198990 isin N such that

sup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) supp119891 sub 1198801198990 lt 12057602 (55)

In view of (54) there exists a pairwise disjoint set 1198611 119861119896in B119886 119909119894 isin 119883 for 119894 = 1 119896 and 119910lowast isin 119861119884lowast such thatsum119896119894=1(1119861119894 otimes 119909119894)120593 le 1 and

1003816100381610038161003816100381610038161003816100381610038161003816119910lowast( 119896sum

119894=1

119898(1198601198990 cap 119861119894) (119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 ge 1205760 (56)

Let 1199040 = sum119896119894=1(11198601198990cap119861119894 otimes 119909119894) Then 1199040120593 le 1 and supp 1199040 sub1198601198990 sub 1198801198990 Then by (55) we get 119879(1199040)119884 lt 12057602On the other hand in view of (56) we have 119879(1199040)119884 ge 1205760

This contradiction establishes that (i) holds

Corollary 14 Assume that Ω is a completely regular Haus-dorff space and (ΩB119886 120583) is complete finitemeasure space Let119879 119871120593(119883) rarr 119884 be a (120574120593 sdot 119884)-continuous linear operator and119898 B119886 rarr L(119883 119884) be its representing measure Then 120593lowast isregular that is for every 119860 isinB119886 and 120576 gt 0 there exist 119885 isinZand 119880 isin P with 119885 sub 119860 sub 119880 such that 120593lowast(119880 119885) le 120576Proof In view of Theorem 12 119898 is 120593lowast-semivariationally 120583-continuous Let 119860 isin B119886 and 120576 gt 0 be given Then byProposition 11 there exists 120575 gt 0 such that 120593lowast(119861) le 120576whenever 119861 isin B119886 and 120583(119861) le 120575 By the regularity of 120583 onecan choose 119885 isin Z and 119880 isin P with 119885 sub 119860 sub 119880 such that120583(119880 119885) le 120575 Hence 120593lowast(119880 119885) le 120576 as desired4 Compact Operatorson Orlicz-Bochner Spaces

The following theorem presents necessary conditions for a(Tand120593 sdot 119884)-continuous operator 119879 119871120593(119883) rarr 119884 to be

compact

Theorem 15 Assume that a Young function 120593 such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot 119884)-continuous linear operator and 119898 Σ119891(120583) rarr L(119883 119884)be its representing measure If 119879 is compact then 119898 is 120593lowast-semivariationally 120583-continuousProof Assume that 119879 is compact and 119898 fails to be 120593lowast-semivariationally 120583-continuous Then there exist 120576 gt 0 and asequence (119860119899) in Σwith1198601198991205830 such that 119879119860119899 = 119898lowast120593(119860119899) gt120576 for 119899 isin N (seeTheorem9)Hence one can choose a sequence(119910lowast119899 ) in 119861119884lowast such that

10038171003817100381710038171003817119910lowast119899 ∘ 11987911986011989910038171003817100381710038171003817lowast120593 ge 120576 forall119899 isin N (57)

By Schauderrsquos theorem the conjugatemapping119879lowast 119884lowast rarr119871120593(119883)lowast is compact Note that 119879lowast(119910lowast119899 ) = 119910lowast119899 ∘ 119879 isin 119871120593(119883)sim119899 forall 119899 isin N where 119871120593(119883)sim119899 is a closed subspace of the Banachspace (119871120593(119883)lowast sdot lowast120593) (see Theorem 2) Then for every 119899 isin N

there exists 119892119899 isin 119871120593lowast(119883lowast 119883) such that

(119910lowast119899 ∘ 119879) (119891) = intΩ⟨119891 (120596) 119892119899 (120596)⟩ 119889120583

for 119891 isin 119871120593 (119883) 1003817100381710038171003817119910lowast119899 ∘ 1198791003817100381710038171003817lowast120593= sup int

Ω

1003817100381710038171003817119891 (120596)1003817100381710038171003817119883 120599 (119892119899) (120596) 119889120583 119891 isin 119861119871120593(119883)= 1003817100381710038171003817120599 (119892119899)1003817100381710038171003817120593lowast

(58)

Hence we obtain that for each 119899 isin N

10038171003817100381710038171003817119910lowast119899 ∘ 11987911986011989910038171003817100381710038171003817lowast120593 = 100381710038171003817100381710038171119860119899120599 (119892119899)10038171003817100381710038171003817120593lowast = 10038171003817100381710038171003817120599 (1119860119899119892119899)10038171003817100381710038171003817120593lowast (59)

8 Journal of Function Spaces

Since 119879lowast(119861119884lowast) is a relatively sequentially compact subset of((119871120593(119883)sim119899 sdot lowast120593) there exist a subsequence (119892119896119899) of (119892119899) and119892 isin 119871120593lowast(119883lowast 119883) such that10038171003817100381710038171003817119865119892119899 minus 11986511989210038171003817100381710038171003817lowast120593 = 10038171003817100381710038171003817120599 (119892119896119899 minus 119892)10038171003817100381710038171003817120593lowast 997888rarr 0 (60)

Choose 119899120576 isin N such that 120599(119892119896119899 minus 119892)120593lowast le 1205762 for 119899 ge 119899120576Hence for 119899 ge 1198991205761003816100381610038161003816100381610038161003816

100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast minus 100381710038171003817100381710038171003817120599 (1119860119896119899119892119896119899)100381710038171003817100381710038171003817120593lowast1003816100381610038161003816100381610038161003816

le 100381710038171003817100381710038171003817120599 (1119860119896119899 (119892119896119899 minus 119892))100381710038171003817100381710038171003817120593lowast = 1003817100381710038171003817100381710038171119860119896119899120599 (119892119896119899 minus 119892)100381710038171003817100381710038171003817120593lowastle 10038171003817100381710038171003817120599 (119892119896119899 minus 119892)10038171003817100381710038171003817120593lowast le 1205762

(61)

Using (57) and (61) for 119899 ge 119899120576 we get120576 le 100381710038171003817100381710038171003817119910lowast ∘ 119879119860119896119899 100381710038171003817100381710038171003817

lowast

120593= 100381710038171003817100381710038171003817120599 (1119860119896119899119892119896119899)100381710038171003817100381710038171003817120593lowast

le 1205762 +100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast

(62)

and hence 1003817100381710038171003817100381710038171119860119896119899120599 (119892)100381710038171003817100381710038171003817120593lowast = 100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast ge 1205762 (63)

On the other hand since 120593lowast is supposed to satisfy theΔ 2-condition we have that 1119860119896119899120599(119892)120593lowast rarr 0 (see [26 Theorem3 pp 58-59]) This contradiction establishes that 119898 is 120593lowast-semivariationally 120583-continuousCorollary 16 Assume that 120593 is a Young function such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operator Then the following statements areequivalent

(i) 119879 is compact(ii) 119879 is (120574120593 sdot 119884)-compact that is there exists a 120574120593-

neighborhood 119881 of 0 in 119871120593(119883) such that 119879(119881) is arelatively norm compact set in 119884

(iii) There exists a Young function 120595 with 120595 ≪ 120593 such thatintΩ119891119889119898 119891 isin 119871120593(119883) 119891120595 le 1 is a relatively norm

compact set in 119884Proof (i)rArr (ii) Assume that (i) holds Then by Theorems 12and 15 119879 is (120574120593 sdot 119884)-continuous Since the space (119871120593(119883) 120574120593)is quasinormable by Grothendieckrsquos classical result (see [15p 429]) we obtain that 119879 is (120574120593 sdot 119884)-compact

(ii)rArr (i) The implication is obvious(ii)hArr (iii) This follows fromTheorem 3

5 Topology Associated withthe 120593lowast-Semivariation ofa Representing Measure

Assume that 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator Let119898 Σ119891(120583) rarrL(119883 119884) be its representingmeasure Let us put

119901119898 (119910lowast) fl (119898119910lowast)120593lowast (Ω) for 119910lowast isin 119884lowast (64)

Note that119901119898 is a seminormon119884lowast Following [22 27] let 120575119898120593lowaststand for the topology on 119861119884lowast defined by the seminorm 119901119898restricted to 119861119884lowast

The following theorem characterizes (Tand120593 sdot 119884)-

continuous compact operators 119879 119871120593(119883) rarr 119884 in terms ofthe topological properties of the space (119861119884lowast 120575119898120593lowast) (see [22Theorem 3])

Theorem 17 Let 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be its representingmeasure Then the following statements are equivalent

(i) The space (119861119884lowast 120575119898120593lowast) is compact

(ii) 119879 is compact

Proof (i) rArr (ii) Assume that (119861119884lowast 120575119898120593lowast) is compact Let(119910lowast119899 ) be a sequence in 119861119884lowast Without loss of generality we canassume that 119910lowast119899 rarr 119910lowast0 in 120575119898120593lowast for some 119910lowast isin 119861119884lowast Then usingTheorem 9 for 119891 isin 119871120593(119883) we have

1003816100381610038161003816(119879lowast (119910lowast119899 ) minus 119879lowast (119910lowast0 )) (119891)1003816100381610038161003816 = 1003816100381610038161003816(119910lowast119899 minus 119910lowast0 ) (119879 (119891))1003816100381610038161003816= 1003816100381610038161003816100381610038161003816intΩ 119891119889119898119910lowast119899 minus119910lowast0

1003816100381610038161003816100381610038161003816 le (119898119910lowast119899 minus119910lowast0 )120593lowast (Ω) 10038171003817100381710038171198911003817100381710038171003817120593 (65)

It follows that 119879lowast(119910lowast119899 ) minus 119879lowast(119910lowast0 )lowast120593 le (119898119910lowast119899 minus119910lowast0 )120593lowast(Ω) where119901119898(119910lowast119899 minus 119910lowast0 ) = (119898119910lowast119899 minus119910lowast0 )120593lowast(Ω) rarr119899 0 This means that 119879lowast iscompact and hence 119879 is compact

(ii)rArr (i) Assume that 119879 is compact and (119910lowast120572 ) is a net in119861119884lowast Since119861119884lowast is 120590(119884lowast 119884)-compact without loss of generalitywe can assume that 119910lowast120572 rarr120572 119910lowast0 in 120590(119884lowast 119884) for some 119910lowast0 isin 119861119884lowast In view of the compactness of the conjugate operator 119879lowast 119884lowast rarr 119871120593(119883)lowast there exists a subset (119910lowast120573 ) of (119910lowast120572 ) and Φ0 isin119871120593(119883)lowast such that 119879lowast(119910lowast120573 ) minus Φ0lowast120593 rarr

1205730 On the other hand

since 119879lowast is (120590(119884lowast 119884) 120590(119871120593(119883)lowast 119871120593(119883)))-continuous we get119879lowast(119910lowast120573 ) rarr120573119879lowast(119910lowast0 ) in 120590(119871120593(119883)lowast 119871120593(119883)) HenceΦ0 = 119879lowast(119910lowast0 )

that is 119879lowast(119910lowast120573 ) minus 119879lowast(119910lowast0 )lowast120593 rarr1205730

Let 120576 gt 0 be givenThen there exist a pairwise disjoint set1198601 119860119899 in Σ119891(120583) and 119909119894 isin 119883 for 119894 = 1 119899 such thatsum119899119894=1(1119860119894 otimes 119909119894)120593 le 1 and

(119898119910lowast120573minus119910lowast0)120593lowast(Ω) le 1003816100381610038161003816100381610038161003816100381610038161003816

119899sum119894=1

(119910lowast120573 minus 119910lowast0 ) (119898 (119860 119894) (119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 + 120576 (66)

Hence

(119898119910lowast120573minus119910lowast0)120593lowast(Ω) le 1003816100381610038161003816100381610038161003816100381610038161003816

119899sum119894=1

(119910lowast120573 minus 119910lowast0 ) (119879 (1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 + 120576

le 1003816100381610038161003816100381610038161003816100381610038161003816(119910lowast120573 minus 119910lowast0 ) 119879(

119899sum119894=1

(1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816

+ 120576

Journal of Function Spaces 9

= 1003816100381610038161003816100381610038161003816100381610038161003816119879lowast (119910lowast120573 minus 119910lowast0 )(

119899sum119894=1

(1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816

+ 120576le 10038171003817100381710038171003817119879lowast (119910lowast120573 minus 119910lowast0 )10038171003817100381710038171003817lowast120593

1003817100381710038171003817100381710038171003817100381710038171003817119899sum119894=1

(1119860119894 otimes 119909119894)1003817100381710038171003817100381710038171003817100381710038171003817120593

+ 120576 le 10038171003817100381710038171003817119879lowast (119910lowast120573) minus 119879 (119910lowast0 )10038171003817100381710038171003817lowast120593 + 120576(67)

Hence 119901119898(119910lowast120573 minus 119910lowast0 ) = (119898119910lowast120573minus119910lowast0)120593lowast(Ω) rarr

1205730 and this means

that the space (119861119884lowast 120575119898120593lowast) is compact

As a consequence of Theorems 17 and 15 we have thefollowing

Corollary 18 Assume that 120593 is a Young function such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be itsrepresenting measure If the space (119861119884lowast 120575119898120593lowast) is compact then119898 is 120593lowast-semivariationally 120583-continuousConflicts of Interest

The author declares that there are no conflicts of interest

References

[1] S Chen and H Hudzik ldquoOn some convexities of Orlicz andOrlicz-Bochner spacesrdquo Commentationes Mathematicae vol29 no 1 pp 13ndash29 1988

[2] P Kolwicz and R Pluciennik ldquo119875-convexity of Orlicz-Bochnerspacesrdquo Proceedings of the American Mathematical Society vol126 no 8 pp 2315ndash2322 1998

[3] S Shang and Y Cui ldquoUniform nonsquareness and locallyuniform nonsquareness in Orlicz-Bochner function spaces andapplicationsrdquo Journal of Functional Analysis vol 267 no 7 pp2056ndash2076 2014

[4] K Feledziak andM Nowak ldquoLocally solid topologies on vectorvalued function spacesrdquo Universitat de Barcelona CollectaneaMathematica vol 48 no 4-6 pp 487ndash511 1997

[5] K Feledziak and M Nowak ldquoIntegral representation of linearoperators on Orlicz-Bochner spacesrdquo Universitat de BarcelonaCollectanea Mathematica vol 61 no 3 pp 277ndash290 2010

[6] M Nowak ldquoLinear operators on vector-valued function spaceswithMackey topologiesrdquo Journal of Convex Analysis vol 15 no1 pp 165ndash178 2008

[7] M Nowak ldquoLebesgue topologies on vector-valued functionspacesrdquoMathematica Japonica vol 52 no 2 pp 171ndash182 2000

[8] A V Buhvalov ldquoThe analytic representation of operators withan abstract normrdquo Izvestiya Vysshikh Uchebnykh ZavedeniiMatematika vol 11 no 162 pp 21ndash32 1975

[9] A V Buhvalov ldquoThe analytic representation of linear operatorsby means of measurable vector-valued functionsrdquo IzvestiyaVysshikh Uchebnykh Zavedenii Matematika vol 7 no 182 pp21ndash31 1977

[10] J Diestel and J J Uhl ldquoVector measuresrdquo vol 15 AmericanMathematical Society Surveys Providence RI USA 1977

[11] M Nowak ldquoDuality theory of vector-valued function spaces IrdquoCommentationes Mathematicae vol 37 pp 195ndash215 1997

[12] A V Bukhvalov and G Ya Lozanovskii ldquoOn sets closed inmeasure in spaces of mesurable functionsrdquo Transactions of theMoscow Mathematical Society vol 2 pp 127ndash148 1978

[13] A Wiweger ldquoLinear spaces with mixed topologyrdquo StudiaMathematica vol 20 pp 47ndash68 1961

[14] K Feledziak ldquoUniformly Lebesgue topologies on Kothe-Bochner spacesrdquoCommentationesMathematicae vol 37 pp 81ndash98 1997

[15] W Ruess ldquo[Weakly] compact operators and DF-spacesrdquo PacificJournal of Mathematics vol 98 no 2 pp 419ndash441 1982

[16] M Nowak ldquoCompactness of Bochner representable operatorson Orlicz spacesrdquo Positivity An International MathematicsJournal Devoted to Theory and Applications of Positivity vol 13no 1 pp 193ndash199 2009

[17] N E Gretsky and J Uhl ldquoBounded linear operators on Banachfunction spaces of vector-valued functionsrdquo Transactions of theAmerican Mathematical Society vol 167 pp 263ndash277 1972

[18] J J Uhl ldquoCompact operators on Orlicz spacesrdquo in Rendicontidel Seminario Matematico vol 42 pp 209ndash219 University ofPadova 1969

[19] N Dinculeanu Vector Measures International Series of Mono-graphs in Pure and Applied Mathematics 95 Pergamon PressNew York NY USA 1967

[20] N Dinculeanu Linear Operators on Lp-Spaces Vector andOperator-Valued Measures and Applications Academic PressNew York NY USA 1973

[21] N Dinculeanu Vector Integration and Stochastic Integration inBanach Spaces John Wiley amp Sons 2000

[22] R A Alo A De Korwin and L E Kunes ldquoTopological aspectsof 119902-regular measuresrdquo Studia Mathematica vol 48 pp 49ndash601973

[23] J Diestel ldquoOn the representation of bounded linear operatorsfrom Orlicz spaces of Lebesgue-Bochner measurable functionsto any Banach spacerdquo Bulletin de lrsquoAcademie Polonaise desSciences Serie des Sciences Mathematiques Astronomiques etPhysiques vol 18 pp 375ndash378 1970

[24] L Vanderputten ldquoRepresentation of operators defined on thespace of Bochner integrable functionsrdquo Extracta Mathematicaevol 16 no 3 pp 383ndash391 2001

[25] R FWheeler ldquoA survey of Bairemeasures and strict topologiesrdquoExpositiones Mathematicae vol 1 no 2 pp 97ndash190 1983

[26] W A Luxemburg Banach function spaces [Thesis] DelftNetherlands 1955

[27] P W Lewis ldquoVector measures and topologyrdquo Revue RoumainedeMathematique Pures et Appliquees vol 16 pp 1201ndash1209 1971

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Journal of Function Spaces 3

Let 120574120593[T120593T0] (briefly 120574120593) denote the natural mixedtopology on 119871120593(119883) that is 120574120593 is the finest linear topologythat agrees with T0 on sdot 120593-bounded sets in 119871120593(119883) (see[5 13 14] for more details) Then 120574120593 is a locally convex-solidHausdorff topology (see [14 Theorem 32]) and 120574120593 and T120593

have the same bounded sets This means that (119871120593(119883) 120574120593)is a generalized DF-space (see [15]) and its follows that(119871120593(119883) 120574120593) is quasinormable (see [15 p 422]) Moreover fora sequence (119891119899) in 119871120593(119883) 119891119899 rarr 0 in 120574120593 if and only if 119891119899 rarr 0inT0 and sup119899119891119899120593 lt infin (see [14 Theorem 31])

We say that a Young function 120593 increases essentially morerapidly than another 120595 (in symbols 120595 ≪ 120593) if for arbitrary119888 gt 0 120595(119888119905)120593(119905) rarr 0 as 119905 rarr 0 and 119905 rarr infin

Theorem 3 Let 120593 be a Young function Then the mixedtopology 120574120593 on 119871120593(119883) is generated by the family of norms sdot 120595|119871120593(119883) 120595 ≪ 120593Proof It is known that the mixed topology 120574120593 on 119871120593 isgenerated by the family of norms sdot 120595|119871120593 120595 ≪ 120593 (see [16Theorem 21]) Since 119891120595 = 119891(sdot)119883120595 for 119891 isin 119871120593(119883) by[14 (54) p 97] the mixed topology 120574120593 on 119871120593(119883) is generatedby the family of norms sdot 120595|119871120593(119883) 120595 ≪ 120593

Since 120595 ≪ 120593 implies 120595 ⊲ 120593 in view of Theorems 2 and 3we get

120574120593 sub Tand120593 (10)

The problem of integral representation of bounded lin-ear operators on Banach function spaces of vector-valuedfunctions to Banach spaces in terms of the correspondingoperator-valued measures has been the object of much study(see [5 17ndash24]) In particular Dinculeanu (see [19 sect 13 Sect3] [20] [21 sect 8 Sect B]) studied the problem of integralrepresentation of bounded linear operators from 119871119901(119883) to aBanach space 119884 It is known that if 1 le 119901 lt infin 120583(Ω) lt infinand an operator measure 119898 Σ rarr L(119883 119884) vanishes on120583-null sets and has the finite 119902-semivariation 119902(Ω) (1 lt119902 le infin 1119901 + 1119902 = 1) then one can define the integralintΩ119891119889119898 for all 119891 isin 119871119901(119883) Moreover if 119879 119871119901(119883) rarr 119884

is a bounded linear operator then the associated operatormeasure 119898 Σ rarr L(119883 119884) has the finite 119902-semivariation119902(Ω) and 119879(119891) = intΩ 119891119889119898 for all 119891 isin 119871119901(119883) (see [19 sect 13Theorem 1 p 259] [20 Theorem 4]) The relationships of the119902-semivariation 119902 to the properties of operators from 119871119901(119883)to 119884 were studied in [22] Diestel [23] found the integralrepresentation of bounded linear operators from an Orlicz-Bochner space 119871120593(119883) to a Banach spaces if 120583(Ω) lt infin and aYoung 120593 satisfies the Δ 2-condition

The present paper is a continuation of [5] where weestablish integral representation of (120574120593 sdot 119884)-continuouslinear operators 119879 119871120593(119883) rarr 119884 We study the problemof integration of functions in 119871120593(119883) with respect to therepresenting operator measures of (Tand

120593 sdot 119884)-continuouslinear operators 119879 119871120593(119883) rarr 119884 An integral representationtheorem for (Tand

120593 sdot 119884)-continuous linear operators 119879 119871120593(119883) rarr 119884 is established (see Theorem 9 below) We study

the relationships between (Tand120593 sdot 119884)-continuous operators119879 119871120593(119883) rarr 119884 and the properties of their representing

measures119898 Σ119891(120583) rarrL(119883 119884)2 120593lowast-Semivariation of Operator Measures

Assume that 119898 Σ119891(120583) rarr L(119883 119884) is an additive measuresuch that119898 ≪ 120583 that is119898(119860) = 0 if 120583(119860) = 0

Let S(Σ119891(120583) 119883) denote the space of all 119883-valued Σ119891(120583)-simple functions on Ω Then 119904 isin S(Σ119891(120583) 119883) if 119904 = sum(1119860119894 otimes119909119894) where (119860 119894) is a finite pairwise disjoint sequence in Σ119891(120583)and 119909119894 isin 119883 For 119904 = sum119899119894=1(1119860119894 otimes 119909119894) isin S(Σ119891(120583) 119883) and 119860 isin Σwe can define the integralint

119860119904 119889119898 by

int119860119904 119889119898 fl

119899sum119894=1

119898(119860 119894 cap 119860) (119909119894) (11)

Note that

int119860119904 119889119898 = int

Ω1119860119904 119889119898 (12)

For 119910lowast isin 119884lowast we define a measure 119898119910lowast Σ119891(120583) rarr 119883lowast by theequality

119898119910lowast (119860) (119909) fl 119910lowast (119898 (119860) (119909))for 119860 isin Σ119891 (120583) 119909 isin 119883 (13)

For 119904 = sum119899119894=1(1119860119894 otimes 119909119894) isin S(Σ119891(120583) 119883) and 119860 isin Σ wedefine the integral int

119860119904 119889119898119910lowast by the equality

int119860119904 119889119898119910lowast fl 119899sum

119894=1

119898119910lowast (119860 119894 cap 119860) (119909119894) (14)

Then

119910lowast (int119860119904 119889119898) = int

119860119904 119889119898119910lowast (15)

Following [23] [19 sect 13] one can define the 120593lowast-semivariation120593lowast(119860) of119898 on 119860 isin Σ by

120593lowast (119860) fl sup1003817100381710038171003817100381710038171003817100381710038171003817119899sum119894=1

119898(119860 cap 119860 119894) (119909119894)1003817100381710038171003817100381710038171003817100381710038171003817119884 (16)

where the supremum is taken over all finite pairwise disjointsets 1198601 119860119899 in Σ119891(120583) and 119909119894 isin 119883 for 119894 = 1 119899 suchthat sum119899119894=1(1119860119894 otimes 119909119894)120593 le 1

One can observe that

120593lowast (119860)= sup1003817100381710038171003817100381710038171003817int119860 119904 119889119898

1003817100381710038171003817100381710038171003817119884 119904 isin S (Σ119891 (120583) 119883) 119904120593 le 1 (17)

Note that

120593lowast (119860) le 120593lowast (119861) if 119860 119861 isin Σ with 119860 sub 119861120593lowast (119860 cup 119861) le 120593lowast (119860) + 120593lowast (119861) for 119860 119861 isin Σ (18)

4 Journal of Function Spaces

Let (119898119910lowast)120593lowast(119860) stand for the 120593lowast-semivariation of119898119910lowast on 119860 isinΣ that is(119898119910lowast)120593lowast (119860)= sup 1003816100381610038161003816100381610038161003816int119860 119904 119889119898119910lowast

1003816100381610038161003816100381610038161003816 119904 isin S (Σ119891 (120583) 119883) 119904120593 le 1 (19)

The following lemma will be useful

Lemma 4 Let 120593 be a Young function and 119898 Σ119891(120583) rarrL(119883 119884) be a measure with 119898 ≪ 120583 and 120593lowast(Ω) lt infin Thenthe following statements hold

(i) If119891 isin 119864120593(119883) then there exists a sdot120593-Cauchy sequence(119904119899) inS(Σ119891(120583) 119883) such that 119904119899(120596)minus119891(120596)119883 rarr 0 120583-ae

(ii) If (119904119899) is a sdot 120593-Cauchy sequence inS(Σ119891(120583) 119883) thenfor119860 isin Σ (int

119860119904119899119889119898) is a Cauchy sequence in a Banach

space119884 and for every119910lowast isin 119884lowast (int119860119904119899119889119898119910lowast) is aCauchy

sequence in R(iii) If 119891 isin 119864120593(119883) and (1199041015840119899) and (11990410158401015840119899 ) are sdot 120593-Cauchy

sequence inS(Σ119891(120583) 119883) such that 1199041015840119899(120596)minus119891(120596)119883 rarr0 120583-ae and 11990410158401015840119899 (120596) minus 119891(120596)119883 rarr 0 120583-ae then for119860 isin Σ one haslimint

1198601199041015840119899119889119898 = limint

11986011990410158401015840119899 119889119898 (20)

and for every 119910lowast isin 119884lowast one haslimint

1198601199041015840119899119889119898119910lowast = limint

11986011990410158401015840119899 119889119898119910lowast (21)

Proof (i) Let 119891 isin 119864120593(119883) Then there exists a sequence (119904119899)in S(Σ119891(120583) 119883) such that 119904119899(120596) minus 119891(120596)119883 rarr 0 120583-ae and119904119899(120596)119883 le 119891(120596)119883 120583-ae for all 119899 isin N (see [21 Theorem 6p 4]) Using the Lebesgue dominated convergence theoremwe obtain that int

Ω120593(120582(119904119899(120596) minus 119891(120596)119883)119889120583 rarr 0 for all 120582 gt 0

so 119904119899 minus 119891120593 rarr 0 Hence (119904119899) is a sdot 120593-Cauchy sequence(ii) Assume that (119904119899) is a sdot 120593-Cauchy sequence in

S(Σ119891(120583) 119883) Hence for 119899 119896 isin N we have

1003817100381710038171003817100381710038171003817int119860 119904119899119889119898 minus int1198601199041198961198891198981003817100381710038171003817100381710038171003817119884 =

1003817100381710038171003817100381710038171003817int119860 (119904119899 minus 119904119896) 1198891198981003817100381710038171003817100381710038171003817119884

le 1003817100381710038171003817119904119899 minus 1199041198961003817100381710038171003817120593 120593lowast (119860) le 1003817100381710038171003817119904119899 minus 1199041198961003817100381710038171003817120593 120593lowast (Ω) (22)

It follows that (int119860119904119899119889119898) is a Cauchy sequence in 119884 Hence in

view of (15) for 119910lowast isin 119884lowast (int119860119904119899119889119898119910lowast) is a Cauchy sequence

in R(iii) Note that (1199041015840119899 minus 11990410158401015840119899 ) is a sdot 120593-Cauchy sequence and1199041015840119899(120596) minus 11990410158401015840119899 (120596)119883 rarr 0 120583-ae Hence there exists ℎ isin 119864120593(119883)

such that (1199041015840119899minus11990410158401015840119899 )minusℎ120593 rarr 0 Note thatT0|119864120593(119883) sub T120593|119864120593(119883)Hence (1199041015840119899minus11990410158401015840119899 )minusℎ rarr 0 inT0 and it follows that there exists asubsequence (1199041015840119896119899minus11990410158401015840119896119899) of (1199041015840119899minus11990410158401015840119899 ) such that (1199041015840119896119899(120596)minus11990410158401015840119896119899(120596))minus

ℎ(120596)119883 rarr 0 120583-ae Then ℎ(120596) = 0 120583-ae so 1199041015840119899 minus 11990410158401015840119899 120593 rarr 0and for 119860 isin Σ we get

1003817100381710038171003817100381710038171003817int119860 1199041015840119899119889119898 minus int11986011990410158401015840119899 1198891198981003817100381710038171003817100381710038171003817119884 =

1003817100381710038171003817100381710038171003817int119860 (1199041015840119899 minus 11990410158401015840119899 ) 1198891198981003817100381710038171003817100381710038171003817119884

le 100381710038171003817100381710038171199041015840119899 minus 11990410158401015840119899 10038171003817100381710038171003817120593 120593lowast (119860) (23)

It follows that

limint1198601199041015840119899119889119898 = limint

11986011990410158401015840119899 119889119898 (24)

and hence in view of (15) for every 119910lowast isin 119884lowast we havelimint

1198601199041015840119899119889119898119910lowast = limint

11986011990410158401015840119899 119889119898119910lowast (25)

Following [21 sect 13 Definition 1 p 254] in view ofLemma 4 we have the following

Definition 5 Let 120593 be a Young function and 119898 Σ119891(120583) rarrL(119883 119884) be an additive measure such that 119898 ≪ 120583 and120593lowast(Ω) lt infin Then for every 119891 isin 119864120593(119883) and 119860 isin Σ wecan define the integralint

119860119891119889119898 by the equality

int119860119891119889119898 fl limint

119860119904119899119889119898 (26)

and for 119910lowast isin 119884lowast we can define the integralint119860119891119889119898119910lowast by the

equality

int119860119891119889119898119910lowast fl limint

119860119904119899119889119898119910lowast (27)

where (119904119899) is an arbitrary sdot 120593-Cauchy sequence inS(Σ119891(120583) 119883) such that 119904119899(120596) minus 119891(120596)119883 rarr 0 120583-ae3 Integral Representation of ContinuousOperators on Orlicz-Bochner Spaces

For a bounded linear operator 119879 119871120593(119883) rarr 119884 let

119879120593 fl sup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) (28)

Proposition 6 Let 119879 119871120593(119883) rarr 119884 be a bounded linearoperator and

119898(119860) (119909) fl 119879 (1119860 otimes 119909) for 119860 isin Σ119891 (120583) 119909 isin 119883 (29)

Then the following statements hold

(i) For 119860 isin Σ119891(120583)119898(119860) isin L(119883 119884) and 119898(119860) le 119879120593 sdot1119860120593(ii) 119898 ≪ 120583(iii) 119898(119860119899) rarr 0 if 119860119899 darr 0 with 119860119899 isin Σ119891(120583)(iv) 119898 Σ119891(120583) rarr L(119883 119884) is countably additive that is119898(⋃infin119899=1 119861119899) = suminfin119899=1119898(119861119899) if (119861119899) is a pairwise disjoint

sequence in Σ119891(120583) with⋃infin119899=1 119861119899 isin Σ119891(120583)(v) 120593lowast(Ω) le 119879120593

Journal of Function Spaces 5

Proof (i) Let 119860 isin Σ119891(120583) Then for 119909 isin 119861119883 we have 1119860 otimes119909120593 le 1119860120593 and hence

119898 (119860) (119909)119884 = 1003817100381710038171003817119879 (1119860 otimes 119909)1003817100381710038171003817119884 le 119879120593 sdot 10038171003817100381710038171119860 otimes 1199091003817100381710038171003817120593le 119879120593 100381710038171003817100381711198601003817100381710038171003817120593 (30)

so 119898(119860) le 119879120593 sdot 1119860120593(ii) This follows from (i) because 1119860120593 = 0 if 120583(119860) = 0(iii) Assume that 119860119899 darr 0 with 119860119899 isin Σ119891(120583) Then

11198601(120596) ge 1119860119899(120596) darr 0 for 120596 isin Ω By the Lebesgue dominatedconvergence theorem we obtain that int

Ω120593(1205821119860119899(120596))119889120583 rarr 0

for every 120582 gt 0 This means that 1119860119899120593 rarr 0 and by (i)119898(119860119899) rarr 0(iv) Assume that (119861119899) is a pairwise disjoint sequence inΣ119891(120583) with 119861 = ⋃infin119899=1 119861119899 isin Σ119891(120583) Let 119860119899 = 119861 ⋃119899119894=1 119861119894 for119899 isin N Then 119860119899 isin Σ119891(120583) and 119860119899 darr 0 Hence by (iii) 119898(119861) minussum119899119894=1119898(119861119894) = 119898(119861) minus 119898(⋃119899119894=1 119861119894) = 119898(119860119899) rarr 0Statement (v) is obvious

Definition 7 Let119879 119871120593(119883) rarr 119884 be a bounded linear operatorand

119898(119860) (119909) fl 119879 (1119860 otimes 119909) for 119860 isin Σ119891 (120583) 119909 isin 119883 (31)

Then the measure 119898 Σ119891(120583) rarr L(119883 119884) will be called arepresenting measure of 119879Proposition 8 Let 119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot 119884)-continuous linear operator and 119898 Σ119891(120583) rarr L(119883 119884) beits representing measure Then there exists a Young function 120595such that 120595 ⊲ 120593 and 120595lowast(Ω) lt infin

Proof According toTheorem 2 there exist a finite set 120595119894 119894 =1 119899 of Young functions with 120595119894 ⊲ 120593 for 119894 = 1 119899 and119886 gt 0 such that1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 le 119886max1le119894le119899

10038171003817100381710038171198911003817100381710038171003817120595119894 forall119891 isin 119871120593 (119883) (32)

Let120595(119905) = max1le119894le119899120595119894(119905) for 119905 ge 0Then120595 is a Young functionwith 120595 ⊲ 120593 and

1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 le 119886 10038171003817100381710038171198911003817100381710038171003817120595 forall119891 isin 119871120593 (119883) (33)

Hence120595lowast (Ω)= sup 119879 (119904)119884 119904 isin S (Σ119891 (120583) 119883) 119904120595 le 1le 119886 lt infin

(34)

For a linear operator 119879 119871120593(119883) rarr 119884 and 119860 isin Σ let119879119860 (119891) fl 119879 (1119860119891) for 119891 isin 119871120593 (119883) (35)

Nowwe can state ourmain result that extends the classicalresults concerning the integral representation of operators onLebesgue-Bochner spaces 119871119901(119883) (1 le 119901 lt infin) (see [19 sect13 Theorem 1 pp 259ndash261]) to operators on Orlicz-Bochnerspaces 119871120593(119883)

Theorem 9 Let 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be its representingmeasure Then for 119860 isin Σ the following statements hold

(i) 119879119860 119871120593(119883) rarr 119884 is a (Tand120593 sdot 119884)-continuous linear

operator(ii) For 119891 isin 119871120593(119883) one has

119879119860 (119891) = int119860119891119889119898 (36)

and for 119910lowast isin 119884lowast one has119910lowast (119879119860 (119891)) = int

119860119891119889119898119910lowast (37)

(iii) For 119891 isin 119871120593(119883) the measure 119898119891 Σ rarr 119884 defined bythe equality

119898119891 (119860) fl int119860119891119889119898 for 119860 isin Σ (38)

is countably additive(iv) 119879119860120593 = 120593lowast(119860)

and for 119910lowast isin 119884lowast 119910lowast ∘ 119879119860lowast120593 = (119910lowast ∘ 119879)119860lowast120593 =(119898119910lowast)120593lowast(119860)(v) 120593lowast(119860) = sup(119898119910lowast)120593lowast(119860) 119910lowast isin 119861119884lowast(vi) For 119891 isin 119871120593(119883) one has

1003817100381710038171003817100381710038171003817int119860 1198911198891198981003817100381710038171003817100381710038171003817119884 le 120593lowast (119860) 10038171003817100381710038171198911003817100381710038171003817120593 (39)

and for 119910lowast isin 119884lowast one has1003816100381610038161003816100381610038161003816int119860 119891119889119898119910lowast

1003816100381610038161003816100381610038161003816 le (119898119910lowast)120593lowast (119860) 10038171003817100381710038171198911003817100381710038171003817120593 (40)

Proof (i) Assume that (119891120572) is a net in 119871120593(119883) such that119891120572 rarr 0inTand

120593 SinceTand120593 is a locally solid topology on 119871120593(119883) we get

1119860119891120572 rarr 0 inTand120593 Hence

1003817100381710038171003817119879119860 (119891120572)1003817100381710038171003817119884 = 1003817100381710038171003817119879 (1119860119891120572)1003817100381710038171003817119884 997888rarr 0 (41)

(ii) In view of Proposition 8 there exists a Young function120595 such that 120595 ⊲ 120593 and 120595lowast(Ω) lt infin Then 119871120593(119883) sub119864120595(119883) Let 119891 isin 119871120593(119883) Then there exists a sequence (119904119899)in S(Σ119891(120583) 119883) such that 119904119899(120596) minus 119891(120596)119883 rarr 0 120583-ae and119904119899(120596)119883 le 119891(120596)119883 120583-ae for all 119899 isin N (see [21 Theorem 6p 4])Then 119904119899 rarr 119891 inTand

120593 becauseTand120593 is a Lebesgue topology

Hence 119904119899 minus 119891120595 rarr 0 In view of Lemma 4 we can define theintegral int

119860119891119889119898 by the equality

int119860119891119889119898 fl limint

119860119904119899119889119898 (42)

6 Journal of Function Spaces

Since 119879119860(119904119899) = int119860119904119899119889119898 and by (i) 119879119860 is (Tand

120593 sdot 119884)-continuous we get

119879119860 (119891) = limint119860119904119899119889119898 (43)

Hence

119879119860 (119891) = int119860119891119889119898 (44)

and for 119910lowast isin 119884lowast we have119910lowast (119879119860 (119891)) = lim119910lowast (int

119860119904119899119889119898) = limint

119860119904119899119889119898119910lowast

= int119860119891119889119898119910lowast

(45)

(iii) Let 119891 isin 119871120593(119883) and (119860119899) be a sequence in Σ suchthat 119860119899 darr 0 Then 1119860119899(120596) darr 0 for 120596 isin Ω andhence 1119860119899(120596)119891(120596)119883 rarr 0 120583-ae and 1119860119899(120596)119891(120596)119883 le119891(120596)119883 120583-ae Hence 1119860119899119891 rarr 0 in Tand

120593 because Tand120593 is a

Lebesgue topology and by (i) we get

10038171003817100381710038171003817119898119891 (119860119899)10038171003817100381710038171003817119884 =100381710038171003817100381710038171003817100381710038171003817int119860119899 119891119889119898

100381710038171003817100381710038171003817100381710038171003817119884 =10038171003817100381710038171003817119879 (1119860119899119891)10038171003817100381710038171003817119884 997888rarr 0 (46)

(iv) Note that 120593lowast(119860) le 119879119860120593 To show that 119879119860120593 le120593lowast(119860) assume that 119891 isin 119861119871120593(119883) Choose a sequence (119904119899)in S(Σ119891(120583) 119883) such that 119904119899(120596) minus 119891(120596)119883 rarr 0 120583-ae and119904119899(120596)119883 le 119891(120596)119883 120583-ae for all 119899 isin N Since Tand

120593 isa Lebesgue topology we have 119904119899 rarr 119891 in Tand

120593 and hence119879119860(119904119899) minus 119879119860(119891)119884 rarr 0 Note that 119879119860(119904119899) = int119860 119904119899119889119898

Let 120576 gt 0 be given Choose 1198990 isin N such that 119879119860(119891) minusint1198601199041198990119889119898119884 le 120576 Then

1003817100381710038171003817119879119860 (119891)1003817100381710038171003817119884 le 1003817100381710038171003817100381710038171003817119879119860 (119891) minus int119860 11990411989901198891198981003817100381710038171003817100381710038171003817119884 +

1003817100381710038171003817100381710038171003817int119860 11990411989901198891198981003817100381710038171003817100381710038171003817119884

le 120576 + 120593lowast (119860) (47)

It follows that 119879119860120593 le 120593lowast(119860) so 120593lowast(119860) = 119879119860120593 Hencefor 119910lowast isin 119884lowast we easily get

1003817100381710038171003817(119910lowast ∘ 119879)1198601003817100381710038171003817lowast120593 = 1003817100381710038171003817119910lowast ∘ 1198791198601003817100381710038171003817lowast120593 = (119898119910lowast)120593lowast (119860) (48)

(v) Using (iv) we have

120593lowast (119860) = 10038171003817100381710038171198791198601003817100381710038171003817120593= sup 1003817100381710038171003817119879119860 (119891)1003817100381710038171003817119884 119891 isin 119871120593 (119883) 10038171003817100381710038171198911003817100381710038171003817120593 le 1= sup119910lowastisin119861119884lowast

1003816100381610038161003816(119910lowast ∘ 119879119860) (119891)1003816100381610038161003816 119891 isin 119871120593 (119883) 10038171003817100381710038171198911003817100381710038171003817120593 le 1= sup119910lowastisin119861119884lowast

1003817100381710038171003817119910lowast ∘ 1198791198601003817100381710038171003817lowast120593 = sup119910lowastisin119861119884lowast

(119898119910lowast)120593lowast (119860) (49)

(vi) This follows from (ii) and (iv)

For a sequence (119860119899) in Σ we will write 1198601198991205830 if 119860119899 darrand 120583(119860119899 cap 119860) rarr 0 for every 119860 isin Σ119891(120583)Definition 10 A measure 119898 Σ119891(120583) rarr L(119883 119884) with 119898 ≪120583 and 120593lowast(Ω) lt infin is said to be 120593lowast-semivariationally 120583-continuous if 120593lowast(119860119899) rarr 0 whenever 1198601198991205830 (119860119899) sub Σ

Using a standard argument we can show the following

Proposition 11 Let119898 Σ rarrL(119883 119884) be an additive measuresuch that 119898 ≪ 120583 and 120593(Ω) lt infin Then the followingstatements are equivalent

(i) 119898 is 120593lowast-semivariationally 120583-continuous(ii) The following two conditions hold simultaneously

(a) For every 120576 gt 0 there exists 120575 gt 0 such that120593lowast(119860) le 120576 whenever 120583(119860) le 120575 119860 isin Σ(b) For every 120576 gt 0 there exists 1198600 isin Σ119891(120583) such that120593lowast(Ω 1198600) le 120576

The following theorem characterizes 120593lowast-semivariation-ally 120583-continuous representing measures

Theorem 12 Let 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be its representingmeasure Then the following statements are equivalent

(i) 119898 is 120593lowast-semivariationally 120583-continuous(ii) 119879 is (120574120593 sdot 119884)-continuous(iii) 119879(119891119899)119884 rarr 0 if 119891119899 rarr 0 inT0 and sup119899119891119899120593 lt infin(iv) 119879119860119899120593 rarr 0 if 1198601198991205830 (119860119899) sub Σ

Proof (i)hArr (ii)hArr (iii) See [5 Corollary 28 and Proposition11]

(i)hArr (iv) This follows fromTheorem 9

Now assume that Ω is a completely regular Hausdorffspace LetB119886 denote the 120590-algebra of Baire sets in Ω whichis the 120590-algebra generated by the class Z of all zero sets ofbounded continuous positive functions on120596 ByPwedenotethe family of all cozero (=positive) in Ω (see [25 p 108])

Let 120583 B119886 rarr [0infin) be a countably additive measureThen 120583 is zero-set regular that is for every119860 isinB119886 and 120576 gt 0there exists 119885 isin Z with 119885 sub 119860 such that 120583(119860 119885) le 120576 (see[25 p 118]) It follows that for every 119860 isin B119886 and 120576 gt 0 thereexist 119880 isin P 119880 sup 119860 such that 120583(119880 119860) le 120576

We can assume that 120583 to be complete (if necessary wecan take the completion (ΩB119886 120583) of the measure space(ΩB119886 120583))Proposition 13 Assume that Ω is a completely regular Haus-dorff space and (ΩB119886 120583) is a complete finite measure spaceLet119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operatorand119898 B119886 rarrL(119883 119884) be its representing measure Then thefollowing statements are equivalent

(i) 119898 is 120593lowast-semivariationally 120583-continuous

Journal of Function Spaces 7

(ii) For every sequence (119860119899) in B119886 such that 119860119899 darr and120583(119860119899) rarr 0 there exists a sequence (119880119899) inPwith119860119899 sub119880119899 darr such that 120593lowast(119880119899) rarr 0(iii) For every sequence (119860119899) in B119886 such that 119860119899 darr and120583(119860119899) rarr 0 there exists a sequence (119880119899) inPwith119860119899 sub119880119899 darr such thatsup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) supp119891 sub 119880119899 997888rarr 0 (50)

Proof (i)rArr (ii) Assume that (i) holds and (119860119899) is a sequencein B119886 such that 119860119899 darr and 120583(119860119899) rarr 0 Then there exists asequence (119880119899) inP such that119860119899 sub 119880119899 darr and120583(119880119899119860119899) le 1119899for 119899 isin N

Let 120576 gt 0 be given Then in view of Proposition 11 thereexists 120575 gt 0 such that 120593lowast(119860) le 1205762 if 120583(119860) le 120575with119860 isinB119886Choose 1198991 isin N such that 120583(119880119899 119860119899) le 120575 for 119899 ge 1198992 Then120593lowast(119880119899 119860119899) le 1205762 for 119899 ge 1198991 Since 120593lowast(119860119899) rarr 0 we canchoose 1198992 isin N such that 120593lowast(119860119899) le 1205762 for 119899 ge 1198992 Then for119899 ge 1198990 = max(1198991 1198992) we get

120593lowast (119880119899) le 120593lowast (119880119899 119860119899) + 120593lowast (119860119899) le 120576 (51)

that is (ii) holds(ii)rArr (iii) Assume that (ii) holds and (119860119899) is a sequence

in B119900 such that 119860119899 darr and 120583(119860119899) rarr 0 Then there exists asequence (119880119899) inP with 119860119899 sub 119880119899 darr such that 120593lowast(119880119899) rarr 0Note that for 119891 isin 119861119871120593(119883) with supp119891 sub 119880119899 for 119899 isin N byTheorem 9 we have

1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 = 1003817100381710038171003817100381710038171003817intΩ 1198911198891198981003817100381710038171003817100381710038171003817119884 =

100381710038171003817100381710038171003817100381710038171003817int119880119899 119891119889119898100381710038171003817100381710038171003817100381710038171003817119884 le 120593lowast (119880119899) (52)

It follows that (iii) holds(iii)rArr (i) Assume that (iii) holds and119860119899 darrwith 120583(119860119899) rarr0Then there exists a sequence (119880119899) inPwith119860119899 sub 119880119899 darr such

that

sup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) supp119891 sub 119880119899 997888rarr 0 (53)

Assume on the contrary that (i) fails to hold Then withoutloss of generality we can assume that

120593lowast (119860119899) gt 1205760 for some 1205760 gt 0 all 119899 isin N (54)

Choose 1198990 isin N such that

sup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) supp119891 sub 1198801198990 lt 12057602 (55)

In view of (54) there exists a pairwise disjoint set 1198611 119861119896in B119886 119909119894 isin 119883 for 119894 = 1 119896 and 119910lowast isin 119861119884lowast such thatsum119896119894=1(1119861119894 otimes 119909119894)120593 le 1 and

1003816100381610038161003816100381610038161003816100381610038161003816119910lowast( 119896sum

119894=1

119898(1198601198990 cap 119861119894) (119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 ge 1205760 (56)

Let 1199040 = sum119896119894=1(11198601198990cap119861119894 otimes 119909119894) Then 1199040120593 le 1 and supp 1199040 sub1198601198990 sub 1198801198990 Then by (55) we get 119879(1199040)119884 lt 12057602On the other hand in view of (56) we have 119879(1199040)119884 ge 1205760

This contradiction establishes that (i) holds

Corollary 14 Assume that Ω is a completely regular Haus-dorff space and (ΩB119886 120583) is complete finitemeasure space Let119879 119871120593(119883) rarr 119884 be a (120574120593 sdot 119884)-continuous linear operator and119898 B119886 rarr L(119883 119884) be its representing measure Then 120593lowast isregular that is for every 119860 isinB119886 and 120576 gt 0 there exist 119885 isinZand 119880 isin P with 119885 sub 119860 sub 119880 such that 120593lowast(119880 119885) le 120576Proof In view of Theorem 12 119898 is 120593lowast-semivariationally 120583-continuous Let 119860 isin B119886 and 120576 gt 0 be given Then byProposition 11 there exists 120575 gt 0 such that 120593lowast(119861) le 120576whenever 119861 isin B119886 and 120583(119861) le 120575 By the regularity of 120583 onecan choose 119885 isin Z and 119880 isin P with 119885 sub 119860 sub 119880 such that120583(119880 119885) le 120575 Hence 120593lowast(119880 119885) le 120576 as desired4 Compact Operatorson Orlicz-Bochner Spaces

The following theorem presents necessary conditions for a(Tand120593 sdot 119884)-continuous operator 119879 119871120593(119883) rarr 119884 to be

compact

Theorem 15 Assume that a Young function 120593 such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot 119884)-continuous linear operator and 119898 Σ119891(120583) rarr L(119883 119884)be its representing measure If 119879 is compact then 119898 is 120593lowast-semivariationally 120583-continuousProof Assume that 119879 is compact and 119898 fails to be 120593lowast-semivariationally 120583-continuous Then there exist 120576 gt 0 and asequence (119860119899) in Σwith1198601198991205830 such that 119879119860119899 = 119898lowast120593(119860119899) gt120576 for 119899 isin N (seeTheorem9)Hence one can choose a sequence(119910lowast119899 ) in 119861119884lowast such that

10038171003817100381710038171003817119910lowast119899 ∘ 11987911986011989910038171003817100381710038171003817lowast120593 ge 120576 forall119899 isin N (57)

By Schauderrsquos theorem the conjugatemapping119879lowast 119884lowast rarr119871120593(119883)lowast is compact Note that 119879lowast(119910lowast119899 ) = 119910lowast119899 ∘ 119879 isin 119871120593(119883)sim119899 forall 119899 isin N where 119871120593(119883)sim119899 is a closed subspace of the Banachspace (119871120593(119883)lowast sdot lowast120593) (see Theorem 2) Then for every 119899 isin N

there exists 119892119899 isin 119871120593lowast(119883lowast 119883) such that

(119910lowast119899 ∘ 119879) (119891) = intΩ⟨119891 (120596) 119892119899 (120596)⟩ 119889120583

for 119891 isin 119871120593 (119883) 1003817100381710038171003817119910lowast119899 ∘ 1198791003817100381710038171003817lowast120593= sup int

Ω

1003817100381710038171003817119891 (120596)1003817100381710038171003817119883 120599 (119892119899) (120596) 119889120583 119891 isin 119861119871120593(119883)= 1003817100381710038171003817120599 (119892119899)1003817100381710038171003817120593lowast

(58)

Hence we obtain that for each 119899 isin N

10038171003817100381710038171003817119910lowast119899 ∘ 11987911986011989910038171003817100381710038171003817lowast120593 = 100381710038171003817100381710038171119860119899120599 (119892119899)10038171003817100381710038171003817120593lowast = 10038171003817100381710038171003817120599 (1119860119899119892119899)10038171003817100381710038171003817120593lowast (59)

8 Journal of Function Spaces

Since 119879lowast(119861119884lowast) is a relatively sequentially compact subset of((119871120593(119883)sim119899 sdot lowast120593) there exist a subsequence (119892119896119899) of (119892119899) and119892 isin 119871120593lowast(119883lowast 119883) such that10038171003817100381710038171003817119865119892119899 minus 11986511989210038171003817100381710038171003817lowast120593 = 10038171003817100381710038171003817120599 (119892119896119899 minus 119892)10038171003817100381710038171003817120593lowast 997888rarr 0 (60)

Choose 119899120576 isin N such that 120599(119892119896119899 minus 119892)120593lowast le 1205762 for 119899 ge 119899120576Hence for 119899 ge 1198991205761003816100381610038161003816100381610038161003816

100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast minus 100381710038171003817100381710038171003817120599 (1119860119896119899119892119896119899)100381710038171003817100381710038171003817120593lowast1003816100381610038161003816100381610038161003816

le 100381710038171003817100381710038171003817120599 (1119860119896119899 (119892119896119899 minus 119892))100381710038171003817100381710038171003817120593lowast = 1003817100381710038171003817100381710038171119860119896119899120599 (119892119896119899 minus 119892)100381710038171003817100381710038171003817120593lowastle 10038171003817100381710038171003817120599 (119892119896119899 minus 119892)10038171003817100381710038171003817120593lowast le 1205762

(61)

Using (57) and (61) for 119899 ge 119899120576 we get120576 le 100381710038171003817100381710038171003817119910lowast ∘ 119879119860119896119899 100381710038171003817100381710038171003817

lowast

120593= 100381710038171003817100381710038171003817120599 (1119860119896119899119892119896119899)100381710038171003817100381710038171003817120593lowast

le 1205762 +100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast

(62)

and hence 1003817100381710038171003817100381710038171119860119896119899120599 (119892)100381710038171003817100381710038171003817120593lowast = 100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast ge 1205762 (63)

On the other hand since 120593lowast is supposed to satisfy theΔ 2-condition we have that 1119860119896119899120599(119892)120593lowast rarr 0 (see [26 Theorem3 pp 58-59]) This contradiction establishes that 119898 is 120593lowast-semivariationally 120583-continuousCorollary 16 Assume that 120593 is a Young function such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operator Then the following statements areequivalent

(i) 119879 is compact(ii) 119879 is (120574120593 sdot 119884)-compact that is there exists a 120574120593-

neighborhood 119881 of 0 in 119871120593(119883) such that 119879(119881) is arelatively norm compact set in 119884

(iii) There exists a Young function 120595 with 120595 ≪ 120593 such thatintΩ119891119889119898 119891 isin 119871120593(119883) 119891120595 le 1 is a relatively norm

compact set in 119884Proof (i)rArr (ii) Assume that (i) holds Then by Theorems 12and 15 119879 is (120574120593 sdot 119884)-continuous Since the space (119871120593(119883) 120574120593)is quasinormable by Grothendieckrsquos classical result (see [15p 429]) we obtain that 119879 is (120574120593 sdot 119884)-compact

(ii)rArr (i) The implication is obvious(ii)hArr (iii) This follows fromTheorem 3

5 Topology Associated withthe 120593lowast-Semivariation ofa Representing Measure

Assume that 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator Let119898 Σ119891(120583) rarrL(119883 119884) be its representingmeasure Let us put

119901119898 (119910lowast) fl (119898119910lowast)120593lowast (Ω) for 119910lowast isin 119884lowast (64)

Note that119901119898 is a seminormon119884lowast Following [22 27] let 120575119898120593lowaststand for the topology on 119861119884lowast defined by the seminorm 119901119898restricted to 119861119884lowast

The following theorem characterizes (Tand120593 sdot 119884)-

continuous compact operators 119879 119871120593(119883) rarr 119884 in terms ofthe topological properties of the space (119861119884lowast 120575119898120593lowast) (see [22Theorem 3])

Theorem 17 Let 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be its representingmeasure Then the following statements are equivalent

(i) The space (119861119884lowast 120575119898120593lowast) is compact

(ii) 119879 is compact

Proof (i) rArr (ii) Assume that (119861119884lowast 120575119898120593lowast) is compact Let(119910lowast119899 ) be a sequence in 119861119884lowast Without loss of generality we canassume that 119910lowast119899 rarr 119910lowast0 in 120575119898120593lowast for some 119910lowast isin 119861119884lowast Then usingTheorem 9 for 119891 isin 119871120593(119883) we have

1003816100381610038161003816(119879lowast (119910lowast119899 ) minus 119879lowast (119910lowast0 )) (119891)1003816100381610038161003816 = 1003816100381610038161003816(119910lowast119899 minus 119910lowast0 ) (119879 (119891))1003816100381610038161003816= 1003816100381610038161003816100381610038161003816intΩ 119891119889119898119910lowast119899 minus119910lowast0

1003816100381610038161003816100381610038161003816 le (119898119910lowast119899 minus119910lowast0 )120593lowast (Ω) 10038171003817100381710038171198911003817100381710038171003817120593 (65)

It follows that 119879lowast(119910lowast119899 ) minus 119879lowast(119910lowast0 )lowast120593 le (119898119910lowast119899 minus119910lowast0 )120593lowast(Ω) where119901119898(119910lowast119899 minus 119910lowast0 ) = (119898119910lowast119899 minus119910lowast0 )120593lowast(Ω) rarr119899 0 This means that 119879lowast iscompact and hence 119879 is compact

(ii)rArr (i) Assume that 119879 is compact and (119910lowast120572 ) is a net in119861119884lowast Since119861119884lowast is 120590(119884lowast 119884)-compact without loss of generalitywe can assume that 119910lowast120572 rarr120572 119910lowast0 in 120590(119884lowast 119884) for some 119910lowast0 isin 119861119884lowast In view of the compactness of the conjugate operator 119879lowast 119884lowast rarr 119871120593(119883)lowast there exists a subset (119910lowast120573 ) of (119910lowast120572 ) and Φ0 isin119871120593(119883)lowast such that 119879lowast(119910lowast120573 ) minus Φ0lowast120593 rarr

1205730 On the other hand

since 119879lowast is (120590(119884lowast 119884) 120590(119871120593(119883)lowast 119871120593(119883)))-continuous we get119879lowast(119910lowast120573 ) rarr120573119879lowast(119910lowast0 ) in 120590(119871120593(119883)lowast 119871120593(119883)) HenceΦ0 = 119879lowast(119910lowast0 )

that is 119879lowast(119910lowast120573 ) minus 119879lowast(119910lowast0 )lowast120593 rarr1205730

Let 120576 gt 0 be givenThen there exist a pairwise disjoint set1198601 119860119899 in Σ119891(120583) and 119909119894 isin 119883 for 119894 = 1 119899 such thatsum119899119894=1(1119860119894 otimes 119909119894)120593 le 1 and

(119898119910lowast120573minus119910lowast0)120593lowast(Ω) le 1003816100381610038161003816100381610038161003816100381610038161003816

119899sum119894=1

(119910lowast120573 minus 119910lowast0 ) (119898 (119860 119894) (119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 + 120576 (66)

Hence

(119898119910lowast120573minus119910lowast0)120593lowast(Ω) le 1003816100381610038161003816100381610038161003816100381610038161003816

119899sum119894=1

(119910lowast120573 minus 119910lowast0 ) (119879 (1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 + 120576

le 1003816100381610038161003816100381610038161003816100381610038161003816(119910lowast120573 minus 119910lowast0 ) 119879(

119899sum119894=1

(1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816

+ 120576

Journal of Function Spaces 9

= 1003816100381610038161003816100381610038161003816100381610038161003816119879lowast (119910lowast120573 minus 119910lowast0 )(

119899sum119894=1

(1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816

+ 120576le 10038171003817100381710038171003817119879lowast (119910lowast120573 minus 119910lowast0 )10038171003817100381710038171003817lowast120593

1003817100381710038171003817100381710038171003817100381710038171003817119899sum119894=1

(1119860119894 otimes 119909119894)1003817100381710038171003817100381710038171003817100381710038171003817120593

+ 120576 le 10038171003817100381710038171003817119879lowast (119910lowast120573) minus 119879 (119910lowast0 )10038171003817100381710038171003817lowast120593 + 120576(67)

Hence 119901119898(119910lowast120573 minus 119910lowast0 ) = (119898119910lowast120573minus119910lowast0)120593lowast(Ω) rarr

1205730 and this means

that the space (119861119884lowast 120575119898120593lowast) is compact

As a consequence of Theorems 17 and 15 we have thefollowing

Corollary 18 Assume that 120593 is a Young function such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be itsrepresenting measure If the space (119861119884lowast 120575119898120593lowast) is compact then119898 is 120593lowast-semivariationally 120583-continuousConflicts of Interest

The author declares that there are no conflicts of interest

References

[1] S Chen and H Hudzik ldquoOn some convexities of Orlicz andOrlicz-Bochner spacesrdquo Commentationes Mathematicae vol29 no 1 pp 13ndash29 1988

[2] P Kolwicz and R Pluciennik ldquo119875-convexity of Orlicz-Bochnerspacesrdquo Proceedings of the American Mathematical Society vol126 no 8 pp 2315ndash2322 1998

[3] S Shang and Y Cui ldquoUniform nonsquareness and locallyuniform nonsquareness in Orlicz-Bochner function spaces andapplicationsrdquo Journal of Functional Analysis vol 267 no 7 pp2056ndash2076 2014

[4] K Feledziak andM Nowak ldquoLocally solid topologies on vectorvalued function spacesrdquo Universitat de Barcelona CollectaneaMathematica vol 48 no 4-6 pp 487ndash511 1997

[5] K Feledziak and M Nowak ldquoIntegral representation of linearoperators on Orlicz-Bochner spacesrdquo Universitat de BarcelonaCollectanea Mathematica vol 61 no 3 pp 277ndash290 2010

[6] M Nowak ldquoLinear operators on vector-valued function spaceswithMackey topologiesrdquo Journal of Convex Analysis vol 15 no1 pp 165ndash178 2008

[7] M Nowak ldquoLebesgue topologies on vector-valued functionspacesrdquoMathematica Japonica vol 52 no 2 pp 171ndash182 2000

[8] A V Buhvalov ldquoThe analytic representation of operators withan abstract normrdquo Izvestiya Vysshikh Uchebnykh ZavedeniiMatematika vol 11 no 162 pp 21ndash32 1975

[9] A V Buhvalov ldquoThe analytic representation of linear operatorsby means of measurable vector-valued functionsrdquo IzvestiyaVysshikh Uchebnykh Zavedenii Matematika vol 7 no 182 pp21ndash31 1977

[10] J Diestel and J J Uhl ldquoVector measuresrdquo vol 15 AmericanMathematical Society Surveys Providence RI USA 1977

[11] M Nowak ldquoDuality theory of vector-valued function spaces IrdquoCommentationes Mathematicae vol 37 pp 195ndash215 1997

[12] A V Bukhvalov and G Ya Lozanovskii ldquoOn sets closed inmeasure in spaces of mesurable functionsrdquo Transactions of theMoscow Mathematical Society vol 2 pp 127ndash148 1978

[13] A Wiweger ldquoLinear spaces with mixed topologyrdquo StudiaMathematica vol 20 pp 47ndash68 1961

[14] K Feledziak ldquoUniformly Lebesgue topologies on Kothe-Bochner spacesrdquoCommentationesMathematicae vol 37 pp 81ndash98 1997

[15] W Ruess ldquo[Weakly] compact operators and DF-spacesrdquo PacificJournal of Mathematics vol 98 no 2 pp 419ndash441 1982

[16] M Nowak ldquoCompactness of Bochner representable operatorson Orlicz spacesrdquo Positivity An International MathematicsJournal Devoted to Theory and Applications of Positivity vol 13no 1 pp 193ndash199 2009

[17] N E Gretsky and J Uhl ldquoBounded linear operators on Banachfunction spaces of vector-valued functionsrdquo Transactions of theAmerican Mathematical Society vol 167 pp 263ndash277 1972

[18] J J Uhl ldquoCompact operators on Orlicz spacesrdquo in Rendicontidel Seminario Matematico vol 42 pp 209ndash219 University ofPadova 1969

[19] N Dinculeanu Vector Measures International Series of Mono-graphs in Pure and Applied Mathematics 95 Pergamon PressNew York NY USA 1967

[20] N Dinculeanu Linear Operators on Lp-Spaces Vector andOperator-Valued Measures and Applications Academic PressNew York NY USA 1973

[21] N Dinculeanu Vector Integration and Stochastic Integration inBanach Spaces John Wiley amp Sons 2000

[22] R A Alo A De Korwin and L E Kunes ldquoTopological aspectsof 119902-regular measuresrdquo Studia Mathematica vol 48 pp 49ndash601973

[23] J Diestel ldquoOn the representation of bounded linear operatorsfrom Orlicz spaces of Lebesgue-Bochner measurable functionsto any Banach spacerdquo Bulletin de lrsquoAcademie Polonaise desSciences Serie des Sciences Mathematiques Astronomiques etPhysiques vol 18 pp 375ndash378 1970

[24] L Vanderputten ldquoRepresentation of operators defined on thespace of Bochner integrable functionsrdquo Extracta Mathematicaevol 16 no 3 pp 383ndash391 2001

[25] R FWheeler ldquoA survey of Bairemeasures and strict topologiesrdquoExpositiones Mathematicae vol 1 no 2 pp 97ndash190 1983

[26] W A Luxemburg Banach function spaces [Thesis] DelftNetherlands 1955

[27] P W Lewis ldquoVector measures and topologyrdquo Revue RoumainedeMathematique Pures et Appliquees vol 16 pp 1201ndash1209 1971

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4 Journal of Function Spaces

Let (119898119910lowast)120593lowast(119860) stand for the 120593lowast-semivariation of119898119910lowast on 119860 isinΣ that is(119898119910lowast)120593lowast (119860)= sup 1003816100381610038161003816100381610038161003816int119860 119904 119889119898119910lowast

1003816100381610038161003816100381610038161003816 119904 isin S (Σ119891 (120583) 119883) 119904120593 le 1 (19)

The following lemma will be useful

Lemma 4 Let 120593 be a Young function and 119898 Σ119891(120583) rarrL(119883 119884) be a measure with 119898 ≪ 120583 and 120593lowast(Ω) lt infin Thenthe following statements hold

(i) If119891 isin 119864120593(119883) then there exists a sdot120593-Cauchy sequence(119904119899) inS(Σ119891(120583) 119883) such that 119904119899(120596)minus119891(120596)119883 rarr 0 120583-ae

(ii) If (119904119899) is a sdot 120593-Cauchy sequence inS(Σ119891(120583) 119883) thenfor119860 isin Σ (int

119860119904119899119889119898) is a Cauchy sequence in a Banach

space119884 and for every119910lowast isin 119884lowast (int119860119904119899119889119898119910lowast) is aCauchy

sequence in R(iii) If 119891 isin 119864120593(119883) and (1199041015840119899) and (11990410158401015840119899 ) are sdot 120593-Cauchy

sequence inS(Σ119891(120583) 119883) such that 1199041015840119899(120596)minus119891(120596)119883 rarr0 120583-ae and 11990410158401015840119899 (120596) minus 119891(120596)119883 rarr 0 120583-ae then for119860 isin Σ one haslimint

1198601199041015840119899119889119898 = limint

11986011990410158401015840119899 119889119898 (20)

and for every 119910lowast isin 119884lowast one haslimint

1198601199041015840119899119889119898119910lowast = limint

11986011990410158401015840119899 119889119898119910lowast (21)

Proof (i) Let 119891 isin 119864120593(119883) Then there exists a sequence (119904119899)in S(Σ119891(120583) 119883) such that 119904119899(120596) minus 119891(120596)119883 rarr 0 120583-ae and119904119899(120596)119883 le 119891(120596)119883 120583-ae for all 119899 isin N (see [21 Theorem 6p 4]) Using the Lebesgue dominated convergence theoremwe obtain that int

Ω120593(120582(119904119899(120596) minus 119891(120596)119883)119889120583 rarr 0 for all 120582 gt 0

so 119904119899 minus 119891120593 rarr 0 Hence (119904119899) is a sdot 120593-Cauchy sequence(ii) Assume that (119904119899) is a sdot 120593-Cauchy sequence in

S(Σ119891(120583) 119883) Hence for 119899 119896 isin N we have

1003817100381710038171003817100381710038171003817int119860 119904119899119889119898 minus int1198601199041198961198891198981003817100381710038171003817100381710038171003817119884 =

1003817100381710038171003817100381710038171003817int119860 (119904119899 minus 119904119896) 1198891198981003817100381710038171003817100381710038171003817119884

le 1003817100381710038171003817119904119899 minus 1199041198961003817100381710038171003817120593 120593lowast (119860) le 1003817100381710038171003817119904119899 minus 1199041198961003817100381710038171003817120593 120593lowast (Ω) (22)

It follows that (int119860119904119899119889119898) is a Cauchy sequence in 119884 Hence in

view of (15) for 119910lowast isin 119884lowast (int119860119904119899119889119898119910lowast) is a Cauchy sequence

in R(iii) Note that (1199041015840119899 minus 11990410158401015840119899 ) is a sdot 120593-Cauchy sequence and1199041015840119899(120596) minus 11990410158401015840119899 (120596)119883 rarr 0 120583-ae Hence there exists ℎ isin 119864120593(119883)

such that (1199041015840119899minus11990410158401015840119899 )minusℎ120593 rarr 0 Note thatT0|119864120593(119883) sub T120593|119864120593(119883)Hence (1199041015840119899minus11990410158401015840119899 )minusℎ rarr 0 inT0 and it follows that there exists asubsequence (1199041015840119896119899minus11990410158401015840119896119899) of (1199041015840119899minus11990410158401015840119899 ) such that (1199041015840119896119899(120596)minus11990410158401015840119896119899(120596))minus

ℎ(120596)119883 rarr 0 120583-ae Then ℎ(120596) = 0 120583-ae so 1199041015840119899 minus 11990410158401015840119899 120593 rarr 0and for 119860 isin Σ we get

1003817100381710038171003817100381710038171003817int119860 1199041015840119899119889119898 minus int11986011990410158401015840119899 1198891198981003817100381710038171003817100381710038171003817119884 =

1003817100381710038171003817100381710038171003817int119860 (1199041015840119899 minus 11990410158401015840119899 ) 1198891198981003817100381710038171003817100381710038171003817119884

le 100381710038171003817100381710038171199041015840119899 minus 11990410158401015840119899 10038171003817100381710038171003817120593 120593lowast (119860) (23)

It follows that

limint1198601199041015840119899119889119898 = limint

11986011990410158401015840119899 119889119898 (24)

and hence in view of (15) for every 119910lowast isin 119884lowast we havelimint

1198601199041015840119899119889119898119910lowast = limint

11986011990410158401015840119899 119889119898119910lowast (25)

Following [21 sect 13 Definition 1 p 254] in view ofLemma 4 we have the following

Definition 5 Let 120593 be a Young function and 119898 Σ119891(120583) rarrL(119883 119884) be an additive measure such that 119898 ≪ 120583 and120593lowast(Ω) lt infin Then for every 119891 isin 119864120593(119883) and 119860 isin Σ wecan define the integralint

119860119891119889119898 by the equality

int119860119891119889119898 fl limint

119860119904119899119889119898 (26)

and for 119910lowast isin 119884lowast we can define the integralint119860119891119889119898119910lowast by the

equality

int119860119891119889119898119910lowast fl limint

119860119904119899119889119898119910lowast (27)

where (119904119899) is an arbitrary sdot 120593-Cauchy sequence inS(Σ119891(120583) 119883) such that 119904119899(120596) minus 119891(120596)119883 rarr 0 120583-ae3 Integral Representation of ContinuousOperators on Orlicz-Bochner Spaces

For a bounded linear operator 119879 119871120593(119883) rarr 119884 let

119879120593 fl sup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) (28)

Proposition 6 Let 119879 119871120593(119883) rarr 119884 be a bounded linearoperator and

119898(119860) (119909) fl 119879 (1119860 otimes 119909) for 119860 isin Σ119891 (120583) 119909 isin 119883 (29)

Then the following statements hold

(i) For 119860 isin Σ119891(120583)119898(119860) isin L(119883 119884) and 119898(119860) le 119879120593 sdot1119860120593(ii) 119898 ≪ 120583(iii) 119898(119860119899) rarr 0 if 119860119899 darr 0 with 119860119899 isin Σ119891(120583)(iv) 119898 Σ119891(120583) rarr L(119883 119884) is countably additive that is119898(⋃infin119899=1 119861119899) = suminfin119899=1119898(119861119899) if (119861119899) is a pairwise disjoint

sequence in Σ119891(120583) with⋃infin119899=1 119861119899 isin Σ119891(120583)(v) 120593lowast(Ω) le 119879120593

Journal of Function Spaces 5

Proof (i) Let 119860 isin Σ119891(120583) Then for 119909 isin 119861119883 we have 1119860 otimes119909120593 le 1119860120593 and hence

119898 (119860) (119909)119884 = 1003817100381710038171003817119879 (1119860 otimes 119909)1003817100381710038171003817119884 le 119879120593 sdot 10038171003817100381710038171119860 otimes 1199091003817100381710038171003817120593le 119879120593 100381710038171003817100381711198601003817100381710038171003817120593 (30)

so 119898(119860) le 119879120593 sdot 1119860120593(ii) This follows from (i) because 1119860120593 = 0 if 120583(119860) = 0(iii) Assume that 119860119899 darr 0 with 119860119899 isin Σ119891(120583) Then

11198601(120596) ge 1119860119899(120596) darr 0 for 120596 isin Ω By the Lebesgue dominatedconvergence theorem we obtain that int

Ω120593(1205821119860119899(120596))119889120583 rarr 0

for every 120582 gt 0 This means that 1119860119899120593 rarr 0 and by (i)119898(119860119899) rarr 0(iv) Assume that (119861119899) is a pairwise disjoint sequence inΣ119891(120583) with 119861 = ⋃infin119899=1 119861119899 isin Σ119891(120583) Let 119860119899 = 119861 ⋃119899119894=1 119861119894 for119899 isin N Then 119860119899 isin Σ119891(120583) and 119860119899 darr 0 Hence by (iii) 119898(119861) minussum119899119894=1119898(119861119894) = 119898(119861) minus 119898(⋃119899119894=1 119861119894) = 119898(119860119899) rarr 0Statement (v) is obvious

Definition 7 Let119879 119871120593(119883) rarr 119884 be a bounded linear operatorand

119898(119860) (119909) fl 119879 (1119860 otimes 119909) for 119860 isin Σ119891 (120583) 119909 isin 119883 (31)

Then the measure 119898 Σ119891(120583) rarr L(119883 119884) will be called arepresenting measure of 119879Proposition 8 Let 119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot 119884)-continuous linear operator and 119898 Σ119891(120583) rarr L(119883 119884) beits representing measure Then there exists a Young function 120595such that 120595 ⊲ 120593 and 120595lowast(Ω) lt infin

Proof According toTheorem 2 there exist a finite set 120595119894 119894 =1 119899 of Young functions with 120595119894 ⊲ 120593 for 119894 = 1 119899 and119886 gt 0 such that1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 le 119886max1le119894le119899

10038171003817100381710038171198911003817100381710038171003817120595119894 forall119891 isin 119871120593 (119883) (32)

Let120595(119905) = max1le119894le119899120595119894(119905) for 119905 ge 0Then120595 is a Young functionwith 120595 ⊲ 120593 and

1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 le 119886 10038171003817100381710038171198911003817100381710038171003817120595 forall119891 isin 119871120593 (119883) (33)

Hence120595lowast (Ω)= sup 119879 (119904)119884 119904 isin S (Σ119891 (120583) 119883) 119904120595 le 1le 119886 lt infin

(34)

For a linear operator 119879 119871120593(119883) rarr 119884 and 119860 isin Σ let119879119860 (119891) fl 119879 (1119860119891) for 119891 isin 119871120593 (119883) (35)

Nowwe can state ourmain result that extends the classicalresults concerning the integral representation of operators onLebesgue-Bochner spaces 119871119901(119883) (1 le 119901 lt infin) (see [19 sect13 Theorem 1 pp 259ndash261]) to operators on Orlicz-Bochnerspaces 119871120593(119883)

Theorem 9 Let 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be its representingmeasure Then for 119860 isin Σ the following statements hold

(i) 119879119860 119871120593(119883) rarr 119884 is a (Tand120593 sdot 119884)-continuous linear

operator(ii) For 119891 isin 119871120593(119883) one has

119879119860 (119891) = int119860119891119889119898 (36)

and for 119910lowast isin 119884lowast one has119910lowast (119879119860 (119891)) = int

119860119891119889119898119910lowast (37)

(iii) For 119891 isin 119871120593(119883) the measure 119898119891 Σ rarr 119884 defined bythe equality

119898119891 (119860) fl int119860119891119889119898 for 119860 isin Σ (38)

is countably additive(iv) 119879119860120593 = 120593lowast(119860)

and for 119910lowast isin 119884lowast 119910lowast ∘ 119879119860lowast120593 = (119910lowast ∘ 119879)119860lowast120593 =(119898119910lowast)120593lowast(119860)(v) 120593lowast(119860) = sup(119898119910lowast)120593lowast(119860) 119910lowast isin 119861119884lowast(vi) For 119891 isin 119871120593(119883) one has

1003817100381710038171003817100381710038171003817int119860 1198911198891198981003817100381710038171003817100381710038171003817119884 le 120593lowast (119860) 10038171003817100381710038171198911003817100381710038171003817120593 (39)

and for 119910lowast isin 119884lowast one has1003816100381610038161003816100381610038161003816int119860 119891119889119898119910lowast

1003816100381610038161003816100381610038161003816 le (119898119910lowast)120593lowast (119860) 10038171003817100381710038171198911003817100381710038171003817120593 (40)

Proof (i) Assume that (119891120572) is a net in 119871120593(119883) such that119891120572 rarr 0inTand

120593 SinceTand120593 is a locally solid topology on 119871120593(119883) we get

1119860119891120572 rarr 0 inTand120593 Hence

1003817100381710038171003817119879119860 (119891120572)1003817100381710038171003817119884 = 1003817100381710038171003817119879 (1119860119891120572)1003817100381710038171003817119884 997888rarr 0 (41)

(ii) In view of Proposition 8 there exists a Young function120595 such that 120595 ⊲ 120593 and 120595lowast(Ω) lt infin Then 119871120593(119883) sub119864120595(119883) Let 119891 isin 119871120593(119883) Then there exists a sequence (119904119899)in S(Σ119891(120583) 119883) such that 119904119899(120596) minus 119891(120596)119883 rarr 0 120583-ae and119904119899(120596)119883 le 119891(120596)119883 120583-ae for all 119899 isin N (see [21 Theorem 6p 4])Then 119904119899 rarr 119891 inTand

120593 becauseTand120593 is a Lebesgue topology

Hence 119904119899 minus 119891120595 rarr 0 In view of Lemma 4 we can define theintegral int

119860119891119889119898 by the equality

int119860119891119889119898 fl limint

119860119904119899119889119898 (42)

6 Journal of Function Spaces

Since 119879119860(119904119899) = int119860119904119899119889119898 and by (i) 119879119860 is (Tand

120593 sdot 119884)-continuous we get

119879119860 (119891) = limint119860119904119899119889119898 (43)

Hence

119879119860 (119891) = int119860119891119889119898 (44)

and for 119910lowast isin 119884lowast we have119910lowast (119879119860 (119891)) = lim119910lowast (int

119860119904119899119889119898) = limint

119860119904119899119889119898119910lowast

= int119860119891119889119898119910lowast

(45)

(iii) Let 119891 isin 119871120593(119883) and (119860119899) be a sequence in Σ suchthat 119860119899 darr 0 Then 1119860119899(120596) darr 0 for 120596 isin Ω andhence 1119860119899(120596)119891(120596)119883 rarr 0 120583-ae and 1119860119899(120596)119891(120596)119883 le119891(120596)119883 120583-ae Hence 1119860119899119891 rarr 0 in Tand

120593 because Tand120593 is a

Lebesgue topology and by (i) we get

10038171003817100381710038171003817119898119891 (119860119899)10038171003817100381710038171003817119884 =100381710038171003817100381710038171003817100381710038171003817int119860119899 119891119889119898

100381710038171003817100381710038171003817100381710038171003817119884 =10038171003817100381710038171003817119879 (1119860119899119891)10038171003817100381710038171003817119884 997888rarr 0 (46)

(iv) Note that 120593lowast(119860) le 119879119860120593 To show that 119879119860120593 le120593lowast(119860) assume that 119891 isin 119861119871120593(119883) Choose a sequence (119904119899)in S(Σ119891(120583) 119883) such that 119904119899(120596) minus 119891(120596)119883 rarr 0 120583-ae and119904119899(120596)119883 le 119891(120596)119883 120583-ae for all 119899 isin N Since Tand

120593 isa Lebesgue topology we have 119904119899 rarr 119891 in Tand

120593 and hence119879119860(119904119899) minus 119879119860(119891)119884 rarr 0 Note that 119879119860(119904119899) = int119860 119904119899119889119898

Let 120576 gt 0 be given Choose 1198990 isin N such that 119879119860(119891) minusint1198601199041198990119889119898119884 le 120576 Then

1003817100381710038171003817119879119860 (119891)1003817100381710038171003817119884 le 1003817100381710038171003817100381710038171003817119879119860 (119891) minus int119860 11990411989901198891198981003817100381710038171003817100381710038171003817119884 +

1003817100381710038171003817100381710038171003817int119860 11990411989901198891198981003817100381710038171003817100381710038171003817119884

le 120576 + 120593lowast (119860) (47)

It follows that 119879119860120593 le 120593lowast(119860) so 120593lowast(119860) = 119879119860120593 Hencefor 119910lowast isin 119884lowast we easily get

1003817100381710038171003817(119910lowast ∘ 119879)1198601003817100381710038171003817lowast120593 = 1003817100381710038171003817119910lowast ∘ 1198791198601003817100381710038171003817lowast120593 = (119898119910lowast)120593lowast (119860) (48)

(v) Using (iv) we have

120593lowast (119860) = 10038171003817100381710038171198791198601003817100381710038171003817120593= sup 1003817100381710038171003817119879119860 (119891)1003817100381710038171003817119884 119891 isin 119871120593 (119883) 10038171003817100381710038171198911003817100381710038171003817120593 le 1= sup119910lowastisin119861119884lowast

1003816100381610038161003816(119910lowast ∘ 119879119860) (119891)1003816100381610038161003816 119891 isin 119871120593 (119883) 10038171003817100381710038171198911003817100381710038171003817120593 le 1= sup119910lowastisin119861119884lowast

1003817100381710038171003817119910lowast ∘ 1198791198601003817100381710038171003817lowast120593 = sup119910lowastisin119861119884lowast

(119898119910lowast)120593lowast (119860) (49)

(vi) This follows from (ii) and (iv)

For a sequence (119860119899) in Σ we will write 1198601198991205830 if 119860119899 darrand 120583(119860119899 cap 119860) rarr 0 for every 119860 isin Σ119891(120583)Definition 10 A measure 119898 Σ119891(120583) rarr L(119883 119884) with 119898 ≪120583 and 120593lowast(Ω) lt infin is said to be 120593lowast-semivariationally 120583-continuous if 120593lowast(119860119899) rarr 0 whenever 1198601198991205830 (119860119899) sub Σ

Using a standard argument we can show the following

Proposition 11 Let119898 Σ rarrL(119883 119884) be an additive measuresuch that 119898 ≪ 120583 and 120593(Ω) lt infin Then the followingstatements are equivalent

(i) 119898 is 120593lowast-semivariationally 120583-continuous(ii) The following two conditions hold simultaneously

(a) For every 120576 gt 0 there exists 120575 gt 0 such that120593lowast(119860) le 120576 whenever 120583(119860) le 120575 119860 isin Σ(b) For every 120576 gt 0 there exists 1198600 isin Σ119891(120583) such that120593lowast(Ω 1198600) le 120576

The following theorem characterizes 120593lowast-semivariation-ally 120583-continuous representing measures

Theorem 12 Let 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be its representingmeasure Then the following statements are equivalent

(i) 119898 is 120593lowast-semivariationally 120583-continuous(ii) 119879 is (120574120593 sdot 119884)-continuous(iii) 119879(119891119899)119884 rarr 0 if 119891119899 rarr 0 inT0 and sup119899119891119899120593 lt infin(iv) 119879119860119899120593 rarr 0 if 1198601198991205830 (119860119899) sub Σ

Proof (i)hArr (ii)hArr (iii) See [5 Corollary 28 and Proposition11]

(i)hArr (iv) This follows fromTheorem 9

Now assume that Ω is a completely regular Hausdorffspace LetB119886 denote the 120590-algebra of Baire sets in Ω whichis the 120590-algebra generated by the class Z of all zero sets ofbounded continuous positive functions on120596 ByPwedenotethe family of all cozero (=positive) in Ω (see [25 p 108])

Let 120583 B119886 rarr [0infin) be a countably additive measureThen 120583 is zero-set regular that is for every119860 isinB119886 and 120576 gt 0there exists 119885 isin Z with 119885 sub 119860 such that 120583(119860 119885) le 120576 (see[25 p 118]) It follows that for every 119860 isin B119886 and 120576 gt 0 thereexist 119880 isin P 119880 sup 119860 such that 120583(119880 119860) le 120576

We can assume that 120583 to be complete (if necessary wecan take the completion (ΩB119886 120583) of the measure space(ΩB119886 120583))Proposition 13 Assume that Ω is a completely regular Haus-dorff space and (ΩB119886 120583) is a complete finite measure spaceLet119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operatorand119898 B119886 rarrL(119883 119884) be its representing measure Then thefollowing statements are equivalent

(i) 119898 is 120593lowast-semivariationally 120583-continuous

Journal of Function Spaces 7

(ii) For every sequence (119860119899) in B119886 such that 119860119899 darr and120583(119860119899) rarr 0 there exists a sequence (119880119899) inPwith119860119899 sub119880119899 darr such that 120593lowast(119880119899) rarr 0(iii) For every sequence (119860119899) in B119886 such that 119860119899 darr and120583(119860119899) rarr 0 there exists a sequence (119880119899) inPwith119860119899 sub119880119899 darr such thatsup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) supp119891 sub 119880119899 997888rarr 0 (50)

Proof (i)rArr (ii) Assume that (i) holds and (119860119899) is a sequencein B119886 such that 119860119899 darr and 120583(119860119899) rarr 0 Then there exists asequence (119880119899) inP such that119860119899 sub 119880119899 darr and120583(119880119899119860119899) le 1119899for 119899 isin N

Let 120576 gt 0 be given Then in view of Proposition 11 thereexists 120575 gt 0 such that 120593lowast(119860) le 1205762 if 120583(119860) le 120575with119860 isinB119886Choose 1198991 isin N such that 120583(119880119899 119860119899) le 120575 for 119899 ge 1198992 Then120593lowast(119880119899 119860119899) le 1205762 for 119899 ge 1198991 Since 120593lowast(119860119899) rarr 0 we canchoose 1198992 isin N such that 120593lowast(119860119899) le 1205762 for 119899 ge 1198992 Then for119899 ge 1198990 = max(1198991 1198992) we get

120593lowast (119880119899) le 120593lowast (119880119899 119860119899) + 120593lowast (119860119899) le 120576 (51)

that is (ii) holds(ii)rArr (iii) Assume that (ii) holds and (119860119899) is a sequence

in B119900 such that 119860119899 darr and 120583(119860119899) rarr 0 Then there exists asequence (119880119899) inP with 119860119899 sub 119880119899 darr such that 120593lowast(119880119899) rarr 0Note that for 119891 isin 119861119871120593(119883) with supp119891 sub 119880119899 for 119899 isin N byTheorem 9 we have

1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 = 1003817100381710038171003817100381710038171003817intΩ 1198911198891198981003817100381710038171003817100381710038171003817119884 =

100381710038171003817100381710038171003817100381710038171003817int119880119899 119891119889119898100381710038171003817100381710038171003817100381710038171003817119884 le 120593lowast (119880119899) (52)

It follows that (iii) holds(iii)rArr (i) Assume that (iii) holds and119860119899 darrwith 120583(119860119899) rarr0Then there exists a sequence (119880119899) inPwith119860119899 sub 119880119899 darr such

that

sup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) supp119891 sub 119880119899 997888rarr 0 (53)

Assume on the contrary that (i) fails to hold Then withoutloss of generality we can assume that

120593lowast (119860119899) gt 1205760 for some 1205760 gt 0 all 119899 isin N (54)

Choose 1198990 isin N such that

sup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) supp119891 sub 1198801198990 lt 12057602 (55)

In view of (54) there exists a pairwise disjoint set 1198611 119861119896in B119886 119909119894 isin 119883 for 119894 = 1 119896 and 119910lowast isin 119861119884lowast such thatsum119896119894=1(1119861119894 otimes 119909119894)120593 le 1 and

1003816100381610038161003816100381610038161003816100381610038161003816119910lowast( 119896sum

119894=1

119898(1198601198990 cap 119861119894) (119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 ge 1205760 (56)

Let 1199040 = sum119896119894=1(11198601198990cap119861119894 otimes 119909119894) Then 1199040120593 le 1 and supp 1199040 sub1198601198990 sub 1198801198990 Then by (55) we get 119879(1199040)119884 lt 12057602On the other hand in view of (56) we have 119879(1199040)119884 ge 1205760

This contradiction establishes that (i) holds

Corollary 14 Assume that Ω is a completely regular Haus-dorff space and (ΩB119886 120583) is complete finitemeasure space Let119879 119871120593(119883) rarr 119884 be a (120574120593 sdot 119884)-continuous linear operator and119898 B119886 rarr L(119883 119884) be its representing measure Then 120593lowast isregular that is for every 119860 isinB119886 and 120576 gt 0 there exist 119885 isinZand 119880 isin P with 119885 sub 119860 sub 119880 such that 120593lowast(119880 119885) le 120576Proof In view of Theorem 12 119898 is 120593lowast-semivariationally 120583-continuous Let 119860 isin B119886 and 120576 gt 0 be given Then byProposition 11 there exists 120575 gt 0 such that 120593lowast(119861) le 120576whenever 119861 isin B119886 and 120583(119861) le 120575 By the regularity of 120583 onecan choose 119885 isin Z and 119880 isin P with 119885 sub 119860 sub 119880 such that120583(119880 119885) le 120575 Hence 120593lowast(119880 119885) le 120576 as desired4 Compact Operatorson Orlicz-Bochner Spaces

The following theorem presents necessary conditions for a(Tand120593 sdot 119884)-continuous operator 119879 119871120593(119883) rarr 119884 to be

compact

Theorem 15 Assume that a Young function 120593 such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot 119884)-continuous linear operator and 119898 Σ119891(120583) rarr L(119883 119884)be its representing measure If 119879 is compact then 119898 is 120593lowast-semivariationally 120583-continuousProof Assume that 119879 is compact and 119898 fails to be 120593lowast-semivariationally 120583-continuous Then there exist 120576 gt 0 and asequence (119860119899) in Σwith1198601198991205830 such that 119879119860119899 = 119898lowast120593(119860119899) gt120576 for 119899 isin N (seeTheorem9)Hence one can choose a sequence(119910lowast119899 ) in 119861119884lowast such that

10038171003817100381710038171003817119910lowast119899 ∘ 11987911986011989910038171003817100381710038171003817lowast120593 ge 120576 forall119899 isin N (57)

By Schauderrsquos theorem the conjugatemapping119879lowast 119884lowast rarr119871120593(119883)lowast is compact Note that 119879lowast(119910lowast119899 ) = 119910lowast119899 ∘ 119879 isin 119871120593(119883)sim119899 forall 119899 isin N where 119871120593(119883)sim119899 is a closed subspace of the Banachspace (119871120593(119883)lowast sdot lowast120593) (see Theorem 2) Then for every 119899 isin N

there exists 119892119899 isin 119871120593lowast(119883lowast 119883) such that

(119910lowast119899 ∘ 119879) (119891) = intΩ⟨119891 (120596) 119892119899 (120596)⟩ 119889120583

for 119891 isin 119871120593 (119883) 1003817100381710038171003817119910lowast119899 ∘ 1198791003817100381710038171003817lowast120593= sup int

Ω

1003817100381710038171003817119891 (120596)1003817100381710038171003817119883 120599 (119892119899) (120596) 119889120583 119891 isin 119861119871120593(119883)= 1003817100381710038171003817120599 (119892119899)1003817100381710038171003817120593lowast

(58)

Hence we obtain that for each 119899 isin N

10038171003817100381710038171003817119910lowast119899 ∘ 11987911986011989910038171003817100381710038171003817lowast120593 = 100381710038171003817100381710038171119860119899120599 (119892119899)10038171003817100381710038171003817120593lowast = 10038171003817100381710038171003817120599 (1119860119899119892119899)10038171003817100381710038171003817120593lowast (59)

8 Journal of Function Spaces

Since 119879lowast(119861119884lowast) is a relatively sequentially compact subset of((119871120593(119883)sim119899 sdot lowast120593) there exist a subsequence (119892119896119899) of (119892119899) and119892 isin 119871120593lowast(119883lowast 119883) such that10038171003817100381710038171003817119865119892119899 minus 11986511989210038171003817100381710038171003817lowast120593 = 10038171003817100381710038171003817120599 (119892119896119899 minus 119892)10038171003817100381710038171003817120593lowast 997888rarr 0 (60)

Choose 119899120576 isin N such that 120599(119892119896119899 minus 119892)120593lowast le 1205762 for 119899 ge 119899120576Hence for 119899 ge 1198991205761003816100381610038161003816100381610038161003816

100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast minus 100381710038171003817100381710038171003817120599 (1119860119896119899119892119896119899)100381710038171003817100381710038171003817120593lowast1003816100381610038161003816100381610038161003816

le 100381710038171003817100381710038171003817120599 (1119860119896119899 (119892119896119899 minus 119892))100381710038171003817100381710038171003817120593lowast = 1003817100381710038171003817100381710038171119860119896119899120599 (119892119896119899 minus 119892)100381710038171003817100381710038171003817120593lowastle 10038171003817100381710038171003817120599 (119892119896119899 minus 119892)10038171003817100381710038171003817120593lowast le 1205762

(61)

Using (57) and (61) for 119899 ge 119899120576 we get120576 le 100381710038171003817100381710038171003817119910lowast ∘ 119879119860119896119899 100381710038171003817100381710038171003817

lowast

120593= 100381710038171003817100381710038171003817120599 (1119860119896119899119892119896119899)100381710038171003817100381710038171003817120593lowast

le 1205762 +100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast

(62)

and hence 1003817100381710038171003817100381710038171119860119896119899120599 (119892)100381710038171003817100381710038171003817120593lowast = 100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast ge 1205762 (63)

On the other hand since 120593lowast is supposed to satisfy theΔ 2-condition we have that 1119860119896119899120599(119892)120593lowast rarr 0 (see [26 Theorem3 pp 58-59]) This contradiction establishes that 119898 is 120593lowast-semivariationally 120583-continuousCorollary 16 Assume that 120593 is a Young function such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operator Then the following statements areequivalent

(i) 119879 is compact(ii) 119879 is (120574120593 sdot 119884)-compact that is there exists a 120574120593-

neighborhood 119881 of 0 in 119871120593(119883) such that 119879(119881) is arelatively norm compact set in 119884

(iii) There exists a Young function 120595 with 120595 ≪ 120593 such thatintΩ119891119889119898 119891 isin 119871120593(119883) 119891120595 le 1 is a relatively norm

compact set in 119884Proof (i)rArr (ii) Assume that (i) holds Then by Theorems 12and 15 119879 is (120574120593 sdot 119884)-continuous Since the space (119871120593(119883) 120574120593)is quasinormable by Grothendieckrsquos classical result (see [15p 429]) we obtain that 119879 is (120574120593 sdot 119884)-compact

(ii)rArr (i) The implication is obvious(ii)hArr (iii) This follows fromTheorem 3

5 Topology Associated withthe 120593lowast-Semivariation ofa Representing Measure

Assume that 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator Let119898 Σ119891(120583) rarrL(119883 119884) be its representingmeasure Let us put

119901119898 (119910lowast) fl (119898119910lowast)120593lowast (Ω) for 119910lowast isin 119884lowast (64)

Note that119901119898 is a seminormon119884lowast Following [22 27] let 120575119898120593lowaststand for the topology on 119861119884lowast defined by the seminorm 119901119898restricted to 119861119884lowast

The following theorem characterizes (Tand120593 sdot 119884)-

continuous compact operators 119879 119871120593(119883) rarr 119884 in terms ofthe topological properties of the space (119861119884lowast 120575119898120593lowast) (see [22Theorem 3])

Theorem 17 Let 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be its representingmeasure Then the following statements are equivalent

(i) The space (119861119884lowast 120575119898120593lowast) is compact

(ii) 119879 is compact

Proof (i) rArr (ii) Assume that (119861119884lowast 120575119898120593lowast) is compact Let(119910lowast119899 ) be a sequence in 119861119884lowast Without loss of generality we canassume that 119910lowast119899 rarr 119910lowast0 in 120575119898120593lowast for some 119910lowast isin 119861119884lowast Then usingTheorem 9 for 119891 isin 119871120593(119883) we have

1003816100381610038161003816(119879lowast (119910lowast119899 ) minus 119879lowast (119910lowast0 )) (119891)1003816100381610038161003816 = 1003816100381610038161003816(119910lowast119899 minus 119910lowast0 ) (119879 (119891))1003816100381610038161003816= 1003816100381610038161003816100381610038161003816intΩ 119891119889119898119910lowast119899 minus119910lowast0

1003816100381610038161003816100381610038161003816 le (119898119910lowast119899 minus119910lowast0 )120593lowast (Ω) 10038171003817100381710038171198911003817100381710038171003817120593 (65)

It follows that 119879lowast(119910lowast119899 ) minus 119879lowast(119910lowast0 )lowast120593 le (119898119910lowast119899 minus119910lowast0 )120593lowast(Ω) where119901119898(119910lowast119899 minus 119910lowast0 ) = (119898119910lowast119899 minus119910lowast0 )120593lowast(Ω) rarr119899 0 This means that 119879lowast iscompact and hence 119879 is compact

(ii)rArr (i) Assume that 119879 is compact and (119910lowast120572 ) is a net in119861119884lowast Since119861119884lowast is 120590(119884lowast 119884)-compact without loss of generalitywe can assume that 119910lowast120572 rarr120572 119910lowast0 in 120590(119884lowast 119884) for some 119910lowast0 isin 119861119884lowast In view of the compactness of the conjugate operator 119879lowast 119884lowast rarr 119871120593(119883)lowast there exists a subset (119910lowast120573 ) of (119910lowast120572 ) and Φ0 isin119871120593(119883)lowast such that 119879lowast(119910lowast120573 ) minus Φ0lowast120593 rarr

1205730 On the other hand

since 119879lowast is (120590(119884lowast 119884) 120590(119871120593(119883)lowast 119871120593(119883)))-continuous we get119879lowast(119910lowast120573 ) rarr120573119879lowast(119910lowast0 ) in 120590(119871120593(119883)lowast 119871120593(119883)) HenceΦ0 = 119879lowast(119910lowast0 )

that is 119879lowast(119910lowast120573 ) minus 119879lowast(119910lowast0 )lowast120593 rarr1205730

Let 120576 gt 0 be givenThen there exist a pairwise disjoint set1198601 119860119899 in Σ119891(120583) and 119909119894 isin 119883 for 119894 = 1 119899 such thatsum119899119894=1(1119860119894 otimes 119909119894)120593 le 1 and

(119898119910lowast120573minus119910lowast0)120593lowast(Ω) le 1003816100381610038161003816100381610038161003816100381610038161003816

119899sum119894=1

(119910lowast120573 minus 119910lowast0 ) (119898 (119860 119894) (119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 + 120576 (66)

Hence

(119898119910lowast120573minus119910lowast0)120593lowast(Ω) le 1003816100381610038161003816100381610038161003816100381610038161003816

119899sum119894=1

(119910lowast120573 minus 119910lowast0 ) (119879 (1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 + 120576

le 1003816100381610038161003816100381610038161003816100381610038161003816(119910lowast120573 minus 119910lowast0 ) 119879(

119899sum119894=1

(1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816

+ 120576

Journal of Function Spaces 9

= 1003816100381610038161003816100381610038161003816100381610038161003816119879lowast (119910lowast120573 minus 119910lowast0 )(

119899sum119894=1

(1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816

+ 120576le 10038171003817100381710038171003817119879lowast (119910lowast120573 minus 119910lowast0 )10038171003817100381710038171003817lowast120593

1003817100381710038171003817100381710038171003817100381710038171003817119899sum119894=1

(1119860119894 otimes 119909119894)1003817100381710038171003817100381710038171003817100381710038171003817120593

+ 120576 le 10038171003817100381710038171003817119879lowast (119910lowast120573) minus 119879 (119910lowast0 )10038171003817100381710038171003817lowast120593 + 120576(67)

Hence 119901119898(119910lowast120573 minus 119910lowast0 ) = (119898119910lowast120573minus119910lowast0)120593lowast(Ω) rarr

1205730 and this means

that the space (119861119884lowast 120575119898120593lowast) is compact

As a consequence of Theorems 17 and 15 we have thefollowing

Corollary 18 Assume that 120593 is a Young function such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be itsrepresenting measure If the space (119861119884lowast 120575119898120593lowast) is compact then119898 is 120593lowast-semivariationally 120583-continuousConflicts of Interest

The author declares that there are no conflicts of interest

References

[1] S Chen and H Hudzik ldquoOn some convexities of Orlicz andOrlicz-Bochner spacesrdquo Commentationes Mathematicae vol29 no 1 pp 13ndash29 1988

[2] P Kolwicz and R Pluciennik ldquo119875-convexity of Orlicz-Bochnerspacesrdquo Proceedings of the American Mathematical Society vol126 no 8 pp 2315ndash2322 1998

[3] S Shang and Y Cui ldquoUniform nonsquareness and locallyuniform nonsquareness in Orlicz-Bochner function spaces andapplicationsrdquo Journal of Functional Analysis vol 267 no 7 pp2056ndash2076 2014

[4] K Feledziak andM Nowak ldquoLocally solid topologies on vectorvalued function spacesrdquo Universitat de Barcelona CollectaneaMathematica vol 48 no 4-6 pp 487ndash511 1997

[5] K Feledziak and M Nowak ldquoIntegral representation of linearoperators on Orlicz-Bochner spacesrdquo Universitat de BarcelonaCollectanea Mathematica vol 61 no 3 pp 277ndash290 2010

[6] M Nowak ldquoLinear operators on vector-valued function spaceswithMackey topologiesrdquo Journal of Convex Analysis vol 15 no1 pp 165ndash178 2008

[7] M Nowak ldquoLebesgue topologies on vector-valued functionspacesrdquoMathematica Japonica vol 52 no 2 pp 171ndash182 2000

[8] A V Buhvalov ldquoThe analytic representation of operators withan abstract normrdquo Izvestiya Vysshikh Uchebnykh ZavedeniiMatematika vol 11 no 162 pp 21ndash32 1975

[9] A V Buhvalov ldquoThe analytic representation of linear operatorsby means of measurable vector-valued functionsrdquo IzvestiyaVysshikh Uchebnykh Zavedenii Matematika vol 7 no 182 pp21ndash31 1977

[10] J Diestel and J J Uhl ldquoVector measuresrdquo vol 15 AmericanMathematical Society Surveys Providence RI USA 1977

[11] M Nowak ldquoDuality theory of vector-valued function spaces IrdquoCommentationes Mathematicae vol 37 pp 195ndash215 1997

[12] A V Bukhvalov and G Ya Lozanovskii ldquoOn sets closed inmeasure in spaces of mesurable functionsrdquo Transactions of theMoscow Mathematical Society vol 2 pp 127ndash148 1978

[13] A Wiweger ldquoLinear spaces with mixed topologyrdquo StudiaMathematica vol 20 pp 47ndash68 1961

[14] K Feledziak ldquoUniformly Lebesgue topologies on Kothe-Bochner spacesrdquoCommentationesMathematicae vol 37 pp 81ndash98 1997

[15] W Ruess ldquo[Weakly] compact operators and DF-spacesrdquo PacificJournal of Mathematics vol 98 no 2 pp 419ndash441 1982

[16] M Nowak ldquoCompactness of Bochner representable operatorson Orlicz spacesrdquo Positivity An International MathematicsJournal Devoted to Theory and Applications of Positivity vol 13no 1 pp 193ndash199 2009

[17] N E Gretsky and J Uhl ldquoBounded linear operators on Banachfunction spaces of vector-valued functionsrdquo Transactions of theAmerican Mathematical Society vol 167 pp 263ndash277 1972

[18] J J Uhl ldquoCompact operators on Orlicz spacesrdquo in Rendicontidel Seminario Matematico vol 42 pp 209ndash219 University ofPadova 1969

[19] N Dinculeanu Vector Measures International Series of Mono-graphs in Pure and Applied Mathematics 95 Pergamon PressNew York NY USA 1967

[20] N Dinculeanu Linear Operators on Lp-Spaces Vector andOperator-Valued Measures and Applications Academic PressNew York NY USA 1973

[21] N Dinculeanu Vector Integration and Stochastic Integration inBanach Spaces John Wiley amp Sons 2000

[22] R A Alo A De Korwin and L E Kunes ldquoTopological aspectsof 119902-regular measuresrdquo Studia Mathematica vol 48 pp 49ndash601973

[23] J Diestel ldquoOn the representation of bounded linear operatorsfrom Orlicz spaces of Lebesgue-Bochner measurable functionsto any Banach spacerdquo Bulletin de lrsquoAcademie Polonaise desSciences Serie des Sciences Mathematiques Astronomiques etPhysiques vol 18 pp 375ndash378 1970

[24] L Vanderputten ldquoRepresentation of operators defined on thespace of Bochner integrable functionsrdquo Extracta Mathematicaevol 16 no 3 pp 383ndash391 2001

[25] R FWheeler ldquoA survey of Bairemeasures and strict topologiesrdquoExpositiones Mathematicae vol 1 no 2 pp 97ndash190 1983

[26] W A Luxemburg Banach function spaces [Thesis] DelftNetherlands 1955

[27] P W Lewis ldquoVector measures and topologyrdquo Revue RoumainedeMathematique Pures et Appliquees vol 16 pp 1201ndash1209 1971

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Journal of Function Spaces 5

Proof (i) Let 119860 isin Σ119891(120583) Then for 119909 isin 119861119883 we have 1119860 otimes119909120593 le 1119860120593 and hence

119898 (119860) (119909)119884 = 1003817100381710038171003817119879 (1119860 otimes 119909)1003817100381710038171003817119884 le 119879120593 sdot 10038171003817100381710038171119860 otimes 1199091003817100381710038171003817120593le 119879120593 100381710038171003817100381711198601003817100381710038171003817120593 (30)

so 119898(119860) le 119879120593 sdot 1119860120593(ii) This follows from (i) because 1119860120593 = 0 if 120583(119860) = 0(iii) Assume that 119860119899 darr 0 with 119860119899 isin Σ119891(120583) Then

11198601(120596) ge 1119860119899(120596) darr 0 for 120596 isin Ω By the Lebesgue dominatedconvergence theorem we obtain that int

Ω120593(1205821119860119899(120596))119889120583 rarr 0

for every 120582 gt 0 This means that 1119860119899120593 rarr 0 and by (i)119898(119860119899) rarr 0(iv) Assume that (119861119899) is a pairwise disjoint sequence inΣ119891(120583) with 119861 = ⋃infin119899=1 119861119899 isin Σ119891(120583) Let 119860119899 = 119861 ⋃119899119894=1 119861119894 for119899 isin N Then 119860119899 isin Σ119891(120583) and 119860119899 darr 0 Hence by (iii) 119898(119861) minussum119899119894=1119898(119861119894) = 119898(119861) minus 119898(⋃119899119894=1 119861119894) = 119898(119860119899) rarr 0Statement (v) is obvious

Definition 7 Let119879 119871120593(119883) rarr 119884 be a bounded linear operatorand

119898(119860) (119909) fl 119879 (1119860 otimes 119909) for 119860 isin Σ119891 (120583) 119909 isin 119883 (31)

Then the measure 119898 Σ119891(120583) rarr L(119883 119884) will be called arepresenting measure of 119879Proposition 8 Let 119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot 119884)-continuous linear operator and 119898 Σ119891(120583) rarr L(119883 119884) beits representing measure Then there exists a Young function 120595such that 120595 ⊲ 120593 and 120595lowast(Ω) lt infin

Proof According toTheorem 2 there exist a finite set 120595119894 119894 =1 119899 of Young functions with 120595119894 ⊲ 120593 for 119894 = 1 119899 and119886 gt 0 such that1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 le 119886max1le119894le119899

10038171003817100381710038171198911003817100381710038171003817120595119894 forall119891 isin 119871120593 (119883) (32)

Let120595(119905) = max1le119894le119899120595119894(119905) for 119905 ge 0Then120595 is a Young functionwith 120595 ⊲ 120593 and

1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 le 119886 10038171003817100381710038171198911003817100381710038171003817120595 forall119891 isin 119871120593 (119883) (33)

Hence120595lowast (Ω)= sup 119879 (119904)119884 119904 isin S (Σ119891 (120583) 119883) 119904120595 le 1le 119886 lt infin

(34)

For a linear operator 119879 119871120593(119883) rarr 119884 and 119860 isin Σ let119879119860 (119891) fl 119879 (1119860119891) for 119891 isin 119871120593 (119883) (35)

Nowwe can state ourmain result that extends the classicalresults concerning the integral representation of operators onLebesgue-Bochner spaces 119871119901(119883) (1 le 119901 lt infin) (see [19 sect13 Theorem 1 pp 259ndash261]) to operators on Orlicz-Bochnerspaces 119871120593(119883)

Theorem 9 Let 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be its representingmeasure Then for 119860 isin Σ the following statements hold

(i) 119879119860 119871120593(119883) rarr 119884 is a (Tand120593 sdot 119884)-continuous linear

operator(ii) For 119891 isin 119871120593(119883) one has

119879119860 (119891) = int119860119891119889119898 (36)

and for 119910lowast isin 119884lowast one has119910lowast (119879119860 (119891)) = int

119860119891119889119898119910lowast (37)

(iii) For 119891 isin 119871120593(119883) the measure 119898119891 Σ rarr 119884 defined bythe equality

119898119891 (119860) fl int119860119891119889119898 for 119860 isin Σ (38)

is countably additive(iv) 119879119860120593 = 120593lowast(119860)

and for 119910lowast isin 119884lowast 119910lowast ∘ 119879119860lowast120593 = (119910lowast ∘ 119879)119860lowast120593 =(119898119910lowast)120593lowast(119860)(v) 120593lowast(119860) = sup(119898119910lowast)120593lowast(119860) 119910lowast isin 119861119884lowast(vi) For 119891 isin 119871120593(119883) one has

1003817100381710038171003817100381710038171003817int119860 1198911198891198981003817100381710038171003817100381710038171003817119884 le 120593lowast (119860) 10038171003817100381710038171198911003817100381710038171003817120593 (39)

and for 119910lowast isin 119884lowast one has1003816100381610038161003816100381610038161003816int119860 119891119889119898119910lowast

1003816100381610038161003816100381610038161003816 le (119898119910lowast)120593lowast (119860) 10038171003817100381710038171198911003817100381710038171003817120593 (40)

Proof (i) Assume that (119891120572) is a net in 119871120593(119883) such that119891120572 rarr 0inTand

120593 SinceTand120593 is a locally solid topology on 119871120593(119883) we get

1119860119891120572 rarr 0 inTand120593 Hence

1003817100381710038171003817119879119860 (119891120572)1003817100381710038171003817119884 = 1003817100381710038171003817119879 (1119860119891120572)1003817100381710038171003817119884 997888rarr 0 (41)

(ii) In view of Proposition 8 there exists a Young function120595 such that 120595 ⊲ 120593 and 120595lowast(Ω) lt infin Then 119871120593(119883) sub119864120595(119883) Let 119891 isin 119871120593(119883) Then there exists a sequence (119904119899)in S(Σ119891(120583) 119883) such that 119904119899(120596) minus 119891(120596)119883 rarr 0 120583-ae and119904119899(120596)119883 le 119891(120596)119883 120583-ae for all 119899 isin N (see [21 Theorem 6p 4])Then 119904119899 rarr 119891 inTand

120593 becauseTand120593 is a Lebesgue topology

Hence 119904119899 minus 119891120595 rarr 0 In view of Lemma 4 we can define theintegral int

119860119891119889119898 by the equality

int119860119891119889119898 fl limint

119860119904119899119889119898 (42)

6 Journal of Function Spaces

Since 119879119860(119904119899) = int119860119904119899119889119898 and by (i) 119879119860 is (Tand

120593 sdot 119884)-continuous we get

119879119860 (119891) = limint119860119904119899119889119898 (43)

Hence

119879119860 (119891) = int119860119891119889119898 (44)

and for 119910lowast isin 119884lowast we have119910lowast (119879119860 (119891)) = lim119910lowast (int

119860119904119899119889119898) = limint

119860119904119899119889119898119910lowast

= int119860119891119889119898119910lowast

(45)

(iii) Let 119891 isin 119871120593(119883) and (119860119899) be a sequence in Σ suchthat 119860119899 darr 0 Then 1119860119899(120596) darr 0 for 120596 isin Ω andhence 1119860119899(120596)119891(120596)119883 rarr 0 120583-ae and 1119860119899(120596)119891(120596)119883 le119891(120596)119883 120583-ae Hence 1119860119899119891 rarr 0 in Tand

120593 because Tand120593 is a

Lebesgue topology and by (i) we get

10038171003817100381710038171003817119898119891 (119860119899)10038171003817100381710038171003817119884 =100381710038171003817100381710038171003817100381710038171003817int119860119899 119891119889119898

100381710038171003817100381710038171003817100381710038171003817119884 =10038171003817100381710038171003817119879 (1119860119899119891)10038171003817100381710038171003817119884 997888rarr 0 (46)

(iv) Note that 120593lowast(119860) le 119879119860120593 To show that 119879119860120593 le120593lowast(119860) assume that 119891 isin 119861119871120593(119883) Choose a sequence (119904119899)in S(Σ119891(120583) 119883) such that 119904119899(120596) minus 119891(120596)119883 rarr 0 120583-ae and119904119899(120596)119883 le 119891(120596)119883 120583-ae for all 119899 isin N Since Tand

120593 isa Lebesgue topology we have 119904119899 rarr 119891 in Tand

120593 and hence119879119860(119904119899) minus 119879119860(119891)119884 rarr 0 Note that 119879119860(119904119899) = int119860 119904119899119889119898

Let 120576 gt 0 be given Choose 1198990 isin N such that 119879119860(119891) minusint1198601199041198990119889119898119884 le 120576 Then

1003817100381710038171003817119879119860 (119891)1003817100381710038171003817119884 le 1003817100381710038171003817100381710038171003817119879119860 (119891) minus int119860 11990411989901198891198981003817100381710038171003817100381710038171003817119884 +

1003817100381710038171003817100381710038171003817int119860 11990411989901198891198981003817100381710038171003817100381710038171003817119884

le 120576 + 120593lowast (119860) (47)

It follows that 119879119860120593 le 120593lowast(119860) so 120593lowast(119860) = 119879119860120593 Hencefor 119910lowast isin 119884lowast we easily get

1003817100381710038171003817(119910lowast ∘ 119879)1198601003817100381710038171003817lowast120593 = 1003817100381710038171003817119910lowast ∘ 1198791198601003817100381710038171003817lowast120593 = (119898119910lowast)120593lowast (119860) (48)

(v) Using (iv) we have

120593lowast (119860) = 10038171003817100381710038171198791198601003817100381710038171003817120593= sup 1003817100381710038171003817119879119860 (119891)1003817100381710038171003817119884 119891 isin 119871120593 (119883) 10038171003817100381710038171198911003817100381710038171003817120593 le 1= sup119910lowastisin119861119884lowast

1003816100381610038161003816(119910lowast ∘ 119879119860) (119891)1003816100381610038161003816 119891 isin 119871120593 (119883) 10038171003817100381710038171198911003817100381710038171003817120593 le 1= sup119910lowastisin119861119884lowast

1003817100381710038171003817119910lowast ∘ 1198791198601003817100381710038171003817lowast120593 = sup119910lowastisin119861119884lowast

(119898119910lowast)120593lowast (119860) (49)

(vi) This follows from (ii) and (iv)

For a sequence (119860119899) in Σ we will write 1198601198991205830 if 119860119899 darrand 120583(119860119899 cap 119860) rarr 0 for every 119860 isin Σ119891(120583)Definition 10 A measure 119898 Σ119891(120583) rarr L(119883 119884) with 119898 ≪120583 and 120593lowast(Ω) lt infin is said to be 120593lowast-semivariationally 120583-continuous if 120593lowast(119860119899) rarr 0 whenever 1198601198991205830 (119860119899) sub Σ

Using a standard argument we can show the following

Proposition 11 Let119898 Σ rarrL(119883 119884) be an additive measuresuch that 119898 ≪ 120583 and 120593(Ω) lt infin Then the followingstatements are equivalent

(i) 119898 is 120593lowast-semivariationally 120583-continuous(ii) The following two conditions hold simultaneously

(a) For every 120576 gt 0 there exists 120575 gt 0 such that120593lowast(119860) le 120576 whenever 120583(119860) le 120575 119860 isin Σ(b) For every 120576 gt 0 there exists 1198600 isin Σ119891(120583) such that120593lowast(Ω 1198600) le 120576

The following theorem characterizes 120593lowast-semivariation-ally 120583-continuous representing measures

Theorem 12 Let 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be its representingmeasure Then the following statements are equivalent

(i) 119898 is 120593lowast-semivariationally 120583-continuous(ii) 119879 is (120574120593 sdot 119884)-continuous(iii) 119879(119891119899)119884 rarr 0 if 119891119899 rarr 0 inT0 and sup119899119891119899120593 lt infin(iv) 119879119860119899120593 rarr 0 if 1198601198991205830 (119860119899) sub Σ

Proof (i)hArr (ii)hArr (iii) See [5 Corollary 28 and Proposition11]

(i)hArr (iv) This follows fromTheorem 9

Now assume that Ω is a completely regular Hausdorffspace LetB119886 denote the 120590-algebra of Baire sets in Ω whichis the 120590-algebra generated by the class Z of all zero sets ofbounded continuous positive functions on120596 ByPwedenotethe family of all cozero (=positive) in Ω (see [25 p 108])

Let 120583 B119886 rarr [0infin) be a countably additive measureThen 120583 is zero-set regular that is for every119860 isinB119886 and 120576 gt 0there exists 119885 isin Z with 119885 sub 119860 such that 120583(119860 119885) le 120576 (see[25 p 118]) It follows that for every 119860 isin B119886 and 120576 gt 0 thereexist 119880 isin P 119880 sup 119860 such that 120583(119880 119860) le 120576

We can assume that 120583 to be complete (if necessary wecan take the completion (ΩB119886 120583) of the measure space(ΩB119886 120583))Proposition 13 Assume that Ω is a completely regular Haus-dorff space and (ΩB119886 120583) is a complete finite measure spaceLet119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operatorand119898 B119886 rarrL(119883 119884) be its representing measure Then thefollowing statements are equivalent

(i) 119898 is 120593lowast-semivariationally 120583-continuous

Journal of Function Spaces 7

(ii) For every sequence (119860119899) in B119886 such that 119860119899 darr and120583(119860119899) rarr 0 there exists a sequence (119880119899) inPwith119860119899 sub119880119899 darr such that 120593lowast(119880119899) rarr 0(iii) For every sequence (119860119899) in B119886 such that 119860119899 darr and120583(119860119899) rarr 0 there exists a sequence (119880119899) inPwith119860119899 sub119880119899 darr such thatsup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) supp119891 sub 119880119899 997888rarr 0 (50)

Proof (i)rArr (ii) Assume that (i) holds and (119860119899) is a sequencein B119886 such that 119860119899 darr and 120583(119860119899) rarr 0 Then there exists asequence (119880119899) inP such that119860119899 sub 119880119899 darr and120583(119880119899119860119899) le 1119899for 119899 isin N

Let 120576 gt 0 be given Then in view of Proposition 11 thereexists 120575 gt 0 such that 120593lowast(119860) le 1205762 if 120583(119860) le 120575with119860 isinB119886Choose 1198991 isin N such that 120583(119880119899 119860119899) le 120575 for 119899 ge 1198992 Then120593lowast(119880119899 119860119899) le 1205762 for 119899 ge 1198991 Since 120593lowast(119860119899) rarr 0 we canchoose 1198992 isin N such that 120593lowast(119860119899) le 1205762 for 119899 ge 1198992 Then for119899 ge 1198990 = max(1198991 1198992) we get

120593lowast (119880119899) le 120593lowast (119880119899 119860119899) + 120593lowast (119860119899) le 120576 (51)

that is (ii) holds(ii)rArr (iii) Assume that (ii) holds and (119860119899) is a sequence

in B119900 such that 119860119899 darr and 120583(119860119899) rarr 0 Then there exists asequence (119880119899) inP with 119860119899 sub 119880119899 darr such that 120593lowast(119880119899) rarr 0Note that for 119891 isin 119861119871120593(119883) with supp119891 sub 119880119899 for 119899 isin N byTheorem 9 we have

1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 = 1003817100381710038171003817100381710038171003817intΩ 1198911198891198981003817100381710038171003817100381710038171003817119884 =

100381710038171003817100381710038171003817100381710038171003817int119880119899 119891119889119898100381710038171003817100381710038171003817100381710038171003817119884 le 120593lowast (119880119899) (52)

It follows that (iii) holds(iii)rArr (i) Assume that (iii) holds and119860119899 darrwith 120583(119860119899) rarr0Then there exists a sequence (119880119899) inPwith119860119899 sub 119880119899 darr such

that

sup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) supp119891 sub 119880119899 997888rarr 0 (53)

Assume on the contrary that (i) fails to hold Then withoutloss of generality we can assume that

120593lowast (119860119899) gt 1205760 for some 1205760 gt 0 all 119899 isin N (54)

Choose 1198990 isin N such that

sup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) supp119891 sub 1198801198990 lt 12057602 (55)

In view of (54) there exists a pairwise disjoint set 1198611 119861119896in B119886 119909119894 isin 119883 for 119894 = 1 119896 and 119910lowast isin 119861119884lowast such thatsum119896119894=1(1119861119894 otimes 119909119894)120593 le 1 and

1003816100381610038161003816100381610038161003816100381610038161003816119910lowast( 119896sum

119894=1

119898(1198601198990 cap 119861119894) (119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 ge 1205760 (56)

Let 1199040 = sum119896119894=1(11198601198990cap119861119894 otimes 119909119894) Then 1199040120593 le 1 and supp 1199040 sub1198601198990 sub 1198801198990 Then by (55) we get 119879(1199040)119884 lt 12057602On the other hand in view of (56) we have 119879(1199040)119884 ge 1205760

This contradiction establishes that (i) holds

Corollary 14 Assume that Ω is a completely regular Haus-dorff space and (ΩB119886 120583) is complete finitemeasure space Let119879 119871120593(119883) rarr 119884 be a (120574120593 sdot 119884)-continuous linear operator and119898 B119886 rarr L(119883 119884) be its representing measure Then 120593lowast isregular that is for every 119860 isinB119886 and 120576 gt 0 there exist 119885 isinZand 119880 isin P with 119885 sub 119860 sub 119880 such that 120593lowast(119880 119885) le 120576Proof In view of Theorem 12 119898 is 120593lowast-semivariationally 120583-continuous Let 119860 isin B119886 and 120576 gt 0 be given Then byProposition 11 there exists 120575 gt 0 such that 120593lowast(119861) le 120576whenever 119861 isin B119886 and 120583(119861) le 120575 By the regularity of 120583 onecan choose 119885 isin Z and 119880 isin P with 119885 sub 119860 sub 119880 such that120583(119880 119885) le 120575 Hence 120593lowast(119880 119885) le 120576 as desired4 Compact Operatorson Orlicz-Bochner Spaces

The following theorem presents necessary conditions for a(Tand120593 sdot 119884)-continuous operator 119879 119871120593(119883) rarr 119884 to be

compact

Theorem 15 Assume that a Young function 120593 such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot 119884)-continuous linear operator and 119898 Σ119891(120583) rarr L(119883 119884)be its representing measure If 119879 is compact then 119898 is 120593lowast-semivariationally 120583-continuousProof Assume that 119879 is compact and 119898 fails to be 120593lowast-semivariationally 120583-continuous Then there exist 120576 gt 0 and asequence (119860119899) in Σwith1198601198991205830 such that 119879119860119899 = 119898lowast120593(119860119899) gt120576 for 119899 isin N (seeTheorem9)Hence one can choose a sequence(119910lowast119899 ) in 119861119884lowast such that

10038171003817100381710038171003817119910lowast119899 ∘ 11987911986011989910038171003817100381710038171003817lowast120593 ge 120576 forall119899 isin N (57)

By Schauderrsquos theorem the conjugatemapping119879lowast 119884lowast rarr119871120593(119883)lowast is compact Note that 119879lowast(119910lowast119899 ) = 119910lowast119899 ∘ 119879 isin 119871120593(119883)sim119899 forall 119899 isin N where 119871120593(119883)sim119899 is a closed subspace of the Banachspace (119871120593(119883)lowast sdot lowast120593) (see Theorem 2) Then for every 119899 isin N

there exists 119892119899 isin 119871120593lowast(119883lowast 119883) such that

(119910lowast119899 ∘ 119879) (119891) = intΩ⟨119891 (120596) 119892119899 (120596)⟩ 119889120583

for 119891 isin 119871120593 (119883) 1003817100381710038171003817119910lowast119899 ∘ 1198791003817100381710038171003817lowast120593= sup int

Ω

1003817100381710038171003817119891 (120596)1003817100381710038171003817119883 120599 (119892119899) (120596) 119889120583 119891 isin 119861119871120593(119883)= 1003817100381710038171003817120599 (119892119899)1003817100381710038171003817120593lowast

(58)

Hence we obtain that for each 119899 isin N

10038171003817100381710038171003817119910lowast119899 ∘ 11987911986011989910038171003817100381710038171003817lowast120593 = 100381710038171003817100381710038171119860119899120599 (119892119899)10038171003817100381710038171003817120593lowast = 10038171003817100381710038171003817120599 (1119860119899119892119899)10038171003817100381710038171003817120593lowast (59)

8 Journal of Function Spaces

Since 119879lowast(119861119884lowast) is a relatively sequentially compact subset of((119871120593(119883)sim119899 sdot lowast120593) there exist a subsequence (119892119896119899) of (119892119899) and119892 isin 119871120593lowast(119883lowast 119883) such that10038171003817100381710038171003817119865119892119899 minus 11986511989210038171003817100381710038171003817lowast120593 = 10038171003817100381710038171003817120599 (119892119896119899 minus 119892)10038171003817100381710038171003817120593lowast 997888rarr 0 (60)

Choose 119899120576 isin N such that 120599(119892119896119899 minus 119892)120593lowast le 1205762 for 119899 ge 119899120576Hence for 119899 ge 1198991205761003816100381610038161003816100381610038161003816

100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast minus 100381710038171003817100381710038171003817120599 (1119860119896119899119892119896119899)100381710038171003817100381710038171003817120593lowast1003816100381610038161003816100381610038161003816

le 100381710038171003817100381710038171003817120599 (1119860119896119899 (119892119896119899 minus 119892))100381710038171003817100381710038171003817120593lowast = 1003817100381710038171003817100381710038171119860119896119899120599 (119892119896119899 minus 119892)100381710038171003817100381710038171003817120593lowastle 10038171003817100381710038171003817120599 (119892119896119899 minus 119892)10038171003817100381710038171003817120593lowast le 1205762

(61)

Using (57) and (61) for 119899 ge 119899120576 we get120576 le 100381710038171003817100381710038171003817119910lowast ∘ 119879119860119896119899 100381710038171003817100381710038171003817

lowast

120593= 100381710038171003817100381710038171003817120599 (1119860119896119899119892119896119899)100381710038171003817100381710038171003817120593lowast

le 1205762 +100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast

(62)

and hence 1003817100381710038171003817100381710038171119860119896119899120599 (119892)100381710038171003817100381710038171003817120593lowast = 100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast ge 1205762 (63)

On the other hand since 120593lowast is supposed to satisfy theΔ 2-condition we have that 1119860119896119899120599(119892)120593lowast rarr 0 (see [26 Theorem3 pp 58-59]) This contradiction establishes that 119898 is 120593lowast-semivariationally 120583-continuousCorollary 16 Assume that 120593 is a Young function such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operator Then the following statements areequivalent

(i) 119879 is compact(ii) 119879 is (120574120593 sdot 119884)-compact that is there exists a 120574120593-

neighborhood 119881 of 0 in 119871120593(119883) such that 119879(119881) is arelatively norm compact set in 119884

(iii) There exists a Young function 120595 with 120595 ≪ 120593 such thatintΩ119891119889119898 119891 isin 119871120593(119883) 119891120595 le 1 is a relatively norm

compact set in 119884Proof (i)rArr (ii) Assume that (i) holds Then by Theorems 12and 15 119879 is (120574120593 sdot 119884)-continuous Since the space (119871120593(119883) 120574120593)is quasinormable by Grothendieckrsquos classical result (see [15p 429]) we obtain that 119879 is (120574120593 sdot 119884)-compact

(ii)rArr (i) The implication is obvious(ii)hArr (iii) This follows fromTheorem 3

5 Topology Associated withthe 120593lowast-Semivariation ofa Representing Measure

Assume that 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator Let119898 Σ119891(120583) rarrL(119883 119884) be its representingmeasure Let us put

119901119898 (119910lowast) fl (119898119910lowast)120593lowast (Ω) for 119910lowast isin 119884lowast (64)

Note that119901119898 is a seminormon119884lowast Following [22 27] let 120575119898120593lowaststand for the topology on 119861119884lowast defined by the seminorm 119901119898restricted to 119861119884lowast

The following theorem characterizes (Tand120593 sdot 119884)-

continuous compact operators 119879 119871120593(119883) rarr 119884 in terms ofthe topological properties of the space (119861119884lowast 120575119898120593lowast) (see [22Theorem 3])

Theorem 17 Let 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be its representingmeasure Then the following statements are equivalent

(i) The space (119861119884lowast 120575119898120593lowast) is compact

(ii) 119879 is compact

Proof (i) rArr (ii) Assume that (119861119884lowast 120575119898120593lowast) is compact Let(119910lowast119899 ) be a sequence in 119861119884lowast Without loss of generality we canassume that 119910lowast119899 rarr 119910lowast0 in 120575119898120593lowast for some 119910lowast isin 119861119884lowast Then usingTheorem 9 for 119891 isin 119871120593(119883) we have

1003816100381610038161003816(119879lowast (119910lowast119899 ) minus 119879lowast (119910lowast0 )) (119891)1003816100381610038161003816 = 1003816100381610038161003816(119910lowast119899 minus 119910lowast0 ) (119879 (119891))1003816100381610038161003816= 1003816100381610038161003816100381610038161003816intΩ 119891119889119898119910lowast119899 minus119910lowast0

1003816100381610038161003816100381610038161003816 le (119898119910lowast119899 minus119910lowast0 )120593lowast (Ω) 10038171003817100381710038171198911003817100381710038171003817120593 (65)

It follows that 119879lowast(119910lowast119899 ) minus 119879lowast(119910lowast0 )lowast120593 le (119898119910lowast119899 minus119910lowast0 )120593lowast(Ω) where119901119898(119910lowast119899 minus 119910lowast0 ) = (119898119910lowast119899 minus119910lowast0 )120593lowast(Ω) rarr119899 0 This means that 119879lowast iscompact and hence 119879 is compact

(ii)rArr (i) Assume that 119879 is compact and (119910lowast120572 ) is a net in119861119884lowast Since119861119884lowast is 120590(119884lowast 119884)-compact without loss of generalitywe can assume that 119910lowast120572 rarr120572 119910lowast0 in 120590(119884lowast 119884) for some 119910lowast0 isin 119861119884lowast In view of the compactness of the conjugate operator 119879lowast 119884lowast rarr 119871120593(119883)lowast there exists a subset (119910lowast120573 ) of (119910lowast120572 ) and Φ0 isin119871120593(119883)lowast such that 119879lowast(119910lowast120573 ) minus Φ0lowast120593 rarr

1205730 On the other hand

since 119879lowast is (120590(119884lowast 119884) 120590(119871120593(119883)lowast 119871120593(119883)))-continuous we get119879lowast(119910lowast120573 ) rarr120573119879lowast(119910lowast0 ) in 120590(119871120593(119883)lowast 119871120593(119883)) HenceΦ0 = 119879lowast(119910lowast0 )

that is 119879lowast(119910lowast120573 ) minus 119879lowast(119910lowast0 )lowast120593 rarr1205730

Let 120576 gt 0 be givenThen there exist a pairwise disjoint set1198601 119860119899 in Σ119891(120583) and 119909119894 isin 119883 for 119894 = 1 119899 such thatsum119899119894=1(1119860119894 otimes 119909119894)120593 le 1 and

(119898119910lowast120573minus119910lowast0)120593lowast(Ω) le 1003816100381610038161003816100381610038161003816100381610038161003816

119899sum119894=1

(119910lowast120573 minus 119910lowast0 ) (119898 (119860 119894) (119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 + 120576 (66)

Hence

(119898119910lowast120573minus119910lowast0)120593lowast(Ω) le 1003816100381610038161003816100381610038161003816100381610038161003816

119899sum119894=1

(119910lowast120573 minus 119910lowast0 ) (119879 (1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 + 120576

le 1003816100381610038161003816100381610038161003816100381610038161003816(119910lowast120573 minus 119910lowast0 ) 119879(

119899sum119894=1

(1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816

+ 120576

Journal of Function Spaces 9

= 1003816100381610038161003816100381610038161003816100381610038161003816119879lowast (119910lowast120573 minus 119910lowast0 )(

119899sum119894=1

(1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816

+ 120576le 10038171003817100381710038171003817119879lowast (119910lowast120573 minus 119910lowast0 )10038171003817100381710038171003817lowast120593

1003817100381710038171003817100381710038171003817100381710038171003817119899sum119894=1

(1119860119894 otimes 119909119894)1003817100381710038171003817100381710038171003817100381710038171003817120593

+ 120576 le 10038171003817100381710038171003817119879lowast (119910lowast120573) minus 119879 (119910lowast0 )10038171003817100381710038171003817lowast120593 + 120576(67)

Hence 119901119898(119910lowast120573 minus 119910lowast0 ) = (119898119910lowast120573minus119910lowast0)120593lowast(Ω) rarr

1205730 and this means

that the space (119861119884lowast 120575119898120593lowast) is compact

As a consequence of Theorems 17 and 15 we have thefollowing

Corollary 18 Assume that 120593 is a Young function such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be itsrepresenting measure If the space (119861119884lowast 120575119898120593lowast) is compact then119898 is 120593lowast-semivariationally 120583-continuousConflicts of Interest

The author declares that there are no conflicts of interest

References

[1] S Chen and H Hudzik ldquoOn some convexities of Orlicz andOrlicz-Bochner spacesrdquo Commentationes Mathematicae vol29 no 1 pp 13ndash29 1988

[2] P Kolwicz and R Pluciennik ldquo119875-convexity of Orlicz-Bochnerspacesrdquo Proceedings of the American Mathematical Society vol126 no 8 pp 2315ndash2322 1998

[3] S Shang and Y Cui ldquoUniform nonsquareness and locallyuniform nonsquareness in Orlicz-Bochner function spaces andapplicationsrdquo Journal of Functional Analysis vol 267 no 7 pp2056ndash2076 2014

[4] K Feledziak andM Nowak ldquoLocally solid topologies on vectorvalued function spacesrdquo Universitat de Barcelona CollectaneaMathematica vol 48 no 4-6 pp 487ndash511 1997

[5] K Feledziak and M Nowak ldquoIntegral representation of linearoperators on Orlicz-Bochner spacesrdquo Universitat de BarcelonaCollectanea Mathematica vol 61 no 3 pp 277ndash290 2010

[6] M Nowak ldquoLinear operators on vector-valued function spaceswithMackey topologiesrdquo Journal of Convex Analysis vol 15 no1 pp 165ndash178 2008

[7] M Nowak ldquoLebesgue topologies on vector-valued functionspacesrdquoMathematica Japonica vol 52 no 2 pp 171ndash182 2000

[8] A V Buhvalov ldquoThe analytic representation of operators withan abstract normrdquo Izvestiya Vysshikh Uchebnykh ZavedeniiMatematika vol 11 no 162 pp 21ndash32 1975

[9] A V Buhvalov ldquoThe analytic representation of linear operatorsby means of measurable vector-valued functionsrdquo IzvestiyaVysshikh Uchebnykh Zavedenii Matematika vol 7 no 182 pp21ndash31 1977

[10] J Diestel and J J Uhl ldquoVector measuresrdquo vol 15 AmericanMathematical Society Surveys Providence RI USA 1977

[11] M Nowak ldquoDuality theory of vector-valued function spaces IrdquoCommentationes Mathematicae vol 37 pp 195ndash215 1997

[12] A V Bukhvalov and G Ya Lozanovskii ldquoOn sets closed inmeasure in spaces of mesurable functionsrdquo Transactions of theMoscow Mathematical Society vol 2 pp 127ndash148 1978

[13] A Wiweger ldquoLinear spaces with mixed topologyrdquo StudiaMathematica vol 20 pp 47ndash68 1961

[14] K Feledziak ldquoUniformly Lebesgue topologies on Kothe-Bochner spacesrdquoCommentationesMathematicae vol 37 pp 81ndash98 1997

[15] W Ruess ldquo[Weakly] compact operators and DF-spacesrdquo PacificJournal of Mathematics vol 98 no 2 pp 419ndash441 1982

[16] M Nowak ldquoCompactness of Bochner representable operatorson Orlicz spacesrdquo Positivity An International MathematicsJournal Devoted to Theory and Applications of Positivity vol 13no 1 pp 193ndash199 2009

[17] N E Gretsky and J Uhl ldquoBounded linear operators on Banachfunction spaces of vector-valued functionsrdquo Transactions of theAmerican Mathematical Society vol 167 pp 263ndash277 1972

[18] J J Uhl ldquoCompact operators on Orlicz spacesrdquo in Rendicontidel Seminario Matematico vol 42 pp 209ndash219 University ofPadova 1969

[19] N Dinculeanu Vector Measures International Series of Mono-graphs in Pure and Applied Mathematics 95 Pergamon PressNew York NY USA 1967

[20] N Dinculeanu Linear Operators on Lp-Spaces Vector andOperator-Valued Measures and Applications Academic PressNew York NY USA 1973

[21] N Dinculeanu Vector Integration and Stochastic Integration inBanach Spaces John Wiley amp Sons 2000

[22] R A Alo A De Korwin and L E Kunes ldquoTopological aspectsof 119902-regular measuresrdquo Studia Mathematica vol 48 pp 49ndash601973

[23] J Diestel ldquoOn the representation of bounded linear operatorsfrom Orlicz spaces of Lebesgue-Bochner measurable functionsto any Banach spacerdquo Bulletin de lrsquoAcademie Polonaise desSciences Serie des Sciences Mathematiques Astronomiques etPhysiques vol 18 pp 375ndash378 1970

[24] L Vanderputten ldquoRepresentation of operators defined on thespace of Bochner integrable functionsrdquo Extracta Mathematicaevol 16 no 3 pp 383ndash391 2001

[25] R FWheeler ldquoA survey of Bairemeasures and strict topologiesrdquoExpositiones Mathematicae vol 1 no 2 pp 97ndash190 1983

[26] W A Luxemburg Banach function spaces [Thesis] DelftNetherlands 1955

[27] P W Lewis ldquoVector measures and topologyrdquo Revue RoumainedeMathematique Pures et Appliquees vol 16 pp 1201ndash1209 1971

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6 Journal of Function Spaces

Since 119879119860(119904119899) = int119860119904119899119889119898 and by (i) 119879119860 is (Tand

120593 sdot 119884)-continuous we get

119879119860 (119891) = limint119860119904119899119889119898 (43)

Hence

119879119860 (119891) = int119860119891119889119898 (44)

and for 119910lowast isin 119884lowast we have119910lowast (119879119860 (119891)) = lim119910lowast (int

119860119904119899119889119898) = limint

119860119904119899119889119898119910lowast

= int119860119891119889119898119910lowast

(45)

(iii) Let 119891 isin 119871120593(119883) and (119860119899) be a sequence in Σ suchthat 119860119899 darr 0 Then 1119860119899(120596) darr 0 for 120596 isin Ω andhence 1119860119899(120596)119891(120596)119883 rarr 0 120583-ae and 1119860119899(120596)119891(120596)119883 le119891(120596)119883 120583-ae Hence 1119860119899119891 rarr 0 in Tand

120593 because Tand120593 is a

Lebesgue topology and by (i) we get

10038171003817100381710038171003817119898119891 (119860119899)10038171003817100381710038171003817119884 =100381710038171003817100381710038171003817100381710038171003817int119860119899 119891119889119898

100381710038171003817100381710038171003817100381710038171003817119884 =10038171003817100381710038171003817119879 (1119860119899119891)10038171003817100381710038171003817119884 997888rarr 0 (46)

(iv) Note that 120593lowast(119860) le 119879119860120593 To show that 119879119860120593 le120593lowast(119860) assume that 119891 isin 119861119871120593(119883) Choose a sequence (119904119899)in S(Σ119891(120583) 119883) such that 119904119899(120596) minus 119891(120596)119883 rarr 0 120583-ae and119904119899(120596)119883 le 119891(120596)119883 120583-ae for all 119899 isin N Since Tand

120593 isa Lebesgue topology we have 119904119899 rarr 119891 in Tand

120593 and hence119879119860(119904119899) minus 119879119860(119891)119884 rarr 0 Note that 119879119860(119904119899) = int119860 119904119899119889119898

Let 120576 gt 0 be given Choose 1198990 isin N such that 119879119860(119891) minusint1198601199041198990119889119898119884 le 120576 Then

1003817100381710038171003817119879119860 (119891)1003817100381710038171003817119884 le 1003817100381710038171003817100381710038171003817119879119860 (119891) minus int119860 11990411989901198891198981003817100381710038171003817100381710038171003817119884 +

1003817100381710038171003817100381710038171003817int119860 11990411989901198891198981003817100381710038171003817100381710038171003817119884

le 120576 + 120593lowast (119860) (47)

It follows that 119879119860120593 le 120593lowast(119860) so 120593lowast(119860) = 119879119860120593 Hencefor 119910lowast isin 119884lowast we easily get

1003817100381710038171003817(119910lowast ∘ 119879)1198601003817100381710038171003817lowast120593 = 1003817100381710038171003817119910lowast ∘ 1198791198601003817100381710038171003817lowast120593 = (119898119910lowast)120593lowast (119860) (48)

(v) Using (iv) we have

120593lowast (119860) = 10038171003817100381710038171198791198601003817100381710038171003817120593= sup 1003817100381710038171003817119879119860 (119891)1003817100381710038171003817119884 119891 isin 119871120593 (119883) 10038171003817100381710038171198911003817100381710038171003817120593 le 1= sup119910lowastisin119861119884lowast

1003816100381610038161003816(119910lowast ∘ 119879119860) (119891)1003816100381610038161003816 119891 isin 119871120593 (119883) 10038171003817100381710038171198911003817100381710038171003817120593 le 1= sup119910lowastisin119861119884lowast

1003817100381710038171003817119910lowast ∘ 1198791198601003817100381710038171003817lowast120593 = sup119910lowastisin119861119884lowast

(119898119910lowast)120593lowast (119860) (49)

(vi) This follows from (ii) and (iv)

For a sequence (119860119899) in Σ we will write 1198601198991205830 if 119860119899 darrand 120583(119860119899 cap 119860) rarr 0 for every 119860 isin Σ119891(120583)Definition 10 A measure 119898 Σ119891(120583) rarr L(119883 119884) with 119898 ≪120583 and 120593lowast(Ω) lt infin is said to be 120593lowast-semivariationally 120583-continuous if 120593lowast(119860119899) rarr 0 whenever 1198601198991205830 (119860119899) sub Σ

Using a standard argument we can show the following

Proposition 11 Let119898 Σ rarrL(119883 119884) be an additive measuresuch that 119898 ≪ 120583 and 120593(Ω) lt infin Then the followingstatements are equivalent

(i) 119898 is 120593lowast-semivariationally 120583-continuous(ii) The following two conditions hold simultaneously

(a) For every 120576 gt 0 there exists 120575 gt 0 such that120593lowast(119860) le 120576 whenever 120583(119860) le 120575 119860 isin Σ(b) For every 120576 gt 0 there exists 1198600 isin Σ119891(120583) such that120593lowast(Ω 1198600) le 120576

The following theorem characterizes 120593lowast-semivariation-ally 120583-continuous representing measures

Theorem 12 Let 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be its representingmeasure Then the following statements are equivalent

(i) 119898 is 120593lowast-semivariationally 120583-continuous(ii) 119879 is (120574120593 sdot 119884)-continuous(iii) 119879(119891119899)119884 rarr 0 if 119891119899 rarr 0 inT0 and sup119899119891119899120593 lt infin(iv) 119879119860119899120593 rarr 0 if 1198601198991205830 (119860119899) sub Σ

Proof (i)hArr (ii)hArr (iii) See [5 Corollary 28 and Proposition11]

(i)hArr (iv) This follows fromTheorem 9

Now assume that Ω is a completely regular Hausdorffspace LetB119886 denote the 120590-algebra of Baire sets in Ω whichis the 120590-algebra generated by the class Z of all zero sets ofbounded continuous positive functions on120596 ByPwedenotethe family of all cozero (=positive) in Ω (see [25 p 108])

Let 120583 B119886 rarr [0infin) be a countably additive measureThen 120583 is zero-set regular that is for every119860 isinB119886 and 120576 gt 0there exists 119885 isin Z with 119885 sub 119860 such that 120583(119860 119885) le 120576 (see[25 p 118]) It follows that for every 119860 isin B119886 and 120576 gt 0 thereexist 119880 isin P 119880 sup 119860 such that 120583(119880 119860) le 120576

We can assume that 120583 to be complete (if necessary wecan take the completion (ΩB119886 120583) of the measure space(ΩB119886 120583))Proposition 13 Assume that Ω is a completely regular Haus-dorff space and (ΩB119886 120583) is a complete finite measure spaceLet119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operatorand119898 B119886 rarrL(119883 119884) be its representing measure Then thefollowing statements are equivalent

(i) 119898 is 120593lowast-semivariationally 120583-continuous

Journal of Function Spaces 7

(ii) For every sequence (119860119899) in B119886 such that 119860119899 darr and120583(119860119899) rarr 0 there exists a sequence (119880119899) inPwith119860119899 sub119880119899 darr such that 120593lowast(119880119899) rarr 0(iii) For every sequence (119860119899) in B119886 such that 119860119899 darr and120583(119860119899) rarr 0 there exists a sequence (119880119899) inPwith119860119899 sub119880119899 darr such thatsup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) supp119891 sub 119880119899 997888rarr 0 (50)

Proof (i)rArr (ii) Assume that (i) holds and (119860119899) is a sequencein B119886 such that 119860119899 darr and 120583(119860119899) rarr 0 Then there exists asequence (119880119899) inP such that119860119899 sub 119880119899 darr and120583(119880119899119860119899) le 1119899for 119899 isin N

Let 120576 gt 0 be given Then in view of Proposition 11 thereexists 120575 gt 0 such that 120593lowast(119860) le 1205762 if 120583(119860) le 120575with119860 isinB119886Choose 1198991 isin N such that 120583(119880119899 119860119899) le 120575 for 119899 ge 1198992 Then120593lowast(119880119899 119860119899) le 1205762 for 119899 ge 1198991 Since 120593lowast(119860119899) rarr 0 we canchoose 1198992 isin N such that 120593lowast(119860119899) le 1205762 for 119899 ge 1198992 Then for119899 ge 1198990 = max(1198991 1198992) we get

120593lowast (119880119899) le 120593lowast (119880119899 119860119899) + 120593lowast (119860119899) le 120576 (51)

that is (ii) holds(ii)rArr (iii) Assume that (ii) holds and (119860119899) is a sequence

in B119900 such that 119860119899 darr and 120583(119860119899) rarr 0 Then there exists asequence (119880119899) inP with 119860119899 sub 119880119899 darr such that 120593lowast(119880119899) rarr 0Note that for 119891 isin 119861119871120593(119883) with supp119891 sub 119880119899 for 119899 isin N byTheorem 9 we have

1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 = 1003817100381710038171003817100381710038171003817intΩ 1198911198891198981003817100381710038171003817100381710038171003817119884 =

100381710038171003817100381710038171003817100381710038171003817int119880119899 119891119889119898100381710038171003817100381710038171003817100381710038171003817119884 le 120593lowast (119880119899) (52)

It follows that (iii) holds(iii)rArr (i) Assume that (iii) holds and119860119899 darrwith 120583(119860119899) rarr0Then there exists a sequence (119880119899) inPwith119860119899 sub 119880119899 darr such

that

sup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) supp119891 sub 119880119899 997888rarr 0 (53)

Assume on the contrary that (i) fails to hold Then withoutloss of generality we can assume that

120593lowast (119860119899) gt 1205760 for some 1205760 gt 0 all 119899 isin N (54)

Choose 1198990 isin N such that

sup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) supp119891 sub 1198801198990 lt 12057602 (55)

In view of (54) there exists a pairwise disjoint set 1198611 119861119896in B119886 119909119894 isin 119883 for 119894 = 1 119896 and 119910lowast isin 119861119884lowast such thatsum119896119894=1(1119861119894 otimes 119909119894)120593 le 1 and

1003816100381610038161003816100381610038161003816100381610038161003816119910lowast( 119896sum

119894=1

119898(1198601198990 cap 119861119894) (119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 ge 1205760 (56)

Let 1199040 = sum119896119894=1(11198601198990cap119861119894 otimes 119909119894) Then 1199040120593 le 1 and supp 1199040 sub1198601198990 sub 1198801198990 Then by (55) we get 119879(1199040)119884 lt 12057602On the other hand in view of (56) we have 119879(1199040)119884 ge 1205760

This contradiction establishes that (i) holds

Corollary 14 Assume that Ω is a completely regular Haus-dorff space and (ΩB119886 120583) is complete finitemeasure space Let119879 119871120593(119883) rarr 119884 be a (120574120593 sdot 119884)-continuous linear operator and119898 B119886 rarr L(119883 119884) be its representing measure Then 120593lowast isregular that is for every 119860 isinB119886 and 120576 gt 0 there exist 119885 isinZand 119880 isin P with 119885 sub 119860 sub 119880 such that 120593lowast(119880 119885) le 120576Proof In view of Theorem 12 119898 is 120593lowast-semivariationally 120583-continuous Let 119860 isin B119886 and 120576 gt 0 be given Then byProposition 11 there exists 120575 gt 0 such that 120593lowast(119861) le 120576whenever 119861 isin B119886 and 120583(119861) le 120575 By the regularity of 120583 onecan choose 119885 isin Z and 119880 isin P with 119885 sub 119860 sub 119880 such that120583(119880 119885) le 120575 Hence 120593lowast(119880 119885) le 120576 as desired4 Compact Operatorson Orlicz-Bochner Spaces

The following theorem presents necessary conditions for a(Tand120593 sdot 119884)-continuous operator 119879 119871120593(119883) rarr 119884 to be

compact

Theorem 15 Assume that a Young function 120593 such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot 119884)-continuous linear operator and 119898 Σ119891(120583) rarr L(119883 119884)be its representing measure If 119879 is compact then 119898 is 120593lowast-semivariationally 120583-continuousProof Assume that 119879 is compact and 119898 fails to be 120593lowast-semivariationally 120583-continuous Then there exist 120576 gt 0 and asequence (119860119899) in Σwith1198601198991205830 such that 119879119860119899 = 119898lowast120593(119860119899) gt120576 for 119899 isin N (seeTheorem9)Hence one can choose a sequence(119910lowast119899 ) in 119861119884lowast such that

10038171003817100381710038171003817119910lowast119899 ∘ 11987911986011989910038171003817100381710038171003817lowast120593 ge 120576 forall119899 isin N (57)

By Schauderrsquos theorem the conjugatemapping119879lowast 119884lowast rarr119871120593(119883)lowast is compact Note that 119879lowast(119910lowast119899 ) = 119910lowast119899 ∘ 119879 isin 119871120593(119883)sim119899 forall 119899 isin N where 119871120593(119883)sim119899 is a closed subspace of the Banachspace (119871120593(119883)lowast sdot lowast120593) (see Theorem 2) Then for every 119899 isin N

there exists 119892119899 isin 119871120593lowast(119883lowast 119883) such that

(119910lowast119899 ∘ 119879) (119891) = intΩ⟨119891 (120596) 119892119899 (120596)⟩ 119889120583

for 119891 isin 119871120593 (119883) 1003817100381710038171003817119910lowast119899 ∘ 1198791003817100381710038171003817lowast120593= sup int

Ω

1003817100381710038171003817119891 (120596)1003817100381710038171003817119883 120599 (119892119899) (120596) 119889120583 119891 isin 119861119871120593(119883)= 1003817100381710038171003817120599 (119892119899)1003817100381710038171003817120593lowast

(58)

Hence we obtain that for each 119899 isin N

10038171003817100381710038171003817119910lowast119899 ∘ 11987911986011989910038171003817100381710038171003817lowast120593 = 100381710038171003817100381710038171119860119899120599 (119892119899)10038171003817100381710038171003817120593lowast = 10038171003817100381710038171003817120599 (1119860119899119892119899)10038171003817100381710038171003817120593lowast (59)

8 Journal of Function Spaces

Since 119879lowast(119861119884lowast) is a relatively sequentially compact subset of((119871120593(119883)sim119899 sdot lowast120593) there exist a subsequence (119892119896119899) of (119892119899) and119892 isin 119871120593lowast(119883lowast 119883) such that10038171003817100381710038171003817119865119892119899 minus 11986511989210038171003817100381710038171003817lowast120593 = 10038171003817100381710038171003817120599 (119892119896119899 minus 119892)10038171003817100381710038171003817120593lowast 997888rarr 0 (60)

Choose 119899120576 isin N such that 120599(119892119896119899 minus 119892)120593lowast le 1205762 for 119899 ge 119899120576Hence for 119899 ge 1198991205761003816100381610038161003816100381610038161003816

100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast minus 100381710038171003817100381710038171003817120599 (1119860119896119899119892119896119899)100381710038171003817100381710038171003817120593lowast1003816100381610038161003816100381610038161003816

le 100381710038171003817100381710038171003817120599 (1119860119896119899 (119892119896119899 minus 119892))100381710038171003817100381710038171003817120593lowast = 1003817100381710038171003817100381710038171119860119896119899120599 (119892119896119899 minus 119892)100381710038171003817100381710038171003817120593lowastle 10038171003817100381710038171003817120599 (119892119896119899 minus 119892)10038171003817100381710038171003817120593lowast le 1205762

(61)

Using (57) and (61) for 119899 ge 119899120576 we get120576 le 100381710038171003817100381710038171003817119910lowast ∘ 119879119860119896119899 100381710038171003817100381710038171003817

lowast

120593= 100381710038171003817100381710038171003817120599 (1119860119896119899119892119896119899)100381710038171003817100381710038171003817120593lowast

le 1205762 +100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast

(62)

and hence 1003817100381710038171003817100381710038171119860119896119899120599 (119892)100381710038171003817100381710038171003817120593lowast = 100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast ge 1205762 (63)

On the other hand since 120593lowast is supposed to satisfy theΔ 2-condition we have that 1119860119896119899120599(119892)120593lowast rarr 0 (see [26 Theorem3 pp 58-59]) This contradiction establishes that 119898 is 120593lowast-semivariationally 120583-continuousCorollary 16 Assume that 120593 is a Young function such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operator Then the following statements areequivalent

(i) 119879 is compact(ii) 119879 is (120574120593 sdot 119884)-compact that is there exists a 120574120593-

neighborhood 119881 of 0 in 119871120593(119883) such that 119879(119881) is arelatively norm compact set in 119884

(iii) There exists a Young function 120595 with 120595 ≪ 120593 such thatintΩ119891119889119898 119891 isin 119871120593(119883) 119891120595 le 1 is a relatively norm

compact set in 119884Proof (i)rArr (ii) Assume that (i) holds Then by Theorems 12and 15 119879 is (120574120593 sdot 119884)-continuous Since the space (119871120593(119883) 120574120593)is quasinormable by Grothendieckrsquos classical result (see [15p 429]) we obtain that 119879 is (120574120593 sdot 119884)-compact

(ii)rArr (i) The implication is obvious(ii)hArr (iii) This follows fromTheorem 3

5 Topology Associated withthe 120593lowast-Semivariation ofa Representing Measure

Assume that 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator Let119898 Σ119891(120583) rarrL(119883 119884) be its representingmeasure Let us put

119901119898 (119910lowast) fl (119898119910lowast)120593lowast (Ω) for 119910lowast isin 119884lowast (64)

Note that119901119898 is a seminormon119884lowast Following [22 27] let 120575119898120593lowaststand for the topology on 119861119884lowast defined by the seminorm 119901119898restricted to 119861119884lowast

The following theorem characterizes (Tand120593 sdot 119884)-

continuous compact operators 119879 119871120593(119883) rarr 119884 in terms ofthe topological properties of the space (119861119884lowast 120575119898120593lowast) (see [22Theorem 3])

Theorem 17 Let 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be its representingmeasure Then the following statements are equivalent

(i) The space (119861119884lowast 120575119898120593lowast) is compact

(ii) 119879 is compact

Proof (i) rArr (ii) Assume that (119861119884lowast 120575119898120593lowast) is compact Let(119910lowast119899 ) be a sequence in 119861119884lowast Without loss of generality we canassume that 119910lowast119899 rarr 119910lowast0 in 120575119898120593lowast for some 119910lowast isin 119861119884lowast Then usingTheorem 9 for 119891 isin 119871120593(119883) we have

1003816100381610038161003816(119879lowast (119910lowast119899 ) minus 119879lowast (119910lowast0 )) (119891)1003816100381610038161003816 = 1003816100381610038161003816(119910lowast119899 minus 119910lowast0 ) (119879 (119891))1003816100381610038161003816= 1003816100381610038161003816100381610038161003816intΩ 119891119889119898119910lowast119899 minus119910lowast0

1003816100381610038161003816100381610038161003816 le (119898119910lowast119899 minus119910lowast0 )120593lowast (Ω) 10038171003817100381710038171198911003817100381710038171003817120593 (65)

It follows that 119879lowast(119910lowast119899 ) minus 119879lowast(119910lowast0 )lowast120593 le (119898119910lowast119899 minus119910lowast0 )120593lowast(Ω) where119901119898(119910lowast119899 minus 119910lowast0 ) = (119898119910lowast119899 minus119910lowast0 )120593lowast(Ω) rarr119899 0 This means that 119879lowast iscompact and hence 119879 is compact

(ii)rArr (i) Assume that 119879 is compact and (119910lowast120572 ) is a net in119861119884lowast Since119861119884lowast is 120590(119884lowast 119884)-compact without loss of generalitywe can assume that 119910lowast120572 rarr120572 119910lowast0 in 120590(119884lowast 119884) for some 119910lowast0 isin 119861119884lowast In view of the compactness of the conjugate operator 119879lowast 119884lowast rarr 119871120593(119883)lowast there exists a subset (119910lowast120573 ) of (119910lowast120572 ) and Φ0 isin119871120593(119883)lowast such that 119879lowast(119910lowast120573 ) minus Φ0lowast120593 rarr

1205730 On the other hand

since 119879lowast is (120590(119884lowast 119884) 120590(119871120593(119883)lowast 119871120593(119883)))-continuous we get119879lowast(119910lowast120573 ) rarr120573119879lowast(119910lowast0 ) in 120590(119871120593(119883)lowast 119871120593(119883)) HenceΦ0 = 119879lowast(119910lowast0 )

that is 119879lowast(119910lowast120573 ) minus 119879lowast(119910lowast0 )lowast120593 rarr1205730

Let 120576 gt 0 be givenThen there exist a pairwise disjoint set1198601 119860119899 in Σ119891(120583) and 119909119894 isin 119883 for 119894 = 1 119899 such thatsum119899119894=1(1119860119894 otimes 119909119894)120593 le 1 and

(119898119910lowast120573minus119910lowast0)120593lowast(Ω) le 1003816100381610038161003816100381610038161003816100381610038161003816

119899sum119894=1

(119910lowast120573 minus 119910lowast0 ) (119898 (119860 119894) (119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 + 120576 (66)

Hence

(119898119910lowast120573minus119910lowast0)120593lowast(Ω) le 1003816100381610038161003816100381610038161003816100381610038161003816

119899sum119894=1

(119910lowast120573 minus 119910lowast0 ) (119879 (1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 + 120576

le 1003816100381610038161003816100381610038161003816100381610038161003816(119910lowast120573 minus 119910lowast0 ) 119879(

119899sum119894=1

(1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816

+ 120576

Journal of Function Spaces 9

= 1003816100381610038161003816100381610038161003816100381610038161003816119879lowast (119910lowast120573 minus 119910lowast0 )(

119899sum119894=1

(1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816

+ 120576le 10038171003817100381710038171003817119879lowast (119910lowast120573 minus 119910lowast0 )10038171003817100381710038171003817lowast120593

1003817100381710038171003817100381710038171003817100381710038171003817119899sum119894=1

(1119860119894 otimes 119909119894)1003817100381710038171003817100381710038171003817100381710038171003817120593

+ 120576 le 10038171003817100381710038171003817119879lowast (119910lowast120573) minus 119879 (119910lowast0 )10038171003817100381710038171003817lowast120593 + 120576(67)

Hence 119901119898(119910lowast120573 minus 119910lowast0 ) = (119898119910lowast120573minus119910lowast0)120593lowast(Ω) rarr

1205730 and this means

that the space (119861119884lowast 120575119898120593lowast) is compact

As a consequence of Theorems 17 and 15 we have thefollowing

Corollary 18 Assume that 120593 is a Young function such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be itsrepresenting measure If the space (119861119884lowast 120575119898120593lowast) is compact then119898 is 120593lowast-semivariationally 120583-continuousConflicts of Interest

The author declares that there are no conflicts of interest

References

[1] S Chen and H Hudzik ldquoOn some convexities of Orlicz andOrlicz-Bochner spacesrdquo Commentationes Mathematicae vol29 no 1 pp 13ndash29 1988

[2] P Kolwicz and R Pluciennik ldquo119875-convexity of Orlicz-Bochnerspacesrdquo Proceedings of the American Mathematical Society vol126 no 8 pp 2315ndash2322 1998

[3] S Shang and Y Cui ldquoUniform nonsquareness and locallyuniform nonsquareness in Orlicz-Bochner function spaces andapplicationsrdquo Journal of Functional Analysis vol 267 no 7 pp2056ndash2076 2014

[4] K Feledziak andM Nowak ldquoLocally solid topologies on vectorvalued function spacesrdquo Universitat de Barcelona CollectaneaMathematica vol 48 no 4-6 pp 487ndash511 1997

[5] K Feledziak and M Nowak ldquoIntegral representation of linearoperators on Orlicz-Bochner spacesrdquo Universitat de BarcelonaCollectanea Mathematica vol 61 no 3 pp 277ndash290 2010

[6] M Nowak ldquoLinear operators on vector-valued function spaceswithMackey topologiesrdquo Journal of Convex Analysis vol 15 no1 pp 165ndash178 2008

[7] M Nowak ldquoLebesgue topologies on vector-valued functionspacesrdquoMathematica Japonica vol 52 no 2 pp 171ndash182 2000

[8] A V Buhvalov ldquoThe analytic representation of operators withan abstract normrdquo Izvestiya Vysshikh Uchebnykh ZavedeniiMatematika vol 11 no 162 pp 21ndash32 1975

[9] A V Buhvalov ldquoThe analytic representation of linear operatorsby means of measurable vector-valued functionsrdquo IzvestiyaVysshikh Uchebnykh Zavedenii Matematika vol 7 no 182 pp21ndash31 1977

[10] J Diestel and J J Uhl ldquoVector measuresrdquo vol 15 AmericanMathematical Society Surveys Providence RI USA 1977

[11] M Nowak ldquoDuality theory of vector-valued function spaces IrdquoCommentationes Mathematicae vol 37 pp 195ndash215 1997

[12] A V Bukhvalov and G Ya Lozanovskii ldquoOn sets closed inmeasure in spaces of mesurable functionsrdquo Transactions of theMoscow Mathematical Society vol 2 pp 127ndash148 1978

[13] A Wiweger ldquoLinear spaces with mixed topologyrdquo StudiaMathematica vol 20 pp 47ndash68 1961

[14] K Feledziak ldquoUniformly Lebesgue topologies on Kothe-Bochner spacesrdquoCommentationesMathematicae vol 37 pp 81ndash98 1997

[15] W Ruess ldquo[Weakly] compact operators and DF-spacesrdquo PacificJournal of Mathematics vol 98 no 2 pp 419ndash441 1982

[16] M Nowak ldquoCompactness of Bochner representable operatorson Orlicz spacesrdquo Positivity An International MathematicsJournal Devoted to Theory and Applications of Positivity vol 13no 1 pp 193ndash199 2009

[17] N E Gretsky and J Uhl ldquoBounded linear operators on Banachfunction spaces of vector-valued functionsrdquo Transactions of theAmerican Mathematical Society vol 167 pp 263ndash277 1972

[18] J J Uhl ldquoCompact operators on Orlicz spacesrdquo in Rendicontidel Seminario Matematico vol 42 pp 209ndash219 University ofPadova 1969

[19] N Dinculeanu Vector Measures International Series of Mono-graphs in Pure and Applied Mathematics 95 Pergamon PressNew York NY USA 1967

[20] N Dinculeanu Linear Operators on Lp-Spaces Vector andOperator-Valued Measures and Applications Academic PressNew York NY USA 1973

[21] N Dinculeanu Vector Integration and Stochastic Integration inBanach Spaces John Wiley amp Sons 2000

[22] R A Alo A De Korwin and L E Kunes ldquoTopological aspectsof 119902-regular measuresrdquo Studia Mathematica vol 48 pp 49ndash601973

[23] J Diestel ldquoOn the representation of bounded linear operatorsfrom Orlicz spaces of Lebesgue-Bochner measurable functionsto any Banach spacerdquo Bulletin de lrsquoAcademie Polonaise desSciences Serie des Sciences Mathematiques Astronomiques etPhysiques vol 18 pp 375ndash378 1970

[24] L Vanderputten ldquoRepresentation of operators defined on thespace of Bochner integrable functionsrdquo Extracta Mathematicaevol 16 no 3 pp 383ndash391 2001

[25] R FWheeler ldquoA survey of Bairemeasures and strict topologiesrdquoExpositiones Mathematicae vol 1 no 2 pp 97ndash190 1983

[26] W A Luxemburg Banach function spaces [Thesis] DelftNetherlands 1955

[27] P W Lewis ldquoVector measures and topologyrdquo Revue RoumainedeMathematique Pures et Appliquees vol 16 pp 1201ndash1209 1971

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Journal of Function Spaces 7

(ii) For every sequence (119860119899) in B119886 such that 119860119899 darr and120583(119860119899) rarr 0 there exists a sequence (119880119899) inPwith119860119899 sub119880119899 darr such that 120593lowast(119880119899) rarr 0(iii) For every sequence (119860119899) in B119886 such that 119860119899 darr and120583(119860119899) rarr 0 there exists a sequence (119880119899) inPwith119860119899 sub119880119899 darr such thatsup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) supp119891 sub 119880119899 997888rarr 0 (50)

Proof (i)rArr (ii) Assume that (i) holds and (119860119899) is a sequencein B119886 such that 119860119899 darr and 120583(119860119899) rarr 0 Then there exists asequence (119880119899) inP such that119860119899 sub 119880119899 darr and120583(119880119899119860119899) le 1119899for 119899 isin N

Let 120576 gt 0 be given Then in view of Proposition 11 thereexists 120575 gt 0 such that 120593lowast(119860) le 1205762 if 120583(119860) le 120575with119860 isinB119886Choose 1198991 isin N such that 120583(119880119899 119860119899) le 120575 for 119899 ge 1198992 Then120593lowast(119880119899 119860119899) le 1205762 for 119899 ge 1198991 Since 120593lowast(119860119899) rarr 0 we canchoose 1198992 isin N such that 120593lowast(119860119899) le 1205762 for 119899 ge 1198992 Then for119899 ge 1198990 = max(1198991 1198992) we get

120593lowast (119880119899) le 120593lowast (119880119899 119860119899) + 120593lowast (119860119899) le 120576 (51)

that is (ii) holds(ii)rArr (iii) Assume that (ii) holds and (119860119899) is a sequence

in B119900 such that 119860119899 darr and 120583(119860119899) rarr 0 Then there exists asequence (119880119899) inP with 119860119899 sub 119880119899 darr such that 120593lowast(119880119899) rarr 0Note that for 119891 isin 119861119871120593(119883) with supp119891 sub 119880119899 for 119899 isin N byTheorem 9 we have

1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 = 1003817100381710038171003817100381710038171003817intΩ 1198911198891198981003817100381710038171003817100381710038171003817119884 =

100381710038171003817100381710038171003817100381710038171003817int119880119899 119891119889119898100381710038171003817100381710038171003817100381710038171003817119884 le 120593lowast (119880119899) (52)

It follows that (iii) holds(iii)rArr (i) Assume that (iii) holds and119860119899 darrwith 120583(119860119899) rarr0Then there exists a sequence (119880119899) inPwith119860119899 sub 119880119899 darr such

that

sup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) supp119891 sub 119880119899 997888rarr 0 (53)

Assume on the contrary that (i) fails to hold Then withoutloss of generality we can assume that

120593lowast (119860119899) gt 1205760 for some 1205760 gt 0 all 119899 isin N (54)

Choose 1198990 isin N such that

sup 1003817100381710038171003817119879 (119891)1003817100381710038171003817119884 119891 isin 119861119871120593(119883) supp119891 sub 1198801198990 lt 12057602 (55)

In view of (54) there exists a pairwise disjoint set 1198611 119861119896in B119886 119909119894 isin 119883 for 119894 = 1 119896 and 119910lowast isin 119861119884lowast such thatsum119896119894=1(1119861119894 otimes 119909119894)120593 le 1 and

1003816100381610038161003816100381610038161003816100381610038161003816119910lowast( 119896sum

119894=1

119898(1198601198990 cap 119861119894) (119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 ge 1205760 (56)

Let 1199040 = sum119896119894=1(11198601198990cap119861119894 otimes 119909119894) Then 1199040120593 le 1 and supp 1199040 sub1198601198990 sub 1198801198990 Then by (55) we get 119879(1199040)119884 lt 12057602On the other hand in view of (56) we have 119879(1199040)119884 ge 1205760

This contradiction establishes that (i) holds

Corollary 14 Assume that Ω is a completely regular Haus-dorff space and (ΩB119886 120583) is complete finitemeasure space Let119879 119871120593(119883) rarr 119884 be a (120574120593 sdot 119884)-continuous linear operator and119898 B119886 rarr L(119883 119884) be its representing measure Then 120593lowast isregular that is for every 119860 isinB119886 and 120576 gt 0 there exist 119885 isinZand 119880 isin P with 119885 sub 119860 sub 119880 such that 120593lowast(119880 119885) le 120576Proof In view of Theorem 12 119898 is 120593lowast-semivariationally 120583-continuous Let 119860 isin B119886 and 120576 gt 0 be given Then byProposition 11 there exists 120575 gt 0 such that 120593lowast(119861) le 120576whenever 119861 isin B119886 and 120583(119861) le 120575 By the regularity of 120583 onecan choose 119885 isin Z and 119880 isin P with 119885 sub 119860 sub 119880 such that120583(119880 119885) le 120575 Hence 120593lowast(119880 119885) le 120576 as desired4 Compact Operatorson Orlicz-Bochner Spaces

The following theorem presents necessary conditions for a(Tand120593 sdot 119884)-continuous operator 119879 119871120593(119883) rarr 119884 to be

compact

Theorem 15 Assume that a Young function 120593 such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot 119884)-continuous linear operator and 119898 Σ119891(120583) rarr L(119883 119884)be its representing measure If 119879 is compact then 119898 is 120593lowast-semivariationally 120583-continuousProof Assume that 119879 is compact and 119898 fails to be 120593lowast-semivariationally 120583-continuous Then there exist 120576 gt 0 and asequence (119860119899) in Σwith1198601198991205830 such that 119879119860119899 = 119898lowast120593(119860119899) gt120576 for 119899 isin N (seeTheorem9)Hence one can choose a sequence(119910lowast119899 ) in 119861119884lowast such that

10038171003817100381710038171003817119910lowast119899 ∘ 11987911986011989910038171003817100381710038171003817lowast120593 ge 120576 forall119899 isin N (57)

By Schauderrsquos theorem the conjugatemapping119879lowast 119884lowast rarr119871120593(119883)lowast is compact Note that 119879lowast(119910lowast119899 ) = 119910lowast119899 ∘ 119879 isin 119871120593(119883)sim119899 forall 119899 isin N where 119871120593(119883)sim119899 is a closed subspace of the Banachspace (119871120593(119883)lowast sdot lowast120593) (see Theorem 2) Then for every 119899 isin N

there exists 119892119899 isin 119871120593lowast(119883lowast 119883) such that

(119910lowast119899 ∘ 119879) (119891) = intΩ⟨119891 (120596) 119892119899 (120596)⟩ 119889120583

for 119891 isin 119871120593 (119883) 1003817100381710038171003817119910lowast119899 ∘ 1198791003817100381710038171003817lowast120593= sup int

Ω

1003817100381710038171003817119891 (120596)1003817100381710038171003817119883 120599 (119892119899) (120596) 119889120583 119891 isin 119861119871120593(119883)= 1003817100381710038171003817120599 (119892119899)1003817100381710038171003817120593lowast

(58)

Hence we obtain that for each 119899 isin N

10038171003817100381710038171003817119910lowast119899 ∘ 11987911986011989910038171003817100381710038171003817lowast120593 = 100381710038171003817100381710038171119860119899120599 (119892119899)10038171003817100381710038171003817120593lowast = 10038171003817100381710038171003817120599 (1119860119899119892119899)10038171003817100381710038171003817120593lowast (59)

8 Journal of Function Spaces

Since 119879lowast(119861119884lowast) is a relatively sequentially compact subset of((119871120593(119883)sim119899 sdot lowast120593) there exist a subsequence (119892119896119899) of (119892119899) and119892 isin 119871120593lowast(119883lowast 119883) such that10038171003817100381710038171003817119865119892119899 minus 11986511989210038171003817100381710038171003817lowast120593 = 10038171003817100381710038171003817120599 (119892119896119899 minus 119892)10038171003817100381710038171003817120593lowast 997888rarr 0 (60)

Choose 119899120576 isin N such that 120599(119892119896119899 minus 119892)120593lowast le 1205762 for 119899 ge 119899120576Hence for 119899 ge 1198991205761003816100381610038161003816100381610038161003816

100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast minus 100381710038171003817100381710038171003817120599 (1119860119896119899119892119896119899)100381710038171003817100381710038171003817120593lowast1003816100381610038161003816100381610038161003816

le 100381710038171003817100381710038171003817120599 (1119860119896119899 (119892119896119899 minus 119892))100381710038171003817100381710038171003817120593lowast = 1003817100381710038171003817100381710038171119860119896119899120599 (119892119896119899 minus 119892)100381710038171003817100381710038171003817120593lowastle 10038171003817100381710038171003817120599 (119892119896119899 minus 119892)10038171003817100381710038171003817120593lowast le 1205762

(61)

Using (57) and (61) for 119899 ge 119899120576 we get120576 le 100381710038171003817100381710038171003817119910lowast ∘ 119879119860119896119899 100381710038171003817100381710038171003817

lowast

120593= 100381710038171003817100381710038171003817120599 (1119860119896119899119892119896119899)100381710038171003817100381710038171003817120593lowast

le 1205762 +100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast

(62)

and hence 1003817100381710038171003817100381710038171119860119896119899120599 (119892)100381710038171003817100381710038171003817120593lowast = 100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast ge 1205762 (63)

On the other hand since 120593lowast is supposed to satisfy theΔ 2-condition we have that 1119860119896119899120599(119892)120593lowast rarr 0 (see [26 Theorem3 pp 58-59]) This contradiction establishes that 119898 is 120593lowast-semivariationally 120583-continuousCorollary 16 Assume that 120593 is a Young function such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operator Then the following statements areequivalent

(i) 119879 is compact(ii) 119879 is (120574120593 sdot 119884)-compact that is there exists a 120574120593-

neighborhood 119881 of 0 in 119871120593(119883) such that 119879(119881) is arelatively norm compact set in 119884

(iii) There exists a Young function 120595 with 120595 ≪ 120593 such thatintΩ119891119889119898 119891 isin 119871120593(119883) 119891120595 le 1 is a relatively norm

compact set in 119884Proof (i)rArr (ii) Assume that (i) holds Then by Theorems 12and 15 119879 is (120574120593 sdot 119884)-continuous Since the space (119871120593(119883) 120574120593)is quasinormable by Grothendieckrsquos classical result (see [15p 429]) we obtain that 119879 is (120574120593 sdot 119884)-compact

(ii)rArr (i) The implication is obvious(ii)hArr (iii) This follows fromTheorem 3

5 Topology Associated withthe 120593lowast-Semivariation ofa Representing Measure

Assume that 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator Let119898 Σ119891(120583) rarrL(119883 119884) be its representingmeasure Let us put

119901119898 (119910lowast) fl (119898119910lowast)120593lowast (Ω) for 119910lowast isin 119884lowast (64)

Note that119901119898 is a seminormon119884lowast Following [22 27] let 120575119898120593lowaststand for the topology on 119861119884lowast defined by the seminorm 119901119898restricted to 119861119884lowast

The following theorem characterizes (Tand120593 sdot 119884)-

continuous compact operators 119879 119871120593(119883) rarr 119884 in terms ofthe topological properties of the space (119861119884lowast 120575119898120593lowast) (see [22Theorem 3])

Theorem 17 Let 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be its representingmeasure Then the following statements are equivalent

(i) The space (119861119884lowast 120575119898120593lowast) is compact

(ii) 119879 is compact

Proof (i) rArr (ii) Assume that (119861119884lowast 120575119898120593lowast) is compact Let(119910lowast119899 ) be a sequence in 119861119884lowast Without loss of generality we canassume that 119910lowast119899 rarr 119910lowast0 in 120575119898120593lowast for some 119910lowast isin 119861119884lowast Then usingTheorem 9 for 119891 isin 119871120593(119883) we have

1003816100381610038161003816(119879lowast (119910lowast119899 ) minus 119879lowast (119910lowast0 )) (119891)1003816100381610038161003816 = 1003816100381610038161003816(119910lowast119899 minus 119910lowast0 ) (119879 (119891))1003816100381610038161003816= 1003816100381610038161003816100381610038161003816intΩ 119891119889119898119910lowast119899 minus119910lowast0

1003816100381610038161003816100381610038161003816 le (119898119910lowast119899 minus119910lowast0 )120593lowast (Ω) 10038171003817100381710038171198911003817100381710038171003817120593 (65)

It follows that 119879lowast(119910lowast119899 ) minus 119879lowast(119910lowast0 )lowast120593 le (119898119910lowast119899 minus119910lowast0 )120593lowast(Ω) where119901119898(119910lowast119899 minus 119910lowast0 ) = (119898119910lowast119899 minus119910lowast0 )120593lowast(Ω) rarr119899 0 This means that 119879lowast iscompact and hence 119879 is compact

(ii)rArr (i) Assume that 119879 is compact and (119910lowast120572 ) is a net in119861119884lowast Since119861119884lowast is 120590(119884lowast 119884)-compact without loss of generalitywe can assume that 119910lowast120572 rarr120572 119910lowast0 in 120590(119884lowast 119884) for some 119910lowast0 isin 119861119884lowast In view of the compactness of the conjugate operator 119879lowast 119884lowast rarr 119871120593(119883)lowast there exists a subset (119910lowast120573 ) of (119910lowast120572 ) and Φ0 isin119871120593(119883)lowast such that 119879lowast(119910lowast120573 ) minus Φ0lowast120593 rarr

1205730 On the other hand

since 119879lowast is (120590(119884lowast 119884) 120590(119871120593(119883)lowast 119871120593(119883)))-continuous we get119879lowast(119910lowast120573 ) rarr120573119879lowast(119910lowast0 ) in 120590(119871120593(119883)lowast 119871120593(119883)) HenceΦ0 = 119879lowast(119910lowast0 )

that is 119879lowast(119910lowast120573 ) minus 119879lowast(119910lowast0 )lowast120593 rarr1205730

Let 120576 gt 0 be givenThen there exist a pairwise disjoint set1198601 119860119899 in Σ119891(120583) and 119909119894 isin 119883 for 119894 = 1 119899 such thatsum119899119894=1(1119860119894 otimes 119909119894)120593 le 1 and

(119898119910lowast120573minus119910lowast0)120593lowast(Ω) le 1003816100381610038161003816100381610038161003816100381610038161003816

119899sum119894=1

(119910lowast120573 minus 119910lowast0 ) (119898 (119860 119894) (119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 + 120576 (66)

Hence

(119898119910lowast120573minus119910lowast0)120593lowast(Ω) le 1003816100381610038161003816100381610038161003816100381610038161003816

119899sum119894=1

(119910lowast120573 minus 119910lowast0 ) (119879 (1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 + 120576

le 1003816100381610038161003816100381610038161003816100381610038161003816(119910lowast120573 minus 119910lowast0 ) 119879(

119899sum119894=1

(1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816

+ 120576

Journal of Function Spaces 9

= 1003816100381610038161003816100381610038161003816100381610038161003816119879lowast (119910lowast120573 minus 119910lowast0 )(

119899sum119894=1

(1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816

+ 120576le 10038171003817100381710038171003817119879lowast (119910lowast120573 minus 119910lowast0 )10038171003817100381710038171003817lowast120593

1003817100381710038171003817100381710038171003817100381710038171003817119899sum119894=1

(1119860119894 otimes 119909119894)1003817100381710038171003817100381710038171003817100381710038171003817120593

+ 120576 le 10038171003817100381710038171003817119879lowast (119910lowast120573) minus 119879 (119910lowast0 )10038171003817100381710038171003817lowast120593 + 120576(67)

Hence 119901119898(119910lowast120573 minus 119910lowast0 ) = (119898119910lowast120573minus119910lowast0)120593lowast(Ω) rarr

1205730 and this means

that the space (119861119884lowast 120575119898120593lowast) is compact

As a consequence of Theorems 17 and 15 we have thefollowing

Corollary 18 Assume that 120593 is a Young function such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be itsrepresenting measure If the space (119861119884lowast 120575119898120593lowast) is compact then119898 is 120593lowast-semivariationally 120583-continuousConflicts of Interest

The author declares that there are no conflicts of interest

References

[1] S Chen and H Hudzik ldquoOn some convexities of Orlicz andOrlicz-Bochner spacesrdquo Commentationes Mathematicae vol29 no 1 pp 13ndash29 1988

[2] P Kolwicz and R Pluciennik ldquo119875-convexity of Orlicz-Bochnerspacesrdquo Proceedings of the American Mathematical Society vol126 no 8 pp 2315ndash2322 1998

[3] S Shang and Y Cui ldquoUniform nonsquareness and locallyuniform nonsquareness in Orlicz-Bochner function spaces andapplicationsrdquo Journal of Functional Analysis vol 267 no 7 pp2056ndash2076 2014

[4] K Feledziak andM Nowak ldquoLocally solid topologies on vectorvalued function spacesrdquo Universitat de Barcelona CollectaneaMathematica vol 48 no 4-6 pp 487ndash511 1997

[5] K Feledziak and M Nowak ldquoIntegral representation of linearoperators on Orlicz-Bochner spacesrdquo Universitat de BarcelonaCollectanea Mathematica vol 61 no 3 pp 277ndash290 2010

[6] M Nowak ldquoLinear operators on vector-valued function spaceswithMackey topologiesrdquo Journal of Convex Analysis vol 15 no1 pp 165ndash178 2008

[7] M Nowak ldquoLebesgue topologies on vector-valued functionspacesrdquoMathematica Japonica vol 52 no 2 pp 171ndash182 2000

[8] A V Buhvalov ldquoThe analytic representation of operators withan abstract normrdquo Izvestiya Vysshikh Uchebnykh ZavedeniiMatematika vol 11 no 162 pp 21ndash32 1975

[9] A V Buhvalov ldquoThe analytic representation of linear operatorsby means of measurable vector-valued functionsrdquo IzvestiyaVysshikh Uchebnykh Zavedenii Matematika vol 7 no 182 pp21ndash31 1977

[10] J Diestel and J J Uhl ldquoVector measuresrdquo vol 15 AmericanMathematical Society Surveys Providence RI USA 1977

[11] M Nowak ldquoDuality theory of vector-valued function spaces IrdquoCommentationes Mathematicae vol 37 pp 195ndash215 1997

[12] A V Bukhvalov and G Ya Lozanovskii ldquoOn sets closed inmeasure in spaces of mesurable functionsrdquo Transactions of theMoscow Mathematical Society vol 2 pp 127ndash148 1978

[13] A Wiweger ldquoLinear spaces with mixed topologyrdquo StudiaMathematica vol 20 pp 47ndash68 1961

[14] K Feledziak ldquoUniformly Lebesgue topologies on Kothe-Bochner spacesrdquoCommentationesMathematicae vol 37 pp 81ndash98 1997

[15] W Ruess ldquo[Weakly] compact operators and DF-spacesrdquo PacificJournal of Mathematics vol 98 no 2 pp 419ndash441 1982

[16] M Nowak ldquoCompactness of Bochner representable operatorson Orlicz spacesrdquo Positivity An International MathematicsJournal Devoted to Theory and Applications of Positivity vol 13no 1 pp 193ndash199 2009

[17] N E Gretsky and J Uhl ldquoBounded linear operators on Banachfunction spaces of vector-valued functionsrdquo Transactions of theAmerican Mathematical Society vol 167 pp 263ndash277 1972

[18] J J Uhl ldquoCompact operators on Orlicz spacesrdquo in Rendicontidel Seminario Matematico vol 42 pp 209ndash219 University ofPadova 1969

[19] N Dinculeanu Vector Measures International Series of Mono-graphs in Pure and Applied Mathematics 95 Pergamon PressNew York NY USA 1967

[20] N Dinculeanu Linear Operators on Lp-Spaces Vector andOperator-Valued Measures and Applications Academic PressNew York NY USA 1973

[21] N Dinculeanu Vector Integration and Stochastic Integration inBanach Spaces John Wiley amp Sons 2000

[22] R A Alo A De Korwin and L E Kunes ldquoTopological aspectsof 119902-regular measuresrdquo Studia Mathematica vol 48 pp 49ndash601973

[23] J Diestel ldquoOn the representation of bounded linear operatorsfrom Orlicz spaces of Lebesgue-Bochner measurable functionsto any Banach spacerdquo Bulletin de lrsquoAcademie Polonaise desSciences Serie des Sciences Mathematiques Astronomiques etPhysiques vol 18 pp 375ndash378 1970

[24] L Vanderputten ldquoRepresentation of operators defined on thespace of Bochner integrable functionsrdquo Extracta Mathematicaevol 16 no 3 pp 383ndash391 2001

[25] R FWheeler ldquoA survey of Bairemeasures and strict topologiesrdquoExpositiones Mathematicae vol 1 no 2 pp 97ndash190 1983

[26] W A Luxemburg Banach function spaces [Thesis] DelftNetherlands 1955

[27] P W Lewis ldquoVector measures and topologyrdquo Revue RoumainedeMathematique Pures et Appliquees vol 16 pp 1201ndash1209 1971

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8 Journal of Function Spaces

Since 119879lowast(119861119884lowast) is a relatively sequentially compact subset of((119871120593(119883)sim119899 sdot lowast120593) there exist a subsequence (119892119896119899) of (119892119899) and119892 isin 119871120593lowast(119883lowast 119883) such that10038171003817100381710038171003817119865119892119899 minus 11986511989210038171003817100381710038171003817lowast120593 = 10038171003817100381710038171003817120599 (119892119896119899 minus 119892)10038171003817100381710038171003817120593lowast 997888rarr 0 (60)

Choose 119899120576 isin N such that 120599(119892119896119899 minus 119892)120593lowast le 1205762 for 119899 ge 119899120576Hence for 119899 ge 1198991205761003816100381610038161003816100381610038161003816

100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast minus 100381710038171003817100381710038171003817120599 (1119860119896119899119892119896119899)100381710038171003817100381710038171003817120593lowast1003816100381610038161003816100381610038161003816

le 100381710038171003817100381710038171003817120599 (1119860119896119899 (119892119896119899 minus 119892))100381710038171003817100381710038171003817120593lowast = 1003817100381710038171003817100381710038171119860119896119899120599 (119892119896119899 minus 119892)100381710038171003817100381710038171003817120593lowastle 10038171003817100381710038171003817120599 (119892119896119899 minus 119892)10038171003817100381710038171003817120593lowast le 1205762

(61)

Using (57) and (61) for 119899 ge 119899120576 we get120576 le 100381710038171003817100381710038171003817119910lowast ∘ 119879119860119896119899 100381710038171003817100381710038171003817

lowast

120593= 100381710038171003817100381710038171003817120599 (1119860119896119899119892119896119899)100381710038171003817100381710038171003817120593lowast

le 1205762 +100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast

(62)

and hence 1003817100381710038171003817100381710038171119860119896119899120599 (119892)100381710038171003817100381710038171003817120593lowast = 100381710038171003817100381710038171003817120599 (1119860119896119899119892)100381710038171003817100381710038171003817120593lowast ge 1205762 (63)

On the other hand since 120593lowast is supposed to satisfy theΔ 2-condition we have that 1119860119896119899120599(119892)120593lowast rarr 0 (see [26 Theorem3 pp 58-59]) This contradiction establishes that 119898 is 120593lowast-semivariationally 120583-continuousCorollary 16 Assume that 120593 is a Young function such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operator Then the following statements areequivalent

(i) 119879 is compact(ii) 119879 is (120574120593 sdot 119884)-compact that is there exists a 120574120593-

neighborhood 119881 of 0 in 119871120593(119883) such that 119879(119881) is arelatively norm compact set in 119884

(iii) There exists a Young function 120595 with 120595 ≪ 120593 such thatintΩ119891119889119898 119891 isin 119871120593(119883) 119891120595 le 1 is a relatively norm

compact set in 119884Proof (i)rArr (ii) Assume that (i) holds Then by Theorems 12and 15 119879 is (120574120593 sdot 119884)-continuous Since the space (119871120593(119883) 120574120593)is quasinormable by Grothendieckrsquos classical result (see [15p 429]) we obtain that 119879 is (120574120593 sdot 119884)-compact

(ii)rArr (i) The implication is obvious(ii)hArr (iii) This follows fromTheorem 3

5 Topology Associated withthe 120593lowast-Semivariation ofa Representing Measure

Assume that 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator Let119898 Σ119891(120583) rarrL(119883 119884) be its representingmeasure Let us put

119901119898 (119910lowast) fl (119898119910lowast)120593lowast (Ω) for 119910lowast isin 119884lowast (64)

Note that119901119898 is a seminormon119884lowast Following [22 27] let 120575119898120593lowaststand for the topology on 119861119884lowast defined by the seminorm 119901119898restricted to 119861119884lowast

The following theorem characterizes (Tand120593 sdot 119884)-

continuous compact operators 119879 119871120593(119883) rarr 119884 in terms ofthe topological properties of the space (119861119884lowast 120575119898120593lowast) (see [22Theorem 3])

Theorem 17 Let 119879 119871120593(119883) rarr 119884 be a (Tand120593 sdot 119884)-continuous

linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be its representingmeasure Then the following statements are equivalent

(i) The space (119861119884lowast 120575119898120593lowast) is compact

(ii) 119879 is compact

Proof (i) rArr (ii) Assume that (119861119884lowast 120575119898120593lowast) is compact Let(119910lowast119899 ) be a sequence in 119861119884lowast Without loss of generality we canassume that 119910lowast119899 rarr 119910lowast0 in 120575119898120593lowast for some 119910lowast isin 119861119884lowast Then usingTheorem 9 for 119891 isin 119871120593(119883) we have

1003816100381610038161003816(119879lowast (119910lowast119899 ) minus 119879lowast (119910lowast0 )) (119891)1003816100381610038161003816 = 1003816100381610038161003816(119910lowast119899 minus 119910lowast0 ) (119879 (119891))1003816100381610038161003816= 1003816100381610038161003816100381610038161003816intΩ 119891119889119898119910lowast119899 minus119910lowast0

1003816100381610038161003816100381610038161003816 le (119898119910lowast119899 minus119910lowast0 )120593lowast (Ω) 10038171003817100381710038171198911003817100381710038171003817120593 (65)

It follows that 119879lowast(119910lowast119899 ) minus 119879lowast(119910lowast0 )lowast120593 le (119898119910lowast119899 minus119910lowast0 )120593lowast(Ω) where119901119898(119910lowast119899 minus 119910lowast0 ) = (119898119910lowast119899 minus119910lowast0 )120593lowast(Ω) rarr119899 0 This means that 119879lowast iscompact and hence 119879 is compact

(ii)rArr (i) Assume that 119879 is compact and (119910lowast120572 ) is a net in119861119884lowast Since119861119884lowast is 120590(119884lowast 119884)-compact without loss of generalitywe can assume that 119910lowast120572 rarr120572 119910lowast0 in 120590(119884lowast 119884) for some 119910lowast0 isin 119861119884lowast In view of the compactness of the conjugate operator 119879lowast 119884lowast rarr 119871120593(119883)lowast there exists a subset (119910lowast120573 ) of (119910lowast120572 ) and Φ0 isin119871120593(119883)lowast such that 119879lowast(119910lowast120573 ) minus Φ0lowast120593 rarr

1205730 On the other hand

since 119879lowast is (120590(119884lowast 119884) 120590(119871120593(119883)lowast 119871120593(119883)))-continuous we get119879lowast(119910lowast120573 ) rarr120573119879lowast(119910lowast0 ) in 120590(119871120593(119883)lowast 119871120593(119883)) HenceΦ0 = 119879lowast(119910lowast0 )

that is 119879lowast(119910lowast120573 ) minus 119879lowast(119910lowast0 )lowast120593 rarr1205730

Let 120576 gt 0 be givenThen there exist a pairwise disjoint set1198601 119860119899 in Σ119891(120583) and 119909119894 isin 119883 for 119894 = 1 119899 such thatsum119899119894=1(1119860119894 otimes 119909119894)120593 le 1 and

(119898119910lowast120573minus119910lowast0)120593lowast(Ω) le 1003816100381610038161003816100381610038161003816100381610038161003816

119899sum119894=1

(119910lowast120573 minus 119910lowast0 ) (119898 (119860 119894) (119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 + 120576 (66)

Hence

(119898119910lowast120573minus119910lowast0)120593lowast(Ω) le 1003816100381610038161003816100381610038161003816100381610038161003816

119899sum119894=1

(119910lowast120573 minus 119910lowast0 ) (119879 (1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816 + 120576

le 1003816100381610038161003816100381610038161003816100381610038161003816(119910lowast120573 minus 119910lowast0 ) 119879(

119899sum119894=1

(1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816

+ 120576

Journal of Function Spaces 9

= 1003816100381610038161003816100381610038161003816100381610038161003816119879lowast (119910lowast120573 minus 119910lowast0 )(

119899sum119894=1

(1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816

+ 120576le 10038171003817100381710038171003817119879lowast (119910lowast120573 minus 119910lowast0 )10038171003817100381710038171003817lowast120593

1003817100381710038171003817100381710038171003817100381710038171003817119899sum119894=1

(1119860119894 otimes 119909119894)1003817100381710038171003817100381710038171003817100381710038171003817120593

+ 120576 le 10038171003817100381710038171003817119879lowast (119910lowast120573) minus 119879 (119910lowast0 )10038171003817100381710038171003817lowast120593 + 120576(67)

Hence 119901119898(119910lowast120573 minus 119910lowast0 ) = (119898119910lowast120573minus119910lowast0)120593lowast(Ω) rarr

1205730 and this means

that the space (119861119884lowast 120575119898120593lowast) is compact

As a consequence of Theorems 17 and 15 we have thefollowing

Corollary 18 Assume that 120593 is a Young function such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be itsrepresenting measure If the space (119861119884lowast 120575119898120593lowast) is compact then119898 is 120593lowast-semivariationally 120583-continuousConflicts of Interest

The author declares that there are no conflicts of interest

References

[1] S Chen and H Hudzik ldquoOn some convexities of Orlicz andOrlicz-Bochner spacesrdquo Commentationes Mathematicae vol29 no 1 pp 13ndash29 1988

[2] P Kolwicz and R Pluciennik ldquo119875-convexity of Orlicz-Bochnerspacesrdquo Proceedings of the American Mathematical Society vol126 no 8 pp 2315ndash2322 1998

[3] S Shang and Y Cui ldquoUniform nonsquareness and locallyuniform nonsquareness in Orlicz-Bochner function spaces andapplicationsrdquo Journal of Functional Analysis vol 267 no 7 pp2056ndash2076 2014

[4] K Feledziak andM Nowak ldquoLocally solid topologies on vectorvalued function spacesrdquo Universitat de Barcelona CollectaneaMathematica vol 48 no 4-6 pp 487ndash511 1997

[5] K Feledziak and M Nowak ldquoIntegral representation of linearoperators on Orlicz-Bochner spacesrdquo Universitat de BarcelonaCollectanea Mathematica vol 61 no 3 pp 277ndash290 2010

[6] M Nowak ldquoLinear operators on vector-valued function spaceswithMackey topologiesrdquo Journal of Convex Analysis vol 15 no1 pp 165ndash178 2008

[7] M Nowak ldquoLebesgue topologies on vector-valued functionspacesrdquoMathematica Japonica vol 52 no 2 pp 171ndash182 2000

[8] A V Buhvalov ldquoThe analytic representation of operators withan abstract normrdquo Izvestiya Vysshikh Uchebnykh ZavedeniiMatematika vol 11 no 162 pp 21ndash32 1975

[9] A V Buhvalov ldquoThe analytic representation of linear operatorsby means of measurable vector-valued functionsrdquo IzvestiyaVysshikh Uchebnykh Zavedenii Matematika vol 7 no 182 pp21ndash31 1977

[10] J Diestel and J J Uhl ldquoVector measuresrdquo vol 15 AmericanMathematical Society Surveys Providence RI USA 1977

[11] M Nowak ldquoDuality theory of vector-valued function spaces IrdquoCommentationes Mathematicae vol 37 pp 195ndash215 1997

[12] A V Bukhvalov and G Ya Lozanovskii ldquoOn sets closed inmeasure in spaces of mesurable functionsrdquo Transactions of theMoscow Mathematical Society vol 2 pp 127ndash148 1978

[13] A Wiweger ldquoLinear spaces with mixed topologyrdquo StudiaMathematica vol 20 pp 47ndash68 1961

[14] K Feledziak ldquoUniformly Lebesgue topologies on Kothe-Bochner spacesrdquoCommentationesMathematicae vol 37 pp 81ndash98 1997

[15] W Ruess ldquo[Weakly] compact operators and DF-spacesrdquo PacificJournal of Mathematics vol 98 no 2 pp 419ndash441 1982

[16] M Nowak ldquoCompactness of Bochner representable operatorson Orlicz spacesrdquo Positivity An International MathematicsJournal Devoted to Theory and Applications of Positivity vol 13no 1 pp 193ndash199 2009

[17] N E Gretsky and J Uhl ldquoBounded linear operators on Banachfunction spaces of vector-valued functionsrdquo Transactions of theAmerican Mathematical Society vol 167 pp 263ndash277 1972

[18] J J Uhl ldquoCompact operators on Orlicz spacesrdquo in Rendicontidel Seminario Matematico vol 42 pp 209ndash219 University ofPadova 1969

[19] N Dinculeanu Vector Measures International Series of Mono-graphs in Pure and Applied Mathematics 95 Pergamon PressNew York NY USA 1967

[20] N Dinculeanu Linear Operators on Lp-Spaces Vector andOperator-Valued Measures and Applications Academic PressNew York NY USA 1973

[21] N Dinculeanu Vector Integration and Stochastic Integration inBanach Spaces John Wiley amp Sons 2000

[22] R A Alo A De Korwin and L E Kunes ldquoTopological aspectsof 119902-regular measuresrdquo Studia Mathematica vol 48 pp 49ndash601973

[23] J Diestel ldquoOn the representation of bounded linear operatorsfrom Orlicz spaces of Lebesgue-Bochner measurable functionsto any Banach spacerdquo Bulletin de lrsquoAcademie Polonaise desSciences Serie des Sciences Mathematiques Astronomiques etPhysiques vol 18 pp 375ndash378 1970

[24] L Vanderputten ldquoRepresentation of operators defined on thespace of Bochner integrable functionsrdquo Extracta Mathematicaevol 16 no 3 pp 383ndash391 2001

[25] R FWheeler ldquoA survey of Bairemeasures and strict topologiesrdquoExpositiones Mathematicae vol 1 no 2 pp 97ndash190 1983

[26] W A Luxemburg Banach function spaces [Thesis] DelftNetherlands 1955

[27] P W Lewis ldquoVector measures and topologyrdquo Revue RoumainedeMathematique Pures et Appliquees vol 16 pp 1201ndash1209 1971

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Journal of Function Spaces 9

= 1003816100381610038161003816100381610038161003816100381610038161003816119879lowast (119910lowast120573 minus 119910lowast0 )(

119899sum119894=1

(1119860119894 otimes 119909119894))1003816100381610038161003816100381610038161003816100381610038161003816

+ 120576le 10038171003817100381710038171003817119879lowast (119910lowast120573 minus 119910lowast0 )10038171003817100381710038171003817lowast120593

1003817100381710038171003817100381710038171003817100381710038171003817119899sum119894=1

(1119860119894 otimes 119909119894)1003817100381710038171003817100381710038171003817100381710038171003817120593

+ 120576 le 10038171003817100381710038171003817119879lowast (119910lowast120573) minus 119879 (119910lowast0 )10038171003817100381710038171003817lowast120593 + 120576(67)

Hence 119901119898(119910lowast120573 minus 119910lowast0 ) = (119898119910lowast120573minus119910lowast0)120593lowast(Ω) rarr

1205730 and this means

that the space (119861119884lowast 120575119898120593lowast) is compact

As a consequence of Theorems 17 and 15 we have thefollowing

Corollary 18 Assume that 120593 is a Young function such that 120593lowastsatisfies theΔ 2-condition Let119879 119871120593(119883) rarr 119884 be a (Tand

120593 sdot119884)-continuous linear operator and 119898 Σ119891(120583) rarr L(119883 119884) be itsrepresenting measure If the space (119861119884lowast 120575119898120593lowast) is compact then119898 is 120593lowast-semivariationally 120583-continuousConflicts of Interest

The author declares that there are no conflicts of interest

References

[1] S Chen and H Hudzik ldquoOn some convexities of Orlicz andOrlicz-Bochner spacesrdquo Commentationes Mathematicae vol29 no 1 pp 13ndash29 1988

[2] P Kolwicz and R Pluciennik ldquo119875-convexity of Orlicz-Bochnerspacesrdquo Proceedings of the American Mathematical Society vol126 no 8 pp 2315ndash2322 1998

[3] S Shang and Y Cui ldquoUniform nonsquareness and locallyuniform nonsquareness in Orlicz-Bochner function spaces andapplicationsrdquo Journal of Functional Analysis vol 267 no 7 pp2056ndash2076 2014

[4] K Feledziak andM Nowak ldquoLocally solid topologies on vectorvalued function spacesrdquo Universitat de Barcelona CollectaneaMathematica vol 48 no 4-6 pp 487ndash511 1997

[5] K Feledziak and M Nowak ldquoIntegral representation of linearoperators on Orlicz-Bochner spacesrdquo Universitat de BarcelonaCollectanea Mathematica vol 61 no 3 pp 277ndash290 2010

[6] M Nowak ldquoLinear operators on vector-valued function spaceswithMackey topologiesrdquo Journal of Convex Analysis vol 15 no1 pp 165ndash178 2008

[7] M Nowak ldquoLebesgue topologies on vector-valued functionspacesrdquoMathematica Japonica vol 52 no 2 pp 171ndash182 2000

[8] A V Buhvalov ldquoThe analytic representation of operators withan abstract normrdquo Izvestiya Vysshikh Uchebnykh ZavedeniiMatematika vol 11 no 162 pp 21ndash32 1975

[9] A V Buhvalov ldquoThe analytic representation of linear operatorsby means of measurable vector-valued functionsrdquo IzvestiyaVysshikh Uchebnykh Zavedenii Matematika vol 7 no 182 pp21ndash31 1977

[10] J Diestel and J J Uhl ldquoVector measuresrdquo vol 15 AmericanMathematical Society Surveys Providence RI USA 1977

[11] M Nowak ldquoDuality theory of vector-valued function spaces IrdquoCommentationes Mathematicae vol 37 pp 195ndash215 1997

[12] A V Bukhvalov and G Ya Lozanovskii ldquoOn sets closed inmeasure in spaces of mesurable functionsrdquo Transactions of theMoscow Mathematical Society vol 2 pp 127ndash148 1978

[13] A Wiweger ldquoLinear spaces with mixed topologyrdquo StudiaMathematica vol 20 pp 47ndash68 1961

[14] K Feledziak ldquoUniformly Lebesgue topologies on Kothe-Bochner spacesrdquoCommentationesMathematicae vol 37 pp 81ndash98 1997

[15] W Ruess ldquo[Weakly] compact operators and DF-spacesrdquo PacificJournal of Mathematics vol 98 no 2 pp 419ndash441 1982

[16] M Nowak ldquoCompactness of Bochner representable operatorson Orlicz spacesrdquo Positivity An International MathematicsJournal Devoted to Theory and Applications of Positivity vol 13no 1 pp 193ndash199 2009

[17] N E Gretsky and J Uhl ldquoBounded linear operators on Banachfunction spaces of vector-valued functionsrdquo Transactions of theAmerican Mathematical Society vol 167 pp 263ndash277 1972

[18] J J Uhl ldquoCompact operators on Orlicz spacesrdquo in Rendicontidel Seminario Matematico vol 42 pp 209ndash219 University ofPadova 1969

[19] N Dinculeanu Vector Measures International Series of Mono-graphs in Pure and Applied Mathematics 95 Pergamon PressNew York NY USA 1967

[20] N Dinculeanu Linear Operators on Lp-Spaces Vector andOperator-Valued Measures and Applications Academic PressNew York NY USA 1973

[21] N Dinculeanu Vector Integration and Stochastic Integration inBanach Spaces John Wiley amp Sons 2000

[22] R A Alo A De Korwin and L E Kunes ldquoTopological aspectsof 119902-regular measuresrdquo Studia Mathematica vol 48 pp 49ndash601973

[23] J Diestel ldquoOn the representation of bounded linear operatorsfrom Orlicz spaces of Lebesgue-Bochner measurable functionsto any Banach spacerdquo Bulletin de lrsquoAcademie Polonaise desSciences Serie des Sciences Mathematiques Astronomiques etPhysiques vol 18 pp 375ndash378 1970

[24] L Vanderputten ldquoRepresentation of operators defined on thespace of Bochner integrable functionsrdquo Extracta Mathematicaevol 16 no 3 pp 383ndash391 2001

[25] R FWheeler ldquoA survey of Bairemeasures and strict topologiesrdquoExpositiones Mathematicae vol 1 no 2 pp 97ndash190 1983

[26] W A Luxemburg Banach function spaces [Thesis] DelftNetherlands 1955

[27] P W Lewis ldquoVector measures and topologyrdquo Revue RoumainedeMathematique Pures et Appliquees vol 16 pp 1201ndash1209 1971

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom