HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS€¦ · Muhammed Ali Alan and Nihat Gökhan Gö§ü³ y z Abstract We introduce new spaces that are extensions of the Hardy spaces

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HACETTEPE JOURNAL OF

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2017

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Hacettepe Journal of Mathematics and Statistics

A Bimonthly Publication � Volume 46 Issue 4 (2017)

ISSN 1303 � 5010

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CONTENTS

Mathematics

Muhammed Ali Alan and Nihat Gökhan Gö§ü³

A new class of Hardy spaces in the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559

M. Lashkarezadeh Bami and H. Sadeghi

A generalization of amenability for topological semigroups and

semigroup algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

Ömür DEVEC and Erdal KARADUMAN

On the Padovan p-numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

Abdoul Salam Diallo, Silas Longwap and Fortuné Massamba

On twisted Riemannian extensions associated with Szabó metrics . . . . . . . . . . . .593

Ramin Kazemi and Ali Behtoei

The �rst Zagreb and forgotten topological indices of d-ary trees . . . . . . . . . . . . . .603

Chris Lennard and Veysel Nezir

Semi-strongly asymptotically non-expansive mappings and

their applications on �xed point theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613

Marian Przemski

Decompositions of continuity for multifunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621

Zhen-Hang Yang

On converses of some comparison inequalities for homogeneous means . . . . . . 629

Statistics

Morad Alizadeh, Faton Merovci and G.G. Hamedani

Generalized transmuted family of distributions:

properties and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645

Al Kadiri M. A.

Linear penalized spline model estimation using

ranked set sampling technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .669

Hatice Oncel Cekim and Hulya Cingi

Some estimator types for population mean using linear transformation

with the help of the minimum and maximum values of

the auxiliary variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685

A.Chatterjee, G.N.Singh, A. Bandyopadhyay and P.Mukhopadhyay

A general procedure for estimating population variance

in successive sampling using fuzzy tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695

Fatih K�z�laslan and Mustafa Nadar

Statistical inference of P (X < Y ) for the Burr Type XII distributionbased on records . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713

Saba Riaz, Giancarlo Diana and Javid Shabbir

Modi�ed classes of estimators in circular systematic sampling . . . . . . . . . . . . . . . 743

M. H. Tahir, M. Zubair, Gauss M. Cordeiro, Ayman Alzaatreh andM. Mansoor

The Weibull-Power Cauchy distribution:

model, properties and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767

MATHEMATICS

Hacettepe Journal of Mathematics and StatisticsVolume 46 (4) (2017), 559 � 565

A new class of Hardy spaces in the plane

Muhammed Ali Alan∗ and Nihat Gökhan Gö§ü³† ‡

Abstract

We introduce new spaces that are extensions of the Hardy spaces andprove a removable singularity result for holomorphic functions withinthese spaces. Additionally we provide non-trivial examples.

Keywords: Hardy space, Removable singularity.

2000 AMS Classi�cation: Primary 30H10, Secondary 30J99 AMS

Received : 01.08.2016 Accepted : 10.12.2016 Doi : 10.15672/HJMS.20174622759

1. Introduction

This paper deals with a construction of a holomorphic function space on an arbitraryopen connected subset of the complex plane C. In this paper we suggest a method ofconstructing a function spaceW p in any arbitrary domain. The de�nition of the norm onW p makes use of growth information of the function locally in the domain. We show thatW p is Banach when p ≥ 1 and prove a removable singularity theorem. This generalizesthe result of M. Parreau in [8]. In the de�nition of W p we make use of the recentlystudied Poletsky-Stessin-Hardy (PSH) spaces. These spaces were introduced in severalcomplex variables context in [9] and recently studied in planar domains in [1] and for thedisk in the papers [10] and [11].

In general, PSH norm depends on the choice of the subharmonic exhaustion functionwhich exists only when the domain is regular with respect to the classical Dirichletproblem. Our motivation for such a construction comes from the question that whichsubspaces of the classical Hardy space Hp can be obtained as a Poletsky-Stessin-Hardyspace. For example the subspace zHp of Hp is not a Poletsky-Stessin-Hardy spacebecause if the function z belongs to this space, then so does the constant function 1.However we show in section 4 that B(z)Hp can be viewed as a W p space when B is a�nite Blaschke product.

∗Syracuse University, Syracuse, NY 13244, USA, Email: [email protected]†Sabanci University, Tuzla , Istanbul 34956 TURKEY, Email: [email protected]‡Corresponding Author.This work was supported by a TUBITAK grant project 113F301.

560

2. Poletsky-Stessin-Hardy spaces

A function u ≤ 0 on a bounded open set G ⊂ C is called an exhaustion on G if the setBc,u := {z ∈ G : u(z) < c}

is relatively compact in G for any c < 0. We denote the class of harmonic functionsand subharmonic functions on a domain G by har(G) and sh(G), respectively. It isknown that there is a subharmonic exhaustion function on G if and only if G is regularwith respect to the classical Dirichlet problem. Let us denote the class of continuoussubharmonic exhaustion functions on a domain G by E(G). If u is an exhaustion andc < 0 is a number, we set

uc := max{u, c}, Sc,u := {z ∈ G : u(z) = c}.Since uc is a continuous subharmonic function the measure ∆uc is well-de�ned. FollowingDemailly [2] we de�ne

µc,u := ∆uc − χG\Bc,u∆u,where χω is the characteristic function of a set ω ⊂ G. We shall call these measures asDemailly measures.

If u is a negative subharmonic exhaustion function on G, then the Demailly-Lelong-Jensen formula takes the form

(2.1)

∫

Sc,u

v dµc,u =

∫

Bc,u

(v∆u− u∆v) + c∫

Bc,u

∆v,

where µc,u is the Demailly measure which is supported in the level sets Sc,u of u andv ∈ sh(G). This formula is the one variable version of the result which was proved byDemailly [2]. Let us recall that by [2] if

∫G

∆u

561

2.1. Theorem. Let G be a bounded domain and u ∈ E(G). Let p > 0. The followingstatements are equivalent:

i. f ∈ Hpu(G).ii. There exists a least harmonic function h in G which belongs to the class shu so

that |f |p ≤ h on G. Furthermore,

‖f‖pu,p =∫

G

h∆u = ‖h‖u.

Now let G be a bounded domain with C1 boundary or a bounded simply connecteddomain with recti�able boundary. Let u ∈ E(G) and p ≥ 1 (p > 0 if G is simplyconnected). Then the space Hpu(G) (thinking of boundary values) is a closed subspaceof the weighted space Lp(Vudσ) on the boundary ∂G, where (see [1]) dσ is the usualLebesgue measure on ∂G and

Vu(ζ) =

∫

G

PG(z, ζ)∆u(z), ζ ∈ ∂G

is the balayage of the positive measure ∆u to the boundary ∂G. Then Vu(ζ) =∂u∂n

(ζ)is the directional derivative of u in the normal direction at a point ζ ∈ ∂G (see [4] and[11]). The next results are restatements from [9] and they establish basic observationson the classes of Hardy spaces.

2.2. Proposition. [9, Corollary 3.2] Let v be a continuous subharmonic exhaustionfunction on a bounded regular domain G and let v(z) = g(z, w) be the Green function.Then shpu(G) ⊂ shpv(G) and there is a constant c such that ‖ϕ‖v ≤ c‖ϕ‖u for everynonnegative subharmonic function ϕ on G.

2.3. Proposition. [9, Corollary 3.2] Let u and v be continuous subharmonic exhaustionfunctions on G and let K be a compact set in G such that bv(z) ≤ u(z) for some constantb > 0 and all z ∈ G\K . Then shv ⊂ shu and ‖ϕ‖u ≤ b‖ϕ‖v for every ϕ ∈ shv.

The following result is basically contained in the proof of [9, Theorem 3.6] takingn = 1.

2.4. Proposition. Let v be a continuous subharmonic exhaustion function on G, K ⊂ Gbe compact and V ⊂⊂ G be an open set containing K. Suppose that there exists aconstant s > 0 so that v(z) ≤ sgG(z, w) for every w ∈ K and z ∈ G\V . Then

ϕ(w) ≤ s2π‖ϕ‖v, w ∈ K

for every nonnegative ϕ ∈ sh(G).

3. Hardy spaces in arbitrary open sets

In this section we propose a way to de�ne weighted Hardy spaces in arbitrary planardomains. For Hardy spaces in multiply connected domains we refer to [3]. Let us set thenotation �rst. Let Ω be a domain, E ⊂ Ω be a compact polar subset and let Ωj be asequence of regular domains so that Ωj ⊂ Ω and the union of all Ωj is the open set Ω\E.Also for each j let uj ∈ E(Ωj), that is, uj is a subharmonic exhaustion function for Ωj .We de�ne the class W p of holomorphic functions on Ω as follows:

W p := {f ∈ hol(Ω) : supj‖f‖uj ,p

562

for any f ∈ W p. We will write W p[uj ,Ω, E] if we wish to emphasize the sequence offunctions uj used in the de�nition or the underlying domain Ω and the polar set E. Beforeshowing that (W p, ‖‖Wp) is a Banach space we need the following removable singularitytheorem for bounded holomorphic functions due to Lelong.

3.1. Theorem. [7, p.35], [5, p. 107] Let E be a relatively closed pluri-polar set and letf be holomorphic in Ω\E. Suppose that f is bounded on Ω\E. Then f has a uniqueholomorphic extension to the whole of Ω.

We prove an auxiliary result.

3.2. Theorem. Let fn be a holomorphic function on a domain Ω and E be a compactpolar set in Ω. Suppose that fn converges uniformly to a function f on compact subsetsof Ω\E. Then the function f can be extended to a holomorphic function on Ω.

Proof. Let Γ be a bounded open region in Ω with piecewise smooth boundary γ so thatE ⊂ Γ ⊂ Γ ⊂ Ω. Since |fn| converges uniformly to |f | on γ, we see that supn |fn| isuniformly bounded on γ, that is, there exists a number M so that |fn| ≤ M on γ forevery n. We write PΓϕ for the Poisson integral of a continuous function ϕ on γ. Then

|fn(z)| ≤ PΓ|fn|(z) ≤Mfor every n for every z ∈ Γ. Therefore |f(z)| ≤ M for every z ∈ Γ\E. By Theorem3.1 f has a holomorphic extension to Γ. Since f is already holomorphic outside of Γ weconclude that f can be extended to a holomorphic function on Ω. �

We can now prove that (W p, ‖‖Wp) is Banach.3.3. Theorem. (W p, ‖‖Wp) is a Banach space for p ≥ 1.

Proof. If ‖f‖Wp = 0, then ‖f‖uj ,p = 0, that is why f = 0 in Ωj for every j. Hence f = 0on Ω\E, and since E is polar, f = 0 on Ω. The other properties of norm can be easilychecked for ‖f‖Wp . So let us prove that it is complete. Take a Cauchy sequence {fn}from W p. This implies �rst that the sequence of holomorphic functions {fn} is Cauchyin Hpuj for every j. We conclude that fn converges uniformly to a function f on every

compact subset of Ωj for each j, hence on every compact subset of Ω\E. By Theorem3.2 f extends to a holomorphic function to the whole of Ω.

To prove that f ∈ W p we will now show that ‖fn − f‖Wp converges to zero. Givenε > 0 there exists an integer N ≥ 1 so that

supj‖fn − fm‖uj ,p < ε

whenever n, m ≥ N . This gives that‖fn − f‖Wp = sup

j‖fn − f‖uj ,p ≤ ε

for every n ≥ N . Therefore ‖fn − f‖Wp converges to zero and f ∈W p. �

It is known that a polar set is a removable singularity for the classical Hardy spaces inthe plane (see [6] and [8]). The next result can be considered as a removable singularitytheorem for theW p spaces. There byW p[uj ,Ω, E]|Ω\E we denote the class of restrictionsof the functions from W p[uj ,Ω, E] to Ω\E.

3.4. Theorem. Let Ωj ⊂ Ωj+1, E ⊂ Ω be a compact polar set for every j and let p > 0.If there exists an open set U ⊂ Ω\E so that supj uj(z) ≤ ` < 0 for every z ∈ U , then

W p[uj ,Ω, E]|Ω\E = W p[uj ,Ω\E, ∅].

563

Proof. The inclusion W p[uj ,Ω, E]|Ω\E ⊂ W p[uj ,Ω\E, ∅] is immediate from the de�ni-tions. To prove the reverse inclusion we claim that if f belongs to the spaceW p[uj ,Ω\E, ∅],then f extends to a holomorphic function to the whole set Ω. Since f ∈ Hpuj (Ωj), ac-cording to Theorem 2.1 we see that the function hj := PΩj (|f |p) has the properties thathj ∈ har ∩ shuj (Ωj) and that |f |p ≤ hj on Ωj . Then hj ≤ hj+1, and thanks to theHarnack theorem the limit h = limhj is a harmonic function on Ω\E unless h = ∞identically everywhere. Let z0 ∈ U and r > 0 so that {z : |z − z0| ≤ r} ⊂ U . We claimthat h(z0) < ∞. There exists a constant s > 0 so that ` < sgΩ(z, z0) ≤ sgΩj (z, z0)for every z ∈ Ωj with |z − z0| = r, j ≥ 1. Harmonicity of the Green's functiongΩj (z, z0) on Ωj\{z : |z − z0| ≤ r} implies that uj(z) ≤ ` < sgΩj (z, z0) for everyz ∈ Ωj\{z : |z − z0| ≤ r} for every j ≥ 1. By Theorem 2.1 and Proposition 2.4 we seethat

2π

shj(z0) ≤ ‖hj‖p,uj = ‖f‖p,uj ≤ ‖f‖Wp[uj ,Ω\E,∅] 0, de�ne a subharmonic exhaustion function ut in ΓR by

u(z) := ut(z) := ut,R(z) := max

{t log

(R

|z|

), log |z|

}.

Some properties of ut are listed below.

(1) ut(z) = 0, if |z| = 1 or |z| = R.(2) We solve t log

(R|z|

)= log |z| to get |z| = Rt/t+1 and hence

ut(z) =

{t log

(R|z|

)if R < |z| ≤ Rt/t+1;

log |z|, if Rt/t+1 < |z| < 1.(3) We compute the measure µu of u.

Vu(eiθ) =

∂u

∂n|z=eiθ = 1

and

Vu(Reiθ) =

∂u

∂n|z=Reiθ = t/R

for every θ ∈ [0, 2π]. Hence, for any positive measurable function ϕ on ∂ΓR wehave∫

∂ΓR

ϕdµu =t

2πR

∫ 2π

0

ϕ(Reiθ)dθ +1

2π

∫ 2π

0

ϕ(eiθ)dθ.

564

Now we are ready to state the main purpose of this example.

4.2. Theorem. Let Hp be the classical Hardy space in the unit disc for p > 0 and k ≥ 1be an integer. Let (Rn) be any sequence of numbers converging to 0 so that 0 < Rn < 1.Take α so that 1− kp ≤ α < 1− kp+ p. Then we have

zkHp = W p[uRαn ,Rn ,D, {0}]and two spaces have equivalent norms.

Proof. Let W p = W p[uRαn ,Rn ,D, {0}]. If h ∈ zkHp, we will show that h ∈ W p. Leth = zkf , where f ∈ Hp. Then

‖zkf‖pWp = supn‖zkf‖puRαn,p

= supn

(Rα+kp−1n

2π

∫ 2π

0

|f(Rneiθ)|pdθ + 12π

∫ 2π

0

|f(eiθ)|pdθ)

≤ supn

(Rα+kp−1n + 1)‖f‖pHp ≤ 2‖f‖pHp 0. Then

‖h‖Wp ≥ supn

Rα+(k−1)p−1n

2π

∫ 2π

0

|f(Rneiθ)|pdθ

≥ supnRα+(k−1)p−1n |f(0)|p =∞.

The contradiction shows that h(z) = zkf(z) for some f ∈ Hp. Hence W p = zkHp. �

Finally we can do the previous construction for �nite Blaschke products.

4.3. Theorem. Let a1, · · · , aN be distinct points in D and let

B(z) :=

N∏

j=1

(z − aj1− ajz

)kj,

where kj ≥ 1 are integers. Let p > 0. Then there exists a sequence {Ωn} ⊂ D ofN + 1-connected domains and functions un ∈ E(Ωn) so that

B(z)Hp = W p[un,D]

and two spaces have equivalent norms.

Proof. Choose R > 0 small enough so that the circles

Cj =

{z :

∣∣∣∣z − aj1− ajz

∣∣∣∣kj

= R

},

j = 1, · · · , N, are pairwise disjoint. Let ΩR be theN+1-connected domain with boundary∂D∪∪Nj=1Cj . For each j choose αj so that −kjp ≤ αj < −kjp+p. Let ψR be the functionde�ned on ∂D by 1, on Cj by Rα. Then ψR is lower semicontinuous on ΩR, ψR ≥ tR > 0for some constant t = tR and by [4, Theorem 2.1] there exists a subharmonic exhaustionuR ∈ E(ΩR) so that ∂uR/∂n = ψR on ∂ΩR.

565

Now let 0 < Rn < R be numbers decreasing to 0 and consider the space Wp =

W p[uRn ,D]. If h = Bf ∈ B(z)Hp, then

‖Bf‖pWp = supn

(N∑

j=1

Rαj+kjpn

∫

Cj

|f(ζ)|pdσj + 12π

∫ 2π

0

|f(eiθ)|pdθ)

≤ supn

(

N∑

j=1

Rαj+kjpn + 1)‖f‖pHp ≤ (N + 1)‖f‖pHp 0 and therefore,

‖h‖Wp ≥ supnRαj+(kj−1)pn

∫

Cj

|f(ζ)|pdσj

≥ C supnRαj+(kj−1)pn |f(aj)|p =∞.

The contradiction shows that h(z) = B(z)f(z) for some f ∈ Hp. Hence W p = B(z)Hp.�

References

[1] Alan, M. A. and Gö§ü³, N. G. Poletsky-Stessin-Hardy spaces in the plane, Complex Anal.Oper. Theory 8 (2014), no. 5, 975�990.

[2] Demailly, J. P. Mesure de Monge-Ampère et mesures plurisousharmonique, Math. Z. 194(1987), 519�564.

[3] Duren, P. L. Theory of Hp spaces, Pure and Applied Mathematics, Vol. 38 Academic Press,New York-London 1970.

[4] Gö§ü³, N. G. Structure of weighted Hardy spaces in the plane, Filomat 30, (2016) no. 2,473-482.

[5] Jarnicki, P. and P�ug, P. Extension of Holomorphic Functions, De Gruyter Expositions inMathematics 34

[6] Järvi, P. Removable singularities for Hp-functions, Proc. Amer. Math. Soc. 86 (1982), 596�598.

[7] Lelong, P. Fonctions plurisousharmoniques et formes di�erentielles positives, Gordon andBreach, New York, 1968.

[8] Parreau, M. Sur les moyennes des fonctions harmoniques et analytiques et la classi�cationdes surfaces de Riemann, Ann. Inst. Fourier (Grenoble) 3 (1951), 103�197.

[9] Poletsky, E. A. and Stessin, M. I. Hardy and Bergman spaces on hyperconvex domains andtheir composition operators, Indiana Univ. Math. J. 57 (2008), no. 5, 2153�2201.

[10] ahin, S., Poletsky-Stessin Hardy spaces on domains bounded by an analytic Jordan curvein C, Complex Var. Elliptic Equ. 60 (2015), no. 8, 1114�1132

[11] Shrestha, K. R. Weighted Hardy spaces on the unit disk, Complex Anal. Oper. Theory 9(2015), no. 6, 1377�1389.


A generalization of amenability for topologicalsemigroups and semigroup algebras

M. Lashkarezadeh Bami∗ and H. Sadeghi†

Abstract

In this paper for two topological semigroups S and T , and a continuoushomomorphism ϕ from S into T , we introduce and study the conceptof (ϕ, T )-derivations on S and ϕ-amenability of T and investigate therelations between these two concepts. For two Banach algebras A and Band a continuous homomorphism ϕ from A into B we also introduce thenotion of (ϕ,B)-amenability of A and show that a foundation semigroupT with identity is ϕ-amenable whenever the Banach algebra Ma(S) is(ϕ̃,Ma(T ))-amenable, where ϕ̃ : M(S) −→ M(T ) denotes the uniqueextension of ϕ. An example is given to show that the converse is nottrue.

Keywords: Continuous homomorphism, semigroup, Banach algebra,

(ϕ, T )-derivation, ϕ-amenable.

2000 AMS Classi�cation: primary 43A07; secondary 43A10, 46H20.


1. Introduction

The concept of amenability for Banach algebras was initiated by Johnson in [9]. Heshowed that a locally compact Hausdor� group G is amenable if and only if the Banachalgebra L1(G) is amenable. This fails to be true for discrete semigroups. Duncan andNomioka [4] proved that if l1(S) is amenable then S is amenable and l1(S) fails to beamenable if ES is in�nite. Johonson proved that H

1(L1(G), X∗) = {0} if and only ifevery G-derivation into X∗ is inner, where X is a neo-unital Banach L1(G)-bimodule(see [2] and [9]).

Recently, Kaniuth, Lau and Pym introduced ϕ-amenability of a Banach algebra Awhere ϕ is a homomorphism from A to C [10]. Here for two Banach algebras A andB we study the concept of (ϕ,B)-amenability of A where ϕ : A −→ B is a continuoushomomorphism. In the case where A = B, A is called ϕ-amenable. Several authors have

∗M. Lashkarizadeh Bami, Email: [email protected]†H. Sadeghi, Email: [email protected]

568

studied ϕ-derivations, and ϕ-amenability of a Banach algebra A (see [7], [8], [15] and[16]).

Authors in [7], introduced the notion of ϕ-amenability for a locally compact group G,where ϕ is a continuous homomorphism on G. They proved that if the group algebraL1(G) is ϕ̃-amenable, then G is ϕ-amenable and when ϕ is an isomorphism on G, theconverse is valid. Here ϕ̃ is the unique extension of ϕ to M(G).

In this paper for two Banach algebras A and B and a continuous homomorphism ϕ :A −→ B, we �rst introduce the notion of (ϕ,B)-amenability of A. This concept reducesto that of φ-amenability introduced by Kaniuth, Lau, and Pym, when B = C. Also fortwo topological semigroups S and T , and a continuous homomorphism ϕ from S into T ,we introduce and study the concept of (ϕ, T )-derivations on S and ϕ-amenability of T andinvestigate the relation between these two concepts. Then we apply our results to the casewhere S and T are foundation semigroups and prove thatMa(S) is (ϕ̃,Ma(T ))-amenableif and only if every (ϕ, T )-derivation on S is ϕ-inner, where ϕ̃ : Ma(S) −→ Ma(T )denotes the extension of ϕ. This extends a known result due to Johonson for groups tofoundation semigroups. Finally, we show that (ϕ̃,Ma(T ))-amenability of Ma(S) impliesϕ-amenability of T , and present an example to show that the converse is not true.

2. Preliminaries

Let A be a Banach algebra, and let X be an A-bimodule. Then X is a BanachA-bimodule if X is a Banach space and there is a constant k > 0 such that

‖a.x‖ ≤ k‖a‖‖x‖, ‖x.a‖ ≤ k‖a‖‖x‖ (a ∈ A, x ∈ X).By renorming, we can suppose that k = 1. For example, A itself is a Banach A-bimodule,and X∗, the dual space of a Banach A-bimodule X, is a Banach A-bimodule if for everya ∈ A and f ∈ X∗ we de�ne

〈x, a.f〉 = 〈x.a, f〉, 〈x, f.a〉 = 〈a.x, f〉 (x ∈ X).We say that X∗ is the dual module of X.

Suppose that A is a Banach algebra and X is a Banach A-bimodule. A derivationfrom A into X is a linear operator D : A −→ X satisfying

D(ab) = D(a)b+ aD(b) (a, b ∈ A).A derivation D is inner if there is x0 ∈ X such that D(a) = a.x0 − x0.a for a ∈ Aand a Banach algebra A is amenable if for any Banach A-bimodule X, every continuousderivation D : A −→ X∗ is inner.

Let A and B be two Banach algebras. The set of continuous homomorphisms from Ainto B is denoted by Hom(A,B). We denote the set Hom(A,A) by Hom(A).

Suppose that ϕ : A −→ B is a continuous homomorphism. A Banach space X over Cis a Banach (ϕ,B)-bimodule if it is two-sided ϕ(A)-module and there is a positive realnumber K such that

‖ϕ(a).x‖ ≤ K‖ϕ(a)‖B‖x‖ ‖x.ϕ(a)‖ ≤ K‖x‖‖ϕ(a)‖B ,for all a ∈ A and x ∈ X.

Let X be a Banach (ϕ,B)-bimodule and ϕ ∈ Hom(A,B), a linear operator D : A −→X is called a (ϕ,B)-derivation if

D(a1a2) = D(a1).ϕ(a2) + ϕ(a1).D(a2) (a1, a2 ∈ A).A (ϕ,B)-derivation D is called (ϕ,B)-inner if there is x ∈ X such that D(a) = ϕ(a).x−x.ϕ(a) (a ∈ A). A Banach algebra A is called (ϕ,B)-amenable if for any Banach (ϕ,B)-bimodule X, every continuous (ϕ,B)-derivation D : A −→ X∗ is (ϕ,B)-inner. In thecase that A = B, D is called ϕ-derivation and A is called ϕ-amenable.

569

3. ϕ-derivation and ϕ-amenable semigroups

We commence this section with the following de�nition:

3.1. De�nition. Let S and T be two topological semigroups and ϕ : S −→ T be acontinuous homomorphism of S into T . We say that the complex Banach space X is aleft Banach ϕ-module if there exists a mapping

ϕ(S)×X −→ X, (ϕ(s), x) −→ ϕ(s).x,having the following properties:

(i) ϕ(s).(x1 + x2) = ϕ(s).x1 + ϕ(s).x2, λ(ϕ(s).x

)= ϕ(s).(λx),

ϕ(s1s2).x = ϕ(s1).(ϕ(s2).x

)for all s, s1, s2 ∈ S, x1, x2 ∈ X and λ ∈ C,

(ii) if sα −→ s in S and x ∈ X, then ϕ(sα).x −→ ϕ(s).x, in the norm topology, and(iii) there is M > 0 such that for every x ∈ X and s ∈ S, we have

‖ϕ(s).x‖ ≤M‖x‖.In the same way, one de�nes a right Banach ϕ-module. The (two sided) Banach ϕ-moduleX is a left and right Banach ϕ-module such that

(ϕ(s1).x

).ϕ(s2) = ϕ(s1).

(x.ϕ(s2)

)(s1, s2 ∈ S, x ∈ X).

Note that if X is a Banach ϕ-module, then X∗, the dual space of X, is also anϕ-module through the following actions:

〈ϕ(s).x∗, x〉 = 〈x∗, x.ϕ(s)〉, 〈x∗.ϕ(s), x〉 = 〈x∗, ϕ(s).x〉 (s ∈ S, x ∈ X,x∗ ∈ X∗).A left (resp. right) action of ϕ(S) on X is trivial if ϕ(s).x = x (s ∈ S, x ∈ X) (resp.x.ϕ(s) = x (s ∈ S, x ∈ X)).

3.2. De�nition. Let S and T be two topological semigroups and ϕ : S −→ T bea continuous homomorphism of S into T . Let X be a Banach ϕ-module. A weak*-continuous map D : S −→ X∗ is called a (ϕ, T )-derivation (ϕ-derivation in the case thatS = T ) if

(i) D(s1s2) = ϕ(s1).D(s2) +D(s1).ϕ(s2) (s1, s2 ∈ S);(ii) sups∈S‖D(s)‖

570

3.4. De�nition. For two topological semigroups S and T , and continuous homomor-phism ϕ : S −→ T , T is called ϕ-left amenable ( resp. ϕ-right amenable) if there is aϕ-left invariant mean ( resp. ϕ-right invariant mean) on LUC(T) (resp. on RUC(T)). Asemigroup T is called ϕ-amenable if it is both ϕ-left and ϕ-right amenable.

The proof of the following lemma is straightforward.

3.5. Lemma. Let S and T be two topological semigroups, and ϕ : S −→ T be a contin-uous homomorphism. If T is amenable then T is ϕ-amenable. The converse is true ifϕ(S) is dense in T .

3.6. Proposition. Let S and T be two topological semigroups, and ϕ : S −→ T bea continuous homomorphism. If for every Banach ϕ-module X, any (ϕ, T )-derivationD : S −→ X∗ is ϕ-inner, then T is ϕ-amenable.

Proof. We �rst note that LUC(T) is a Banach ϕ-module through the following actionsgiven by

ϕ(s).f = f,(f.ϕ(s)

)(t′) = f

(ϕ(s)t′

) (s ∈ S, t′ ∈ T, f ∈ LUC(T )

).

Let n ∈ LUC(T)∗ such that 〈1, n〉=1. De�ne d : S −→ LUC(T)∗ by d(s) = ϕ(s).n − n.It is easy to see that C1T ia a closed submodule of LUC(T). Let X = LUC(T)C1T . Sincefor each s ∈ S, 〈1, d(s)〉 = 0, there exists a (ϕ, T )-derivation D : S −→ X∗ such thatπ∗ ◦D(s) = d(s)(s ∈ S), where π is the canonical map from LUC(T) onto X. Thus thereexists g ∈ X∗ such that D(s) = ϕ(s).g − g (s ∈ S). Hence

π∗ ◦D(s) = π∗(ϕ(s).g)− π∗g = ϕ(s).n− n.So

ϕ(s).n− π∗(ϕ(s).g) = n− π∗g.(3.1)Let ñ = n− π∗g. Then ñ ∈ LUC(T)∗. From (3.1), it follows that

ϕ(s).ñ = ϕ(s).n− ϕ(s).(π∗g) = ϕ(s).n− π∗(ϕ(s).g) = n− π∗g = ñ (s ∈ S).Since LUC(T) is a commutative C∗-algebra with identity, there exists a compact Hausdor�space ∆ such that C(∆) and LUC(T) are isometrically ∗−isomorphic C∗-algebras. Thus wecan consider ñ as a ϕ-left invariant complex Borel regular measure on ∆. Let m = |ñ|‖ñ‖ ,

then ϕ(s).m = m. Therefore for every f ∈ LUC(T) and s ∈ S〈lϕ(s)f,m〉 = 〈f.ϕ(s),m〉 = 〈f, ϕ(s).m〉 = 〈f,m〉.

Hence T is ϕ-left amenable. Similarly we can show that T is ϕ-right amenable. ThereforeT is ϕ-amenable �

The following proposition provides a converse for Proposition 3.6 in a special case.

3.7. Proposition. Let S be a topological semigroup, and ϕ : S −→ S be a continuoushomomorphism such that ϕ(S) is dense in S. If S is ϕ-left amenable, then for everyBanach ϕ-module X with trivial left action, any ϕ-derivation D : S −→ X∗ is ϕ-inner.

Proof. Suppose S is ϕ-left amenable and X is a Banach ϕ(S)-module with trivial leftaction, and D : S −→ X∗ is a ϕ-derivation. For every x ∈ X we de�ne fx : S −→ C byfx(s) = 〈x,D(s)〉 (s ∈ S). Thus

‖fx‖∞ = sups∈S|fx(s)| ≤ sup

s∈S‖D(s)‖‖x‖ ≤M‖x‖,

571

where M > 0 is a uniform bound for D, clearlly, fx is continiuous. We claim thatfx ∈ LUC(S). To see this let sα −→ s in S, then

‖lsαfx − lsfx‖ = sups′∈S|fx(sαs′)− fx(ss′)|

= sups′∈S|〈x,D(sαs′)〉 − 〈x,D(ss′)〉|

≤ sups′∈S|〈x,D(sα).ϕ(s′)−D(s).ϕ(s′)〉|

+ sups′∈S|〈x, ϕ(sα).D(s′)− ϕ(s).D(s′)〉|

= |〈x,D(sα)−D(s)〉|+ sups′∈S|〈x.ϕ(sα)− x.ϕ(s), D(s′)〉|.

Since D is weak*-continuous, we infer that |〈x,D(sα)−D(s)〉| −→ 0. Alsosups′∈S|〈x.ϕ(sα)− x.ϕ(s), D(s′)〉| ≤M‖x.ϕ(sα)− x.ϕ(s)‖,

and by de�nition 3.1 (ii), ‖x.ϕ(sα) − x.ϕ(s)‖ −→ 0. Thus ‖ltαfx − ltfx‖ −→ 0. Sofx ∈ LUC(S). Let m ∈ LUC(S)∗ be such that 〈1,m〉 = 1 and m(lϕ(s)f) = m(f) (s ∈ S, f ∈LUC(S)), and de�ne a linear functional f on X by 〈x, f〉 = 〈fx,m〉 (x ∈ X). For everyx ∈ X and s, s′ ∈ S , we have

fx.ϕ(s)(s′) = 〈x.ϕ(s), D(s′)〉 = 〈x, ϕ(s).D(s′)〉

= 〈x,D(ss′)〉 − 〈x,D(s).ϕ(s′)〉= 〈x,D(ss′)〉 − 〈x,D(s)〉= fx

(ss′)− 〈x,D(s)〉1S(s′).

Therefore fx.ϕ(s) = lsfx−〈x,D(s)〉1S . Since ϕ(S) is dense in S it follows that there existsa net {sα} in S such that limα ϕ(sα) = s, and limα lϕ(sα)fx = lsfx by the de�nition ofLUC(S). Thus

〈x, f − ϕ(s).f〉 = 〈x, f〉 − 〈x.ϕ(s), f〉 = 〈fx − fx.ϕ(s),m〉=〈fx − lsfx + 〈x,D(s)〉1S ,m

〉

=〈fx − lim

αlϕ(sα)fx + 〈x,D(s)〉1S ,m

〉

= limα

〈fx − lϕ(sα)fx + 〈x,D(s)〉1S ,m

〉

= 〈x,D(s)〉.Hence D(s) = f − ϕ(s).f (s ∈ S). Therefore D is ϕ-inner. �

By a similar argument one can prove the following proposition:

3.8. Proposition. Let S be topological semigroup, and ϕ : S −→ S be a continuoushomomorphism such that ϕ(S) is dense in S. If S is ϕ-right amenable, then for everyBanach ϕ-module X with trivial right action, any ϕ-derivation D : S −→ X∗ is ϕ-inner.

4. ϕ̃- amenability of Ma(S)

We start this section with the following.For a topological semigroup S let M(S) denote the space of all bounded, regular,

complex Borel measures on S. This space with the convolution product and norm ‖µ‖ =|µ|(S) is a Banach algebra. The space of all measures µ ∈M(S) for which the mappingsx → |µ| ∗ δx, and x → δx ∗ |µ| from S into M(S) are weakly continuous is denote byMa(S) (L(S), as in [1]). A Hausdor� locally compact topological semigroup S is called afoundation semigroup if S coincides with the closure of

⋃{supp(µ) : µ ∈ Ma(S)}. Note

572

that in the case where S is a foundation semigroup with identity, for every µ ∈ Ma(S)both mappings x → |µ| ∗ δx, and x → δx ∗ |µ| from S into M(S) are norm continuousand Ma(S) has a bounded approximate identity(see [6]).Finally, for any topological semigroup S, µ ∈M(S) and f ∈ Cb(S), we de�ne the complexvalued funcions µ ◦ f and f ◦ µ on S by

µ ◦ f(x) =∫

S

f(yx))dµ(y), f ◦ µ(x) =∫

S

f(xy)dµ(y).(4.1)

Lemma 1.3.4 of [6] , shows that µ◦ f and f ◦µ are in Cb(S). Also for every µ, ν ∈Ma(S)and f ∈ Cb(S) 〈µ ∗ ν, f〉 = 〈µ, f ◦ ν〉 = 〈ν, µ ◦ f〉.4.1. De�nition. If a Banach algebra A is contained in a Banach algebra B as a closedideal, then the strict topology or strong operator topology (so) on B with respect to Ais de�ned through the family of seminorms (pa)a∈A, where

pa(b) := ‖ba‖+ ‖ab‖ (b ∈ B).For a topological semigroup S the strict topology on M(S) with respect to Ma(S) is

simply called so topology or the strict topology on M(S).

4.2. Lemma. Let S and T be two foundation semigroups with identity, and let ϕ : S −→T be a continuous homomorphism. De�ne ϕ̃ : M(S) −→M(T ) by

〈ϕ̃(µ), f〉 =∫

S

f(ϕ(x))dµ(x) (f ∈ C0(T )).

Then ϕ̃ is a continuous homomorphism (with respect to the strict topology on M(S)) thatextends ϕ uniquely and ϕ̃

(Ma(S)

)⊆Ma(T ).

Proof. It is easy to see that ϕ̃ is continuous. By using (4.1), for every f ∈ C0(T ) andµ1, µ2 ∈M(S), we have

〈ϕ̃(µ1) ∗ ϕ̃(µ2), f〉 = 〈ϕ̃(µ1), f ◦ ϕ̃(µ2)〉

=

∫

S

f ◦ ϕ̃(µ2)(ϕ(x)

)dµ1(x)

=

∫

S

∫

S

f(ϕ(x)y

)dϕ̃(µ2)(y)dµ1(x)

=

∫

S

∫

S

f(ϕ(x)ϕ(y)

)dµ2(y)dµ1(x)

=

∫

S

∫

S

f(ϕ(x)ϕ(y)

)dµ1(x)dµ2(y)

=

∫

S

f(ϕ(x)

)dµ1 ∗ µ2(x)

= 〈ϕ̃(µ1 ∗ µ2), f〉.Therefore ϕ̃ is a continuous homomorphism. Let ϕ be another extension of ϕ and letµ ∈M(S). By Theorem 3.3 of [14], µ is the s-lim (strict-lim) of a net (µi) such that eachµi is a combination of point masses. So,

ϕ̃(µ) = ϕ̃(s− limiµi) = lim

iϕ̃(µi) = lim

iϕ(µi) = ϕ(s− lim

iµi) = ϕ(µ).

Thus ϕ̃ = ϕ.To complete the prove, let (µα) be a bounded approximate identity for Ma(T ), then

as in Lemma 2.1 of [11],

‖µα ◦ f − f‖∞ −→ 0 (f ∈ C0(T )).

573

Now for every µ ∈Ma(T ), we obtain

‖ϕ̃(µ) ∗ µα − ϕ̃(µ)‖ = supf∈C0(T ),‖f‖∞≤1

∣∣∣〈ϕ̃(µ) ∗ µα, f〉 − 〈ϕ̃(µ), f〉∣∣∣

= supf∈C0(T ),‖f‖∞≤1

∣∣∣〈ϕ̃(µ), µα ◦ f〉 − 〈ϕ̃(µ), f〉∣∣∣

= supf∈C0(T ),‖f‖∞≤1

∣∣∣〈ϕ̃(µ), µα ◦ f − f〉∣∣∣

≤ supf∈C0(T ),‖f‖∞≤1

‖ϕ̃(µ)‖‖µα ◦ f − f‖∞ −→ 0.

This means that ϕ̃(µ) ∗µα −→ ϕ̃(µ) in norm. So ϕ̃(µ) ∈Ma(T ). Therefore ϕ̃(Ma(S)

)⊆

Ma(T ). �

4.3. De�nition. Let A and B be two Banach algebras and ϕ : A −→ B be a continuoushomomorphism. A Banach (ϕ,B)-bimodule X is called ϕ-pseudo-unital if

X = {ϕ(a1).x.ϕ(a2) : a1, a2 ∈ A, x ∈ X}.

The proof of the following proposition is omitted, since it can be proved in the samedirection of Proposition 2.1.3 of [18].

4.4. Proposition. Let A and B be two Banach algebras which A has a bounded rightapproximate identity, and ϕ : A −→ B be a continuous homomorphism. Let X be aBanach (ϕ,B)-bimodule such that ϕ(A).X = {0}. Then every (ϕ,B)-derivation on A is(ϕ,B)-inner.

Similarly, we can proof above proposition for a Banach algebra A with a bounded leftapproximate identity, where the module action from the right is trivial.

By using above proposition and similar argument as in the proof of the Proposition2.1.5 of [18], we can proof following proposition.

4.5. Proposition. Let A and B be two Banach algebras with bounded approximate iden-tity, and ϕ : A −→ B be a continuous homomorphism. Then the following two conditionare equivalent:

(i) For each Banach (ϕ,B)-bimodule X, any continuous (ϕ,B)-derivation on A is(ϕ,B)-inner.

(ii) For each ϕ-pseudo-unital Banach (ϕ,B)-bimodule X, any continuous (ϕ,B)-derivation on A is (ϕ,B)-inner.

The following proposition generalizes Proposition 2.1.6 of [18].

4.6. Proposition. Let A1 and A2 be two Banach algebras with bounded approximateidentity which are closed ideals of Banach algebras B1 and B2, respectively. Let ϕ :A1 −→ A2 be a continuous homomorphism and X be a ϕ-pseudo-unital Banach (ϕ,A2)-bimodule, and ϕ̃ : B1 −→ B2 be a continuous homomorphism such that ϕ̃ |A1= ϕ. LetD : A1 −→ X∗ be a (ϕ,A2)-derivation, then X is a Banach (ϕ̃, B2)-bimodule and thereis a unique (ϕ̃, B2) derivation D̃ : B1 −→ X∗ satisfying the following:

(i) D̃ |A1= D;(ii) D̃ is continuous with respect to the strict topology on B1 and the w*-topology on

X∗.

Proof. For x ∈ X, let ϕ(a1) ∈ A2 and y ∈ X be such that x = ϕ(a1).y. For b1 ∈ B1,de�ne ϕ̃(b1).x := ϕ̃(b1a1).y. We claim that ϕ̃(b1).x is well de�ne. Let ϕ(a

′1) ∈ A2 and

574

y′ ∈ X be such that x = ϕ(a′1).y′, and let (fβ)β be a bounded approximate identity forA2. Then

ϕ̃(b1a1).y = limβϕ̃(b1)fβϕ(a1).y = lim

βϕ̃(b1)fβϕ(a

′1).y

′ = ϕ̃(b1a′1).y

′ (b1 ∈ B1).

It is obvious that this operation of ϕ̃(B1) on X turns X into a left ϕ̃(B1)-bimodule.Similarly, one de�nes a right Banach ϕ̃(B1)-module structure on X, so that X becomes

a Banach (ϕ̃, B2)-bimodule. Now we de�ne D̃ : B1 −→ X∗ by

D̃(b1) = w∗ − lim

α

(D(b1eα)− ϕ̃(b1).D(eα)

),(4.2)

where (eα)α is a bounded approximate identity for A1. By the similar argument as in

the proof of Proposition 3.1 of [7], one can show that D̃ is de�ne a ϕ-derivation on B1where D̃ |A1= D and D̃ is continuous with respect to the strict topology on B1 and thew*-topology on X∗.

�

4.7. Theorem. Let S and T be two foundation semigroups with identity, and ϕ : S −→ Tbe a continuous homomorphism, and ϕ̃ be as in Lemma 4.2. Then Ma(S) is (ϕ̃,Ma(T ))-amenable if and only if every (ϕ, T )-derivation on S is ϕ-inner.

Proof. Suppose Ma(S) is (ϕ̃,Ma(T ))-amenable, and D : S −→ X∗ is a (ϕ, T )-derivationon S for some ϕ-module X. For every µ ∈Ma(S) and x ∈ X we de�ne

ϕ̃(µ).x =

∫

ϕ(S)

t.xdϕ̃(µ)(t), x.ϕ̃(µ) =

∫

ϕ(S)

x.tdϕ̃(µ)(t).(4.3)

So for some k > 0,

∫

ϕ(S)

‖t.x‖d|ϕ̃(µ)|(t) ≤ k‖x‖‖ϕ̃(µ)‖

575

Clearly, D̃ is continuous. By using (4.4), for every µ1, µ2 ∈Ma(S) and x ∈ X, we have

〈x, D̃(µ1 ∗ µ2)〉 =∫

S

〈x,D(s)〉dµ1 ∗ µ2(s)

=

∫

S

∫

S

〈x,D(s1s2)〉dµ1(s1)dµ2(s2)

=

∫

S

∫

S

〈x,D(s1).ϕ(s2) + ϕ(s1).D(s2)〉dµ1(s1)dµ2(s2)

=

∫

S

∫

S

〈ϕ(s2).x,D(s1)〉dµ1(s1)dµ2(s2)

+

∫

S

∫

S

〈x.ϕ(s1), D(s2)〉dµ1(s1)dµ2(s2)

=

∫

S

〈∫

S

ϕ(s2).xdµ2(s2), D(s1)〉dµ1(s1)

+

∫

S

〈∫

S

x.ϕ(s1)dµ1(s1), D(s2)〉dµ2(s2)

=

∫

S

〈ϕ̃(µ2).x,D(s1)〉dµ1(s1) +∫

S

〈x.ϕ̃(µ1), D(s2)〉dµ2(s2)

= 〈ϕ̃(µ2).x, D̃(µ1)〉+ 〈x.ϕ̃(µ1), D̃(µ2)〉= 〈x, D̃(µ1).ϕ̃(µ2) + ϕ̃(µ1).D̃(µ2)〉.

That is D̃(µ1 ∗ µ2) = D̃(µ1).ϕ̃(µ2) + ϕ̃(µ1).D̃(µ2). Hence D̃ is a (ϕ̃,Ma(T ))-derivation.From the (ϕ̃,Ma(T ))-amenability ofMa(S) it follows that there exists x

∗ ∈ X∗ such thatD̃(µ) = ϕ̃(µ).x∗ − x∗.ϕ̃(µ) (µ ∈ Ma(S)). Moreover, for every x ∈ X and µ ∈ Ma(S), wehave ∫

S

〈x,D(s)〉dµ(s) = 〈x, D̃(µ)〉

= 〈x, ϕ̃(µ).x∗ − x∗.ϕ̃(µ)〉= 〈x, ϕ̃(µ).x∗〉 − 〈x, x∗.ϕ̃(µ)〉

= 〈x,∫

S

ϕ(s).x∗dµ(s)〉 − 〈x,∫

S

x∗.ϕ(s)dµ(s)〉

=

∫

S

〈x, ϕ(s).x∗ − x∗.ϕ(s)〉dµ(s).

So 〈x,D(s)〉 = 〈x, ϕ(s).x∗ − x∗.ϕ(s)〉 (x ∈ X), by Lemma 2.2 of [12]. HenceD(s) = ϕ(s).x∗ − x∗.ϕ(s) (s ∈ S).

Therefore D is ϕ-inner.Conversely, suppose every (ϕ, T )-derivation on S is ϕ-inner. Let D : Ma(S) −→ X∗ be

a (ϕ̃,Ma(T ))-derivation for some Banach (ϕ̃,Ma(T ))-bimodule X. By Proposition 4.5,there is no loss of generality if we suppose that X is ϕ̃-pseudo-unital. So by Proposition4.6, X is a Banach (ϕ̃,M(T ))-bimodule and there is a unique (ϕ̃,M(T ))-derivation D̃ :M(S) −→ X∗ that extends D and is continuous with respect to the strict topology onM(S) and the w∗-topology on X∗. We consider the following module actions ϕ(S) on Xby

ϕ(s).x := δϕ(s).x, x.ϕ(s) := x.δϕ(s) (s ∈ S, x ∈ X),and de�ne DS : S −→ X∗ by DS(s) = D̃(δs) (s ∈ S). It is easy to see that DS de�nes a(ϕ, T )-derivation. So there exists x∗ ∈ X∗ such that

DS(s) = ϕ(s).x∗ − x∗.ϕ(s) (s ∈ S).

576

Consequently, for every s ∈ SD̃(δs) = DS(s) = ϕ(s).x

∗ − x∗.ϕ(s) = ϕ̃(δs).x∗ − x∗.ϕ̃(δs).Since every measure µ inM(S) is the s-lim of a net (µi) such that each µi is a combinationof point masses (see Theorem 3.3 of [14]), from the de�nition of the strict topology itfollows that ν ∗ µi −→ ν ∗ µ (ν ∈ Ma(S)) and µi ∗ ν −→ µ ∗ ν (ν ∈ Ma(S)) in the normtopology. Let x ∈ X, and ϕ̃(ν) ∈Ma(T ) and y ∈ X be such that x = y.ϕ̃(ν). Hence

|〈x, ϕ̃(µi).x∗〉 − 〈x, ϕ̃(µ).x∗〉| = |〈y.ϕ̃(ν), ϕ̃(µi).x∗〉 − 〈y.ϕ̃(ν), ϕ̃(µ).x∗〉|= |〈y.ϕ̃(ν ∗ µi)− y.ϕ̃(ν ∗ µ), x∗〉|6 k‖y‖‖ϕ̃‖‖ν ∗ µi − ν ∗ µ‖‖x∗‖ −→ 0.

This means that w∗-limi ϕ̃(µi).x∗ = ϕ̃(µ).x∗. Similarly, w∗-limi x

∗.ϕ̃(µi) = x∗.ϕ̃(µ).

Now for every µ ∈M(S), we obtainD̃(µ) = D̃(s− lim

iµi) = w

∗ − limiD̃(µi)

= w∗ − limi

(ϕ̃(µi).x

∗ − x∗.ϕ̃(µi))

= ϕ̃(µ).x∗ − x∗.ϕ̃(µ).Thus D̃ is a (ϕ̃,M(T ))-inner derivation and so D is (ϕ̃,Ma(T ))-inner derivation. There-fore Ma(S) is (ϕ̃,Ma(T ))-amenable. �

A combination of Proposition 3.6 and Theorem 4.7, gives the following result.

4.8. Theorem. Let S and T be two foundation semigroups with identity, and let ϕ :S −→ T be a continuous homomorphism, and ϕ̃ be as in Lemma 4.2. If Ma(S) is(ϕ̃,Ma(T ))-amenable, then T is ϕ-amenable.

Before turning the next result, we �rst need to prove the following proposition.

4.9. Proposition. Let A and B be two Banach algebras and let ϕ : A −→ B be acontinuous homomorphism. If ϕ(A) is dense in B and A is (ϕ,B)-amenable, then B isamenable.

Proof. Let D : B −→ X∗ be a continuous derivation for a Banach B-bimodule X, andD̃ = D ◦ ϕ. Obviously D̃ de�ne a continuous (ϕ,B)-derivation from A into X∗. By(ϕ,B)-amenability of A there exists f ∈ X∗ such that

D̃(a) = ϕ(a).f − f.ϕ(a) (a ∈ A).Let b ∈ B, since ϕ(A) is dense in B, there exists a net {aα} ⊂ A such that limα ϕ(aα) = b.Hence

D(b) = limαD(ϕ(aα)) = lim

αD̃(aα) = lim

α

(ϕ(aα).f − f.ϕ(aα)

)= b.f − f.b.

Thus D is an inner derivation. This completes the proof. �

Note that if S is a discrete semigroup then S is a foundation semigroup with LUC(S) =l∞(S), and M(S) = Ma(S) = l

1(S).The next example shown that the converses of the Theorem 4.8 and Proposition 3.6

are not true.

4.10. Example. Let S be the set Z = {· · · ,−2,−1, 0, 1, 2, · · · }, with the product(m,n) −→ m ∨ n = max{m,n}, Z× Z −→ Z.

De�ne ϕ : S −→ S by ϕ(s) = s + 1 (s ∈ S). Then it is easy to check that ϕ is ahomomorphism on S. Since S is an abelian semigroup it follows S is amenable (see [3])and so by Lemma 3.5, S is ϕ-amenable. Since E(S) = S, from Corollary 1 of [5], the

577

convolution semigroup algebra l1(S) is not amenable. Also since ϕ is an epimorphism onS it follows that ϕ̃ is an epimorphism on l1(S). Therefore by Proposition 4.9, l1(S) isnot ϕ̃-amenable. So the converse of the Theorem 4.8 is not valid. Also by Theorem 4.7,we conclude that the converse of the Proposition 3.6 is not true.

Acknowledgement. We are thankful to the referees for their valuable suggestions andcomments and grateful to the o�ce of Graduate studies of the university of Isfahan fortheir support.

References

1. A. C. Baker, J. W. Baker, Algebra of measures on a locally compact semigroup III, J. LondonMath. Soc. (4), 685-695, 1972.

2. H. G. Dales, Banach slgebras and automatic continuity, Clarendon Press, Oxford, 2000.3. M. M. Day, Ergodic theorems for Abelian semigroups, Trans. Amer. Math. Soc., 51, 399-412,

1972.4. J. Duncan, I. Namioka, Amenability of inverse semigroups and thier semigroup algebras,

Proceeding of the Royal Society of Edinburgh 80 A, 309-321, 1998.5. J. Duncan, A. L. T. Paterson, Amenability for discrete convolution semigroup algebras,

Math. Scand. 66, 141-146, 1990.6. H. A. M. Dzinotyiweyi, The analoge of the group algebra for topological semigroups, Research

Notes in Mathematics, 98, Pitman, NewYork, 1984.7. Z. Ghorbani, M. Lashkarizadeh Bami, ϕ-Approximate bi�at and ϕ-amenabel Banach alge-

bras, Proceedings of the Romanian Academy, Series A, 13 (1), 3-10, 2012.8. Z. Ghorbani, M. Lashkarizadeh Bami, ϕ-Amenable and ϕ-bi�at Banach algebras, Bull.

Iranian Math. Soc. 39 (3), 507-515, 2013.9. B. E. Johnson, Cohomology in Banach algebras, harmonic problems, Memoirs Amer. Math.

Soc. 127, 1972.10. E. Kaniuth, A. Lau, J. Pym, On ϕ-amenability of Banach algebras, Math. Proc. Camp.

Phil. Soc., 144, 85-96, 2008.11. M. Lashkarizadeh Bami, Ideals of M(S) as ideals of LUC(S)∗ of a compactly cancellative

semigroup S, Math. Japon., 48, 363-366, 1998.12. M. Lashkarizadeh Bami, Representations of foundation semigroups and their algebras, Cana-

dian J. Math, 37, 29-47, 1985.13. A. T. M. Lau, Amenability of semigroups, the analytical and topological theory of semi-

groups, trends and developments K. H. Hofman, J. D. Lawson and J. S. Pym, eds., Walterde Gruyter and Co., 1990.

14. M. Lashkarizadeh Bami, B. Mohammadzadeh and H. Samea, Derivations on certain semi-group algebras, Journal of Sciences Islamic Republic of Iran, 18 (4), 339-345, 2007.

15. M. Mirzavaziri and M. S. Moslehian, σ-derivations in Banach algebras, Bull. Iranian Math.Soc. 32 (1), 65-78, 2006.

16. M. Mirzavaziri and M.S. Moslehian, Automatic continuity of σ-derivations in C∗-algebras,Proc. Amer. Math. Soc. 134 (11), 3319-3327, 2006.

17. I. Namioka, On certain actions of semigroups on L-spaces, Studia Math, 29, 63-77, 1967.18. V. Runde, Lectures on Amenability, Lecture Notes in Mathematics 1774, Springer-Verlag,

Berlin-Heidelberg-New York, 2002.


On the Padovan p-numbers

Ömür DEVEC ∗ and Erdal KARADUMAN†

Abstract

In this paper, we de�ne the Padovan p-numbers and then we obtaintheir miscellaneous properties such as the generating matrix, the Binetformula, the generating function, the exponential representation, thecombinatorial representations, the sums and permanental representa-tion. Also, we study the Padovan p-numbers modulo m. Furthermore,we de�ne Padovan p-orbit of a �nite group and then, we obtain thelength of the Padovan p-orbits of the quaternion group Q2n , (n ≥ 3).

Keywords: The Padovan p-numbers, Matrix, Sequence, Group, Length.

2000 AMS Classi�cation: 11B50, 05A15, 11C20, 20D60, 20F05.


1. Introduction and Preliminaries

The well-known the Fibonacci sequence is de�ned by initial values F0 = 0 , F1 = 1 andrecurrence relation

Fn+2 = Fn+1 + Fn for n ≥ 0.The Padovan sequence is the sequence of integers P (n) de�ned by initial values P (0) =P (1) = P (2) = 1 and recurrence relation

P (n) = P (n− 2) + P (n− 3) for n ≥ 3.The Padovan sequence is

1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, . . . .

∗Department of Mathematics, Faculty of Science and Letters, Kafkas University, 36100 Kars,TURKEYEmail : [email protected]†Department of Mathematics, Faculty of Science, Atatürk University , 25240 Erzurum,

TURKEYEmail : [email protected]

580

The Padovan numbers are generated by a matrix Q,

Q=

0 1 00 0 11 1 0

,

The powers of Q give

Qn=

P (n− 5) P (n− 3) P (n− 4)P (n− 4) P (n− 2) P (n− 3)P (n− 3) P (n− 1) P (n− 2)

.

For more information on this sequence, see [13].Kalman [14] mentioned that these sequences are special cases of a sequence which isde�ned recursively as a linear combination of the preceding k terms:

an+k = c0an + c1an+1 + · · ·+ ck−1an+k−1,where c0, c1, · · · , ck−1 are real constants. In [14], Kalman derived a number of closed-formformulas for the generalized sequence by companion matrix method as follows:

Ak =

0 1 0 · · · 0 00 0 1 · · · 0 0

0 0 0. . . 0 0

......

......

...0 0 0 · · · 0 1c0 c1 c2 · · · ck−2 ck−1

.

Then by an inductive argument he obtained that

Ank

a0a1...

ak−1

=

anan+1...

an+k−1

.

Many of the numbers obtained by using homogeneous linear recurrence relations and theirmiscellaneous properties have been studied by some authors; see for example, [9, 10, 11,12, 15, 16, 19, 20, 21, 22, 24, 25, 27, 28, 29, 31]. In Section 2, we de�ne the Padovanp-numbers. Then we obtain the generating matrix, the Binet formula, the generatingfunction, the exponential representation, the combinatorial representations, the sums andpermanental representation of the Padovan p-numbers. The study of the linear recurrencesequences in algebraic structures began with the earlier work of Wall [30] where theordinary Fibonacci sequences in cyclic groups were investigated. Recently, many authorshave studied some special linear recurrence sequences in algebraic structures; see forexample, [1, 3, 5, 6, 7, 8, 17, 23, 26]. In Section 3, we study the Padovan p-numbersmodulo m. Also in this section, we give the de�nition of Padovan p-sequences in groupsgenerated by two or more elements. Then we examine these sequences in �nite groups.Furthermore, we obtain the periods of the Padovan p-sequences of the quaternion groupQ2n , (n ≥ 3) as the applications of obtained results in Section 3.

2. The Padovan p-Numbers

Now we de�ne the Padovan p-numbers by the following homogeneous linear recurrencerelation for any given p (p = 2, 3, 4, . . .) and n ≥ 1(2.1) Pap (n+ p+ 2) = Pap (n+ p) + Pap (n)

581

with initial conditions Pap (1) = Pap (2) = · · · = Pap (p) = 0, Pap (p+ 1)= 1 andPap (p+ 2)= 0.When p = 2 in (2.1), we obtain Pa2 (2n+ 1) = Fn for n ≥ 1.By equation (2.1), we have

Pap (n+ p+ 2)Pap (n+ p+ 1)

...Pap (n+ 2)Pap (n+ 1)

=

0 1 0 · · · 0 11 0 0 · · · 0 00 1 0 · · · 0 0

0 0 1. . . 0 0

......

. . .. . .

...0 0 · · · 0 1 0

Pap (n+ p+ 1)Pap (n+ p)

...Pap (n+ 1)Pap (n)

for the sequence of the Padovan p-numbers. Letting

M =

0 1 0 · · · 0 11 0 0 · · · 0 00 1 0 · · · 0 0

0 0 1. . . 0 0

......

. . .. . .

...0 0 · · · 0 1 0

.

The matrix M is said to be a Padovan p-matrix. Also, we obtain that

(2.2) Mn =

Pap(n+p+1) Pap(n+p+2) Pap(n+1) Pap(n+2) ··· Pap(n+p)Pap(n+p) Pap(n+p+1) Pap(n) Pap(n+1) ··· Pap(n+p−1)

......

......

......

Pap(n+1) Pap(n+2) Pap(n−p+1) Pap(n−p+2) ··· Pap(n)Pap(n) Pap(n+1) Pap(n−p) Pap(n−p+1) ··· Pap(n−1)

for n ≥ 1, which can be proved by mathematical induction. We easily derive that(2.3) detM = (−1)p+1 .By equation (2.3), we have detMn = (−1)np+n. Now, we can give a formula for Padovanp-numbers (n ≥ 1 ) by using this determinantal representation.2.1. Lemma. The characteristic equation of the Padovan p-numbers xp+2 − xp − 1 = 0does not have multiple roots.

Proof. Let α be a root of f (x) = 0 where f (x) = xp+2 − xp − 1 so that α /∈ {0, 1}. Ifpossible, α is a multiple root in which case f (α) = f

′(α) = 0. Now f

′(α) = 0 and α 6= 0

give α2 = pp+2

while f (α) = 0 shows αp(α2 − 1

)−1 = 0 so that

(pp+2

) p2 ·(−2p+2

)= 1, an

impossibility since the left hand side is less than 1 for p ≥ 2. This contradiction provesthe Lemma. �

Let f (u) be the characteristic polynomial of the Padovan p-matrix M , then f (u) =up+2 − up − 1. If u1, u2, . . . , up+2 are eigenvalues of the matrix M , then by Lemma 2.1,they are distinct. Let Vp be a (p+ 2)× (p+ 2) Vandermonde matrix such that

(2.4) Vp =

up+11 up+12 · · · up+1p+2

up1 up2 · · · upp+2

...... · · ·

...u1 u2 · · · up+21 1 · · · 1

.

582

Let V(i,j)p be a (p+ 2)× (p+ 2) matrix obtained from Vp by replacing the jth column of

Vp by Aip, where, A

ip is a (p+ 2)× 1 matrix as follows:

Aip =

un+p+2−i1un+p+2−i2...

un+p+2−ip+2

.

Now we consider the Binet formula for the Padovan p-numbers. We give the followingTheorem.

2.2. Theorem. Let Mn = [mij ] . Then, mij =det

(V

(i,j)p

)

det(Vp).

Proof. Since the eigenvalues of the matrixM are distinct, the matrixM is diagonalizable.Let D = (u1, u2, . . . , up+2), then it is easy to see thatMVp = VpD. Since Vp is invertible,

(Vp)−1MVp = D. Thus, the matrix M is similar to D. So we get M

nVp = VpDn for

n ≥ 1. Then we have the following linear system of equations for n ≥ 1:mi1u

p+11 +mi2u

p1 + · · ·+mip+2 = un+p+2−i1

mi1up+12 +mi2u

p2 + · · ·+mip+2 = un+p+2−i2

...

mi1up+1p+2 +mi2u

pp+2 + · · ·+mip+2 = un+p+2−ip+2 .

So, for each i, j = 1, 2, · · · , p+ 2, we obtain mij as follows:

mij =det(V

(i,j)p

)

det (Vp).

Theorem 2.1 gives immediately: �

2.3. Corollary. Pap (n) =det

(V

(p+2,1)p

)

det(Vp)=

det(V

(2,3)p

)

det(Vp)=

det(V

(p+1,p+2)p

)

det(Vp).

Now we give the generating function of the Padovan p-numbers and an exponentialrepresentation for the Padovan p-numbers with the following Theorem.

2.4. Theorem. The generating function g (x) of the Padovan p-numbers is given by

g (x) =1

1− x2 − xp+2for 0 ≤ x2 + xp+2 < 1 and it has exponential representation

g (x) = exp

( ∞∑

i=1

x2i

i(1 + xp)i

).

Proof. Let g (x) be a generating function for the Padovan p-numbers. Then

(2.5)g (x) = Pap (p+ 1) + Pap (p+ 2)x+ Pap (p+ 3)x2 + · · ·+

+Pap (p+ n+ 1)xn + Pap (p+ n+ 2)xn+1 + · · · .By the de�nition of the Padovan p-numbers, we can write

g (x)− x2g (x)− xp+2g (x) = Pap (p+ 1) = 1.So we get

g (x) =1

1− x2 − xp+2

583

for 0 ≤ x2 + xp+2 < 1. Also by a simple calculation, we obtain

ln g (x) = − ln{1− x2 (1 + xp)

}=

∞∑

i=1

x2i

i(1 + xp)i .

Thus the proof is complete. �

Now we give a combinatorial representation for the Padovan p-numbers by the followingTheorem.

2.5. Theorem.

Pap (n+ p+ 1) =∑

np+2≤m≤n

(mj

)

where j = n−2mp

.

Proof. From (2.5), it is clear that the coe�cient of xn in g (x) is Pap (p+ n+ 1). Since

g (x) =1

1− x2 − xp+2 =1

1− (x2 + xp+2) = 1 +(x2 + xp+2

)+

+(x2 + xp+2

)2+ · · ·+

(x2 + xp+2

)n+ · · ·

= 1 + x2 (1 + xp) + x42∑

j=0

(2j

)xpj + · · ·+ x2n

n∑

j=0

(nj

)xpj + · · · ,

we only consider the �rst n + 1 terms on the right-side. By the binomial theorem, wecan write

(x2 + xp+2

)m=(x2 (1 + xp)

)m= x2m

m∑

j=0

(mj

)xpj .

Then by the above equation we see that the coe�cient of xn in(x2 + xp+2

)mfor positive

m and n is(mj

)

where j = n−2mp

. Thus the proof is complete. �

Let E (e1, e2, . . . , el) be the l × lcompanion matrix

E (e1, e2, · · · , el) =

e1 e2 e3 · · · el1 0 0 · · · 00 1 0 · · · 0...

. . .. . .

. . ....

0 · · · 0 1 0

.

2.6. Theorem. (Chen and Louck [4]). The (i, j) entry e(n)ij

(e1, e2, . . . , el) in the

matrix En (e1, e2, . . . , el) is given by the following formula:

(2.6)

e(n)ij

(e1, e2, . . . , el) =∑

(t1,t2,··· ,tl)

tj + tj+1 + · · ·+ tlt1 + t2 + · · ·+ tl

×(t1 + t2 + · · ·+ tlt1, t2, . . . , tl

)et11 · · · etll

where the summation is over nonnegative integers satisfying t1+2t2+ · · ·+ ltl = n− i+j,(t1 + t2 + · · ·+ tlt1, t2, . . . , tl

)= (t1+t2+···+tl)!

t1!t2!···tl! is a multinomial coe�cient, and the coe�cients

in (2.6) are de�ned to be 1 if n = i− j.

584

Now we give other combinatorial representations than the above for the Padovan p-numbers.

2.7. Corollary. i.

Pap (n) =∑

(t1,t2,...,tp+2)

(t1 + t2 + · · ·+ tp+2t1, t2, · · · , tp+2

)

where the summation is over nonnegative integers satisfying t1+2t2+ · · ·+(p+ 2) tp+2 =n− p− 1.

ii.

Pap (n) =∑

(t1,t2,...,tp+2)

tp+2t1 + t2 + · · ·+ tp+2

(t1 + t2 + · · ·+ tp+2t1, t2, . . . , tp+2

)

where the summation is over nonnegative integers satisfying t1+2t2+ · · ·+(p+ 2) tp+2 =n+ 1.

iii.

Pap (n) =∑

(t1,t2,··· ,tp+2)

t3 + t4 + · · ·+ tp+2t1 + t2 + · · ·+ tp+2

(t1 + t2 + · · ·+ tp+2t1, t2, . . . , tp+2

)

where the summation is over nonnegative integers satisfying t1+2t2+ · · ·+(p+ 2) tp+2 =n+ 1.

Proof. If we take i = p+2 and j = 1 for case i., i = p+1 and j = p+2 for case ii. andi = 2 and j = 3 for case iii. in Theorem 2.4, then we can directly see the conclusionsfrom equation (2.2). �

Let the sums of the Padovan p-numbers from p+ 1 to p+ n be denoted by Sn, that is,

(2.7) Sn =n∑

i=1

Pap (p+ i)

and let T and Kn be the (p+ 3)× (p+ 3) matrices

T =

1 0 0 0 · · · 0 01 0 1 0 · · · 0 10 1 0 0 · · · 0 00 0 1 0 · · · 0 0...

.... . .

. . .. . .

......

.... . .

. . .. . .

...0 0 0 · · · 0 1 0

=

1 0 0 · · · 0 0100 M...0

and

Kn =

1 0 0 · · · 0 0SnSn−1

... Mn

...Sn−p−1

.

Then by induction on n, it is easy to see that Kn = Tn.

Now we can give the sums of the Padovan p-numbers by the following Theorem.

585

2.8. Theorem. Let the sums of the Padovan p-numbers from p + 1 to p + n, Sn be asin (2.7). Then

Sn =

p+1∑

i=0

Pap (n+ p+ 2− i)− 1.

Proof. Let U and D1 be the (p+ 3)× (p+ 3) matrices

U =

1 0 0 · · · 0−1 up+11 up+12 · · · up+1p+2−1 up1 up2 · · · upp+2...

...... · · ·

...−1 u1 u2 · · · up+2−1 1 1 · · · 1

and

D1 =

1u1

. . .

up+2

where u1, u2, . . . , up+2 are the roots of the equation up+2 − up − 1 = 0. Expanding

det (U) by the Laplace expansion of the determinant with respect to the �rst row gives usdet (U) = det (Vp) where Vp is as in (2.4). It is easy to see that (1− u)

(up+2 − up − 1

)=

0 is the characteristic equation of the matrix U and the eigenvalues of U are 1, u1, u2, . . . , up+2.By Lemma 2.1, it is known that 1, u1, u2, · · · , up+2 are distinct. Hence, the matrix Uis diagonalizable. Also, we can write TU = UD1. Since the matrix U is invertible,U−1TU = D1. Thus, the matrix T is similar to the matrix D1. Then T

nU = UDn1 , andhence KnU = UD

n1 . Since Sn = k2,1 where Kn = [kij ], by using matrix multiplication,

we can write

Sn −(p+1∑

i=0

Pap (n+ p+ 2− i))

= −1.

So we get

Sn =

p+1∑

i=0

Pap (n+ p+ 2− i)− 1.

�

Now we consider the relationship between the Padovan p-numbers and the permanentof a certain matrix which is obtained using the Padovan p-matrix M .

2.9. De�nition. An n ×m real matrix C = [cij ] is called a contractible matrix in theαth column (resp. row.) if the αth column (resp. row.) contains exactly two non-zeroentries.

Let u1, u2, . . . , un be row vectors of the matrix C and let C be contractible in the αth

column with ciα 6= 0, cjα 6= 0 and i 6= j. Then the (n− 1) × (m− 1) matrix Cij:αobtained from C by replacing the ith row with ciαuj + cjαui and deleting the j

th rowand the αth column is called the contraction in the αth column relative to the ith rowand the jth row.In [2], Brualdi and Gibson showed that per (A) = per (B) if A is a real matrix of ordern > 1 and B is a contraction of A.

586

Let p be a �xed integer such that p ≥ 2 and let Anp = [aij ] be the n × n super-diagonalmatrix with ai,i+1 = ai+1,i = ai,i+p+1 = 1 and for all i and 0 otherwise, that is,

(p+ 2) th↓

Anp =

0 1 0 · · · 0 1 0 · · · 01 0 1 0 · · · 0 1 00 1 0 1 0 · · · 0 1 00 0 1 0 1 0 · · · 0 10 0 0 1 0 1 0 · · · 0...

......

. . .. . .

. . ....

......

. . .. . .

. . .

0 0 0 · · · 0 0 1 0 10 0 0 0 · · · 0 0 1 0

.

Note that if n = 1, A1p = 0.

2.10. Theorem. The permanent of Anp (n ≥ 1, p ≥ 2) is Pap (n+ p+ 1).

Proof. We prove this by induction. First, let us consider the case n < p + 2. Fromthe de�nitions of the matrix Anp and the Padovan p-numbers it is clear that perA

1p =

Pap (p+ 2) = 0 and perA2p = Pap (p+ 3) = 1. Also, we obtain the following matrix for3 ≤ λ ≤ p+ 1

Aλp =

0 1 01 0 1

. . .. . .

. . .

1 0 10 1 0

.

Then

perAλp =

{1 if λ is even,0, if λ is odd.

Furthermore, we know that

Pap (λ+ p+ 1) =

{1 if λ is even,0, if λ is odd,

for 3 ≤ λ ≤ p+ 1.

So we get perAnp = Pap (n+ p+ 1) for 1 ≤ n ≤ p+ 1.Now, let us consider the case n ≥ p+ 2. Suppose that the equation holds for n ≥ p+ 2.Then we show that the equation holds for n+1. If we expand the perAnp by the Laplaceexpansion of the permanent, we obtain

perAn+1p = perAn−1p + perA

n−p−1p .

Since perAn−1p = Pap (n+ p) and perAn−p−1p = Pap (n), we get

perAn+1p = Pap (n+ p+ 2).So, the proof is complete. �

587

3. The Padovan p-Sequences in Groups

We consider the Padovan p-numbers modulo m.Reducing the Padovan p-sequence {Pap (n)} by a modulus m, we can get the repeatingsequence, denoted by

{Papm (n)} = { Papm (1) , Papm (2) , . . . , Papm (p+ 2) , . . . , Papm (i) , . . .}where Papm (i) ≡ Pap (i) (mod m). It has the same recurrence relation as in (2.1).A sequence is periodic if, after a certain point, it consists only of repetitions of a �xedsubsequence. The number of elements in the shortest repeating subsequence is calledthe period of the sequence. For example, the sequence a, b, c, d, b, c, d, b, c, d, ... is periodicafter the initial element a and has period 3. A sequence is simply periodic with period kif the �rst k elements in the sequence form a repeating subsequence. For example, thesequence a, b, c, d, a, b, c, d, a, b, c, d, ... is simply periodic with period 4.

3.1. Theorem. The sequence {Papm (n)} is simply periodic.

Proof. Let Xp = { (x1, x2, · · · , xp+2)| xi′s are integers such that 0 ≤ xi ≤ m− 1}, then|Xp| = mp+2. Since there are mp+2 distinct (p+ 2)-tuples of elements of Zm, at leastone of the (p+ 2)-tuples appears twice in the sequence {Papm (n)}. Therefore, the sub-sequence following this (p+ 2)-tuple repeats; hence, the sequence {Papm (n)} is periodic.So if

Papm (u+ 1) = Papm (v + 1) , Papm (u+ 2) = Papm (v + 2) , . . . ,

Papm (u+ p+ 2) = Papm (v + p+ 2)

and v > u, then v ≡ u (mod p+ 2). From the de�nition, we can easily derive thatPap (n) = Pap (n+ p+ 2)− Pap (n+ p) .

Thus we obtain

Papm (u) = Papm (v) , Papm (u− 1) = Papm (v − 1) , . . . ,Papm (2) = Papm (v − u+ 2) , Papm (1) = Papm (v − u+ 1) ,

which implies that the sequence is simply periodic. �

We denote the period of the sequence {Papm (n)} by hPapm.

3.2. Example. The sequence {Pa32 (n)} is{0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, . . .} .

Since Pa32 (32) = Pa32 (1) = 0, Pa32 (33) = Pa32 (2) = 0, Pa32 (34) = Pa32 (3) = 0,Pa32 (35) = Pa32 (4) = 1 and Pa32 (36) = Pa32 (5) = 0, the sequence is simply periodicwith period hPa32 = 31.

Given an integer matrix A = [aij ], A (mod m) means that all entries of A are reducedmodulo m, that is, A (mod m) = (aij (mod m)). Let us consider the set 〈A〉m ={Ai (mod m)

∣∣ i ≥ 0}. If gcd (m, detA) = 1, then the set 〈A〉m is a cyclic group. Let

the notation∣∣〈A〉m

∣∣ denote the order of the set 〈A〉m. By equation (2.2), it is clear thatthe set 〈M〉m is a cyclic group for every positive integer m.Now we can give the relationship between the Padovan p-matrix M and the periodhPapm by the following Theorem.

3.3. Theorem. If m has the prime factorization m = tα where t is prime and α is apositive integer, then hPaptα =

∣∣〈M〉tα∣∣.

588

Proof. Let∣∣〈M〉tα

∣∣ = u. Then by equation (2.2), it is easy to see that Pap (u+ 1) ≡Pap (u+ 2) ≡ · · · ≡ Pap (u+ p) ≡ 0 (mod tα) , Pap (u+ p+ 1) ≡ 1 (mod tα) andPap (u+ p+ 2) ≡ 0 (mod tα), that is, Pap (u+ 1) ≡ Pap (1) (mod tα), Pap (u+ 2) ≡Pap (2) (mod tα), . . . , Pap (u+ p) ≡ Pap (p) (mod tα), Pap (u+ p+ 1) ≡ Pap (p+ 1) (mod tα)and Pap (u+ p+ 2) ≡ Pap (p+ 2) (mod tα). Since hPaptα is the period of the sequence{Paptα (n)}, we obtain hPaptα |u. Now we need only to prove that hPaptα is divis-ible by

∣∣〈M〉tα∣∣. From equation (2.2), we obtain MhPaptα (mod tα) ≡ I, where I is

the (p+ 2) × (p+ 2) identity matrix. So we get∣∣〈M〉tα

∣∣∣∣hPaptα . Thus the proof iscomplete. �

The auxiliary equation for the Padovan p-numbers can be written as

xp+2 = xp + 1.

3.4. Lemma. Let n ≥ p+ 2 and p ≥ 2, then

(3.1) xn = Pap (n)xp+1 + Pap (n+ 1)xp +

p−1∑

i=0

Pap (n− 1− i)xi.

Proof. This follows directly from induction on n. �

Let t be a prime and let Gtα

p ={xn (modtα) : n ∈ Z, xp+2 = xp + 1

}such that α is a

positive integer. Then, it is clear that the set Gtα

p is a cyclic group.Now we can give a relationship between the characteristic equation of the Padovan p-numbers and the period hPapm by the following Theorem.

3.5. Theorem. The cyclic group Gtα

p is isomorphic to the cyclic group 〈M〉tα , where tis prime and α is a positive integer.

Proof. Let t be a prime and let α be a positive integer. It is clear that hPaptα >2p + 2. Then by equation (3.1) we see that xhPaptα ≡ 1 (mod tα). Thus we obtain∣∣∣Gtαp

∣∣∣ =hPaptα . So by Theorem 3.3 we have Gtα

p∼= 〈M〉tα . �

Now we give some properties of the period hPapm by the following Theorem.

3.6. Theorem. i. Let t be a prime and let u be the smallest positive integer wherehPaptu+1 6= hPaptu , then hPaptσ = tσ−u ·hPaptu for every integer σ > u. In particular,if hPapt 6= hPapt2 , then hPaptσ = tσ−1 · hPapt holds for every integer σ > 1.

ii. If m =∏vi=1 t

eii , (v ≥ 1) where ti's are distinct primes, then

hPapm = lcm[hPapte11

, hPapte22, . . . , hPaptevv

].

Proof. i. By Theorem 3.3 we see that for each positive integer a,MhPapta+1 ≡ I(mod ta+1

),

hence MhPapta+1 ≡ I (mod ta), which means that hPapta divides hPapta+1 . On theother hand, writing MhPapta = I +

(m

(a)ij · ta

), by the binomial theorem, we obtain

MhPapta ·t =(I +

(m

(a)ij · ta

))t=

t∑

i=0

(ti

)(m

(a)ij · ta

)i≡ I

(mod ta+1

),

which implies that hPapta+1 divides hPapta ·t. Thus, hPapta+1 = hPapta or hPapta+1 =hPapta ·t, and the latter holds if and only if there is am(a)ij which is not divisible by t. Dueto fact that we assume u is the smallest positive integer such that hPaptu+1 6= hPaptu ,there is an m

(u)ij which is not divisible by t. Since there is an m

(u)ij such that t does not

dividem(u)ij , it is easy to see that there is anm

(u+1)ij which is not divisible by t. This shows

589

that hPaptu+2 6= hPaptu+1 . Then we see that hPaptu+2 = t · hPaptu+1 = t2 · hPaptu .So by induction on u we obtain hPaptσ = t

σ−u · hPaptu for every integer σ > u. Also,if u = 1, then hPaptσ = t

σ−1 · hPapt for every integer σ > 1.ii. Since hPapteii

is the length of the period of the sequence{Papteii

(n)}, the se-

quence{Papteii

(n)}

repeats only after blocks of length λ · hPapteii , (λ ∈ N). Also,hPapm is the length of the period {Papm (n)}, which implies that

{Papteii

(n)}

re-

peats after hPapm terms for all values i. Thus, hPapm is the form λ · hPapteii forall values of i, and since any such number gives a period of {Papm (n)}. So we gethPapm =lcm

[hPapte11

, hPapte22, . . . , hPaptevv

]. �

We consider the Padovan p-sequences in p-generated groups such that p ≥ 2.Let G be a �nite p-generator group and letX = {(x1, x2, . . . , xp) ∈G×G× · · · ×G︸︷︷︸

p

|< {x1, x2, . . . , xp} >= G}. We call (x1, x2, . . . , xp)

a generating p-tuple for G.

3.7. De�nition. For a p-tuple (x1, x2, . . . , xp) ∈ X, we de�ne the Padovan p-orbitPap (G;x1, x2, . . . , xp) = {ai} by

a0 = e, a1 = x1, a2 = x2, . . . , ap= xp, ap+1 = e, an+p+1 = an−1 · an+p−1,n ≥ 1.

3.8. Theorem. A Padovan p-orbit of a �nite group is simply periodic.

Proof. Let n be the order of G. Since there are np+2 distinct (p+2)-tuples of elementsof G, at least one of the (p+2)-tuples appears twice in a Padovan p-orbit of G. There-fore, the subsequence following this (p+2)-tuple repeats; hence, the Padovan p-orbit isperiodic. �

Since the Padovan p-sequence is periodic, there exist natural numbers u and v, withu > v, such that

au+1 = av+1, au+2 = av+2, . . . , au+p+2 = av+p+2.

By de�nition 3.7, we know that

au = (au+p+2) · (au+p)−1 and av = (av+p+2) · (av+p)−1 .Therefore, au = av, and hence,

au−v = av−v = a0, au−v+1 = av−v+1 = a1, . . . , au−v+p+1 = av−v+p+1 = ap+1,

which implies that the Padovan p-orbit is simply periodic.We denote the length of the period of the Padovan p-orbit Pap (G;x1, x2, . . . , xp) byLPap (G;x1, x2, . . . , xp) and we call this length the Padovan p-length with respect tothe generating p-tuple (x1, x2, . . . , xp). From the de�nition it is clear that the Padovanp-length of a �nite group depends on the chosen generating set and the order in whichthe assignments of x1, x2, . . . , xp are made.The classic Padovan p-sequence in a cyclic group C = 〈x〉 can be written as Pap(C;x, x, . . . , x︸︷︷︸

p+2

).

It is clear that the period of hPapm is the period of the Padovan p-sequence in the cyclicgroup of order m.We will now address the lengths of the Padovan p-orbits of the quaternion group Q2n .

590

The quaternion group Q2n , (n ≥ 3) is de�ned by the presentation

Q2n =〈x, y : x2

n−1= e, y2 = x2

n−2, y−1xy = x−1

〉.

Note that |Q2n | = 2n, |x| = 2n−1 and |y| = 4.3.9. Theorem. Consider the quaternion group Q2n , (n ≥ 3) with generators x, y. Thenthe lengths of the periods of the Padovan 2-orbits LPa2 (Q2n ;x, y) and LPa2 (Q2n ; y, x)are 2n−1 · 3.Proof. We prove the result by direct calculation. We �rst note that xy = yx−1 and yx =x−1y.

First, let us consider the Padovan 2 -orbit Pa2 (Q2n ;x, y). Then we have the sequence

e, x, y, e, y, x, y2, x, y3, x2, y, x3, e, x5, y, x8,y, x13, y2, x21, y3, x34, y, x55, e, x89, y, x144, . . . .

Using the above, the sequence becomes:

a0 = e, a1 = x, a2 = y, e, . . . ,a12 = e, a13 = x

5, a14 = y, a15 = x8, . . . ,

a24 = e, a25 = x89, a26 = y, a27 = x

144, . . . ,a48 = e, a49 = x

28657, a50 = y, a51 = x46368, . . . ,

a12·2i = e, a12·2i+1 = x2i+2·λ1+1, a12·2i+2 = y, a12·2i+3 = x

2i+3·λ2 , . . . ,

where λ1 and λ2 are odd integers and i is an nonnegative integer. So we need thesmallest integer i such that 2n−1

∣∣ 2i+2 for n ≥ 3. If we choose i = n − 3, we obtainx2n−1·3 = e,x2n−1·3+1 = x x2n−1·3+2 = y, x2n−1·3+3 = e . Since the elements succeedingx2n−1·3, x2n−1·3+1, x2n−1·3+2 and x2n−1·3+3 depend on e, x, y and e for their values, the

cycle begins again with the(2n−1 · 3

)ndelement. Thus, LPa2 (Q2n ;x, y) = 2

n−1 · 3.Now consider the Padovan 2 -orbit Pa2 (Q2n ; y, x). Then we have the sequence

e, y, x, e, x, y, x2, y, x3, y2, x5, y3, x8, y, x13, e,x21, y, x34, y, x55, y2, x89, y3, x144, y, x233, e, . . . .

Using the above, the sequence becomes:

a0 = e, a1 = y, a2 = x, e, . . . ,a12 = x

8, a13 = y, a14 = x13, a15 = e, . . . ,

a24 = x144, a25 = y, a26 = x

233, a27 = e, . . . ,a48 = x

46368, a49 = y, a50 = x75025, a51 = e, . . . ,

a12·2i = x2i+3·β1 , a12·2i+1 = y, a12·2i+2 = x

2i+2·β2+1, a12·2i+3 = e, . . . ,

where β1 and β2 are odd integers and i is an integer such that i ≥ 0. Similar to theabove, we obtain that LPa2 (Q2n ; y, x) = 2

n−1 · 3. �

4. Further Questions

There are many open problems in this area. Below are a few of them:

(1) Does there exist a relationship among the Padovan sequence and the consideredsequences in this paper?

(2) Does there exist a formula for calculating the period hPapm?(3) As it is known that the quaternion group Q2n is a special class of the binary

polyhedral group 〈l,m, n〉, the polyhedral group (l,m, n) is a factor group of〈l,m, n〉 and (n, 2, 2) is isomorphic to the dihedral group Dn. Furthermore, thequaternion group Q2n is known as dicyclic group and it is a metacyclic group.Due to these relations and goal for contributing further researches, we select thequaternion group Q2n for applications of the Padovan p-sequences in groups. In

591

terms of a further research, one can consider the question �What are the lengthsof Padovan p-orbits of the groups which are related to the quaternion groupQ2n �.

(4) What general theories can be obtained regarding the length of the period of thePadovan p-orbit of a general group? For example does there exist a decisionprocess to determine whether, or not, a given group has �nite length?

(5) Let us consider in�nite groups such that the lengths of the periods of the Padovanp-orbits of these groups are �nite. To �nd these lengths it would be useful tohave a program. This would possibly rely on using the Knuth-Bendix method,see [18].

Acknowledgement. The authors thank the referees for their valuable suggestions whichimproved the presentation of the paper. This Project was supported by the Commissionfor the Scienti�c Research Projects of Kafkas University. The Project number. 2014-FEF-34.

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[3] Campbell C. M., Doostie H. and Robertson E. F., Fibonacci length of generating pairsin groups in Applications of Fibonacci Numbers, Vol. 3 Eds. G. E. Bergum et al. KluwerAcademic Publishers, 27-35, 1990.

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Math. Sciences, 4 (34), 1685-1694, 2009.


On twisted Riemannian extensions associated withSzabó metrics

Abdoul Salam Diallo∗, Silas Longwap† and Fortuné Massamba‡

Abstract

Let M be an n-dimensional manifold with a torsion free a�ne connec-tion∇ and let T ∗M be the cotangent bundle. In this paper, we considersome of the geometric aspects of a twisted Riemannian extension whichprovide a link between the a�ne geometry of (M,∇) and the neutralsignature pseudo-Riemannian geometry of T ∗M . We investigate thespectral geometry of the Szabó operator on M and on T ∗M .

Keywords: A�ne connection, Cyclic parallel, Szabó manifold, Twisted Rie-

mannian extension.

2000 AMS Classi�cation: 53B05; 53B20

Received : 22.03.2016 Accepted : 24.11.2016 Doi : 10.15672/HJMS.2017.427

1. Introduction

Let M be an n-dimensional manifold with a torsion free a�ne connection and letT ∗M be the cotangent bundle. In [11], Patterson and Walker introduced the notion ofRiemann extensions and showed how a pseudo-Riemannian metric can be given to the 2n-dimensional cotangent bundle of an n-dimensional manifold with given non-Riemannianstructure. They showed that Riemann extensions provide a solution of the general prob-lem of embedding a manifold M carrying a given structure in a manifold N carryinganother structure. The embedding is carried out in such a way that the structure onN induces in a natural way the given structure on M . The Riemann extensions can beconstructed with the help of the coe�cients of the a�ne connection.

∗School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Pri-vate Bag X01, Scottsville 3209, South Africa, Email: [email protected]†School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Pri-

vate Bag X01, Scottsville 3209, South Africa, Email: [email protected]‡School of Mathematics, Statistics and Computer Science, University of KwaZulu-

Natal, Private Bag X01, Scottsville 3209, South Africa, Email: [email protected],[email protected]

Corresponding Author.

594

The Riemann extensions which are pseudo-Riemannian metrics of neutral neutral sig-nature show its importance in relation to the Osserman manifolds [7], Walker manifolds[1] and non-Lorentzian geometry. In [3], the authors generalized the usual Riemannianextensions to the so-called twisted Riemannian extensions. The latter is also called de-formed Riemannian extension (see [6] for more details). In [1, 6], the authors studied thespectral geometry of the Jacobi operator and skew-symmetric curvature operator both onM and on T ∗M . The results on these operators are detailed, for instance, in [6, Theorem2.15].

In this paper, we shall consider some of the geometric aspects of twisted Riemannianextensions and we will investigate the spectral geometry of the Szabó operator onM andon T ∗M . Note that the Szabó operator has not been deeply studied like the Jacobi andskew-symmetric curvature operators.

Our paper is organized as follows. In the section 2, we recall some basic de�nitionsand results on the classical Riemannian extension and the twisted Riemannian extensiondeveloped in the books [1, 6]. Finally in section 3, we investigates the spectral geometryof the Szabó operator on M and on T ∗M , and we construct two examples of pseudo-Riemannian Szabó metrics of signature (3, 3), using the classical and twisted Riemannianextensions, whose Szabó operators are nilpotent.

Throughout this paper, all manifolds, tensors �elds and connections are always as-sumed to be di�erentiable of class C∞.

2. Twisted Riemannian extension

Let (M,∇) be an n-dimensional a�ne manifold and T ∗M be its cotangent bundleand let π : T ∗M → M be the natural projection de�ned by π(p, ω) = p ∈ M and(p, ω) ∈ T ∗M . A system of local coordinates (U, ui), i = 1, · · · , n around p ∈M inducesa system of local coordinates (π−1(U), ui, ui′ = ωi), i

′ = n + i = n + 1, · · · , 2n around(p, ω) ∈ T ∗M , where ui′ = ωi are components of covectors ω in each cotangent spaceT ∗pM , p ∈ U with respect to the natural coframe {dui}. If we use the notation ∂i = ∂∂uiand ∂i′ =

∂∂ωi

, i = i, · · · , n then at each point (p, ω) ∈ T ∗M , its follows that

{(∂1)(p,ω), · · · , (∂n)(p,ω), (∂1′)(p,ω), · · · , (∂n′)(p,ω)},is a basis for the tangent space (T ∗M)(p,ω).

For each vector �eld X on M , de�

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