Interpolation spaces and interpolation methods

Interpolation spaces and i,~terpolation methods (9.

by ~N. A.RONSZAJN AND E. GAGLIAI~DO

~m.mary. - (See iv, troductio~).

I n t r o d u c t i o n . - The beg inn ing of what we now call in terpola t ion methods be tween BANAC~ spaces was the convexi ty theorem of )][. RIESZ [13] in the late 1920's. This theorem gives an in te rpo la t ion met lmd for couples [LP(di~), Lq(d~t)]. In the late 1930's, MAlCCINKIEWleZ [11] obta ined an ex tens ion of the M. RIESz in te rpo la t ion method to couples formed by weak LP-spaces. In the 1950's E. )I. S~EIN and G. WEISS [14] ex tended the method fu r t he r by admi t t ing couples [Lv(d~), Lq(dv)] with d i f fe ren t measu re s l~ and v. Unt i l the late 1950's, the research in in te rpo la t ion methods be tween sp~ees r ema ined essent ia l ly wi th in the f rame of couples of L ~ spaces. At the end of 1958, J . M . LIOZ,~s gave the f i rs t proof of the interpola t ion theorem for quadra t i c in te rpola t ion be tween HILBER~' spaces t2). This work gave impetus to the research on in te rpo la t ion methods for a rb i t r a ry couples of BA~c~tOK spaces. Since then several au thors have in t roduced and developed a n u m b e r of d i f fe ren t in te rpo la t ion methods for couples of genera l B,t~ACI-I spaces. We ment ion here, for example , the methods of E. GAeLIARDO [3], [4], [5], ft. L. LIONS, [6], [7], [8], [9], A. CALDERON [2], ~. PEETRE [t2], and S . G . KREIN [10].

[n the presence of so m a n y d i f fe ren t in te rpo la t ion methods it seemed t imely to s tudy the genera l s t ruc tu re of alt possible me thods : to de t e rmine all of them and to analyze the proper t ies which are common to all. This is essen t ia l ly the subject of the p resen t paper . The r e sea rch was p rompted by the fol lowing specif ic considera t ion .

(t} Research done under 5TSF Grant G17057 snd OSR Contract N. 583 (!3). i~') Note by N. Aron.szajn. I take this occasion to correct a misunderstanding which

seems to have arisen in connection with references to my work in the first paper of J. L. Ltoxs ([6]). Althoagh foL ~ some time [ was aware of the importaace of quadratic interpolation in the theory of B~SSEr~ potentials and in their applications (see [1l), I did n ~t have the proof of the relevant interpolation theorem, In the spring of 1958 I told J . L . LIoNs about the problem~ and in the fall of 1958 he sent me the proof.

52 N. ARo~-szAJ~ - E. GA( ;L~O: lnterpotatio~ spaoes, etc.

For some applicat ions it becomes apparent that it is not a lways of importance to develope a specific method, but ra ther to determine if a general interpolation method (~) exists which assigns to a given couple of B ~ A c ~ spaces a given intermediate space. In other words, the problem is to characterize those intermediate spaces between two given BANAC~ spaces for which such a general interpolation method exists.

A solution to this problem is given in Corollary [14. VII i . The relevant intermediate spaces are those which we call (~interpolation spaces ~ between two given BA~Ac~ spaces. This result led to a thorough study of interpo. lation spaces which comprises the greater part of the paper and is presented in Chapter II.

A number of unexpected facts concerning interpolation spaces were discovered and will be indicated in the brief summary of the paper which follows.

The first three sections of Chapter I - P r e l i m i n a r i e s , contains material famil iar to all who have worked in the domain, but it is collected here in order to"give a precise and intrinsic definition of the notion of <<compatible BA~AC~ spaces)>. Our aim i~ere is to get rid of the redundant topological vector space in which the BA~AC~ spaces are usual ly supposed to be cont inuously imbedded. There is only one new theorem in this chapter, i.e. Theorem [4.I] of §4. In the first version of the paper this theorem played an essential role in the proof of the main theorems in § 14. However , after the introduct ion of the complete lattice of normalized B-~NAC~ subspaces in § 5, the theorem was no longer needed. We include it in view of its intrinsic interest, and also because it may still be of use for special interpolat ion methods.

Chapter I [ deals with interpolat ion spaces. In § 5 we introduce the normalized BANAC]/[ subspaces of a given B~_NAc~ space. The main theorem is that the latt ice of these subspaees is complete and we give the construct ion of the jo int and meet for an arbi t rary class of such subspaces.

In § 6 we introduce the intermediate and interpolat ion spaces, their normalizations and the B~_~'~Acl~ algebra ~;[V, W] for a compatible couple of BA~ACK spaces V and W; ~;[V, ~¥] is composed of all l inear operators on V + W transforming boundedly V into V and W into W.

(3) We say that a method is "general" if it is defined for all compatible couples of BA~ACH spaces (or~ if we want to avoict any possible antinomies in se~ theory we should rather say ~defined on arbitrarily large classes of couples of compatible B~NACg spacos).

g. ARoyszAa~ - E. Gh(~'LL~m)O: Interpolations. ,spaces, etc. 53

§ 7 deals with the case when V N W is not dense in V or W. We describe the restrict ions which this hypothesis imposes on the interpolat ion spaces.

In § 8 we consider the ease when the conjugate spaces of V and W form, in a natural way, a compatible couple. This is the case when V G W is dense both in V and W. W e then study the relat ionship between the interpolat ion spaces (~f IV, W] and [V*~ W*]. The adjoints of operators in ¢g[V, W] form a suba |gebra ~* of ~[F*, W*]. We obtain the rather unexpected result that ~ ; * ~ ~[V*, W*] i f and only if V N W is reflexive.

In § 9 we consider the propert ies of, and the relat ionships be tween the BANAC]t algebras ~[V, W], ~A (this is the algebra of bounded linear opera- tots on a n interpolation space A) and ~;A[V, W] (the subalgebra of ¢g[V, W] which is the closure of the set of operators such that T ( V + WtcA}.

In § 10 we consider for a B~.NAc~ subspace A of a BA~'AC~ space E,

the space A ~E)- the complet ion o[ A tel. E. (This notion was illtroduced

by E. GAGL][ARDO [4]). We give a formula iTheorem [10. V]) expressing A-(I~') in terms of the conjugate spaces of A and E. The main source of appli. cations o[ the notion of relative completion to the stud~ ~ of interpolat ion spaces is Lemma [10. X]. By using this lemma we obtain the f o l l o w i n g - again, rather unexpected resul t : whereas V-J- ~ , and V N W are always interpolat ion spaces between Vand W, V and W, with some trivial exceptions, are never interpolat ion space.s between V-~ W and V N IV. This result indicates t~aat it is not an ~unesseutiaI>> restrict ion when one considers only couples IV, W] with WC V.

In § 11 we come back to interpolat ion spaces. We construct the minimal interpolat ion spaces containing a g iven intermediate space. In particular, for any element u e V-{- W we construct tile minimal interpolation. space ~;(u) containing u: ~(u) is formed b y all elements Tu for Te~[V, W] For u :~0, V N W is always contained in ~(u} but never dense in it {unless u e V N W when VA W--~;(u)). By using a normalized version ~ , ~ of ~;(~) we can construct all interpolat ion spaces between V and W (see Theorem [11. VII).

In § ][2 we analzze the s tructure of the minimal subspaces ~(u). For u ~ V N I,V, ~(u) always contains [u]--[- V N W, i.e., the subspace generated by V N W and u. The last s tatement in this section is to the effect that ~(u) -- [u] -{- V N W only in ver Z exceptional cases, namely when

u ~ ( v n W)(V+ w). That this case can actual ly arise is shown in an example constructed in the last Remark of the section.

54 :N. A R O N S Z A J N - E . GAGL:[ARDO: Interpolatio~ spaces, etc.

Chapter I I I deals with our main results concerning interpolation methods and interpolat ion theorems.

In § 1 3 we consider triples IV, W; A] i . e . , compatible couples of BA~Ac~ spaces fIT, W] with an intermediate space A between them, and study the unilateral or bi lateral interpolat ion theorems between two triples. The notions and exact formulas needed for the main theorem in § 14 are introduced. In addition, we consider the question of optimal interpolat ion theorems between two triples and obtain, somewhat unexpectedly, that in any such optimal interpolat ion theorem the intermediate spaces in each triple must be, actually, interpolat ion spaces.

In § 14 we pass to interpolat ion methods on a class of compatible BANACH couples. We define uniform and normalized interpolat ion methods. Each uniform method can be normalized. There are no known examples of methods which cannot be rendered uniform. An interpolat ion method assigns to each couple of B¢r~cAc~: spaces an intermediate space. This intermediate space must be an interpolat ion space. Corollary [14. VII] gives the result mentioned previously in this introduction, that for a given interpolat ion space A between V and W (A can be assumed normalized) there exist general interpolation methods (normalized} which assign A to the couple [V, W]. The more general Main Theorem [14. VI] states that if we

have any two classes J{Cg~ of compatible BA~AcI~ couples, then any normalized method defined on J{ can be extended to a normalized method

defined on ~ , and among all these extensions there are two extreme ones.

The original iutention of the authors was to include a study of conjugate and se l f -conjugate interpolat ion methods, and also a study of exist ing interpolat ion methods with reference to the results of this paper. However , the present work has already greatly exceeded its expected length and we had to abandon these plans.

CnAPTrm [.

P r e l i m i n ~ r i e s ,

§ 1. - Linear c . u p l e s . - We say that W is l inearly contained in V (or

V l inearly contains W):

W C V (or V D W t l Z

if V and W are vector spaces, I~ 7 is a subspace of V and the identity

mapping of W into V is linear.

N. Aao~sz.~a~ - E. GAGLIARDO: Interpolation spaces, etc. 55

We will use the convention that a subspace is never empty (contains at least the origin) and hence a l inear mapping always has a non-empty domain.

For any two abstrac~ sets A and B, their intersection, A n B, deter. mines the identification mapping from A to B.

Consider V c E and W c E ; they obviously satisfy the following

property :

( l . 1 ) - The identification mapping z front V into W is a linear isomer. phism.

[1.I] DEFINITIO:N. - Two vector spaces V and W form a linear couple [V, W] if the relation {l.1) holds.

Let IV, W] and IV1, W1] be l inear couples and S a mapping of VU W into V1 t2 W1 which transforms l inearly V into 171 and W into W~. W e say then that S is a l inear mapping of [V, W] into [ V , I¥~]. If the mapping S is one-one and on|o V~ U W1 we say that it is an isomorphism. The inverse mapping is ti~ea also an isomorphism and the two couples are called isomorphic (by S). If V~C E and IiV~CE, then an isomorphism S of [17, WJ onto [V,, ~z~] is called an embedding into E and [I7, W] is then embedded (by SI into E.

Consider a l inear couple [17.. W] and the direct sum, V + ~5 Define the subspace Z of the direct sum as follows:

( i .2 ) z = [{v, = -

where z is the identification mapping from V into W. Then the mapping defined on V and W by

(L3 ) v {v, o ) + z , u,}+z

is obviously an embedding of [V, W] into the quotient space (Vq-W)/Z .

In the remainder of the paper we will always identify the elements of V or W with their images (1.3} and therefore we will write

(1.4j (v-i- w ) / z = v + w

56 ~N. A R O N S Z A J I ~ ~ - E . G A G L I A I C D O : D~terpolatio~:~ spaces, etc.

If V c E and W C E and we form Che sum of V and W i n E - - t e m p o -

rarily denoted tV & W)z - - then thel:e is. an obvi,)us canonical extension of the identi ty mapping of V/_) W into E to an- isomorphism of V + W onto I V + W)E which allows us to identify canonical ly I V + W}~ with V + W. In the following therefore I V + W}E will be denoted by V + 1/V.

The proof of the following statement is obvious.

[1.Ii] - I f S is a linear mapping of IV, W] into [V~, W~] then it has a unique linear extension to a mapping of V + W into V~-~-W~. I f S is an isomorphism onlo. then the extension becomes an isomorfism ol V + W onto V~+ W~. If, in addition, S is an embedding into E, then the extension becomes an isomorphism of V + W into E.

Consider two arbitrary vector spaces V and W. Their identification mapping is not necessar i ly linear (its domain may even be empty). Some- times it is possible to extend the identification mapping to a l inear isomorphism x from V into W. We can then idelitify the elements of V and W such that +~:-~ xv. W e thus obtain h'om V and W a linear couple [I/~, W,]

with identification mapping ~, defined as follows: in V + W consider the subspace ),~ of elements { v , - - x v } ; then the canonical isomorphisms v - - : (v, 0} + Z:, w--+ Iv, 0} + Z:, t ransform V and W o~lto subspaces V+

and W+ of tV~- W}/Z~.

A linear couple IV', W+] is called an extension of a l inear couple {V, W] if V C V' and W C W'. In terms of the identificatior~ mappings

1 l

and ~' of the two couples we have the following obvious statement.

[l.III] - In order that the four vector spaces V, W, V', W' form linear couples lIT. W] and [g'~ W] s~eh that the second be an extension of the first, it is necessary and sufficient that V C V', W C W', that ~ and x' be linear isomorphisms and ~' be an exte~sion of x outside of [i f, W] (i.e., i f w : ~ ' v , then either w - - ~ v or v~ V and sv~ W).

§ 2. Compatible couples. - Let E be aH:USDORFF vector spaee and V a subspace of E with its own topology (not necessari ly the one induced by E}.

W e write

V'CE c

if the identi ty mapping V - ~ E is continuous. We will say then that V is continuously eontai~ed in E ,or that V is a topological subspace of E. If

~-. fl~.RONSZAJN - E . GAGLIARDO: Interpolation 8paees~ e t c . 57

V C E and W C E , we consider on V N W and V-I- W canonical tope- e

logics defined as follows: 10 . on V(~ IV we take the coarsest topology among those Which are finer than the ones induced by V and W; 20 . on ~'--k W we take the finest among those which induce on V and W topologies coarser than the one of V and W respectively. An immediate argument shows that with these topologies Vfq W C E and Vq- W C E .

We will be mostly interested in normed topologies. If the norms on V and W are II V]lv and II ~v ilw, the canonical topologies on V('t W and V q - W are given by

('24) liuilvnw=,nax[i:ulJv, lluijw ], I lu l jv÷w-- inf (ji .vjjr-Fllwjiw)(').

If Y and W are normed spaces and VCE, and WCE~ then obviously c c

we have

(2.2) lu . IC VN W~ Ilu,,--vllv~0, I l u . - ~ v l w ~ O = > v - w e v n W.

This property has a meaning for an arbi trary l inear couple of normed vector spaces and leads to the definition

[2.I] - DEFI~SITIO~Z. - A linear couple IV, W] of normed vector spaces is calted a compatible couple if proper ty (2.2)holds.

Let [V. H'] and [V1, W1] be two linear couples of HAUSDOI~FF vector spaces, and let S be a l inear m:~pping of the first into the second. If S maps cont inuously V into V~ and W into W~ then we say that S is a continuous mapping of IV, W] into [V~, H,q].

If S is an embedding of [V, W] into E (with E a HAUSDOm~Y vector space) which maps continuously V and W into E, we say that S is a continuous embedding. In this case if we t ransfer the topologies of V and W by the isomorphism S onto S(V) and S(W) respectively, we obviously get S(V) C E and S(W) C E .

One proves immediately, by using the preceding considerations, the following theorem :

(4) Othe r (equivalent} no rms are somet imes p re fe rab le . F o r ins tance , w h e n the n o r m s in V a n d W are quadra t i c , t he f o l l o w i n g n o r m s on V fl W and V-t- t} are also q u a d r a t i c :

::u ~ n w _ [ f i u I ~' A 1112 Ul!v_~,W = inf [t lv,lv+li 'vv ,w~ '0 -~-T/,~---*'bl~

Anna t i d4. Ma~ematiea 8

58 .N~. ARONSZAJN - E . GAGLIARD0: I~terpolatioJ~ spaces~ etc.

[2 . I I ] . - TI~EO]~]~)I.- a) Let IV, W] be a compatible couple. Then the canonical embedding (1 .3)of IV, W] into (V-~ W ) / Z (with norm (2.1) on V-~ W) is continuous.

b) Let the normed linear couple [I/, W] be continuously embedded into the Hausdor]7 vector space E. Then [V, W] is a compatible couple and the induced isomorphism of V ~- W into E is continuous.

[2.II1]. REMARK. - I f IV, W] is a compatible couple and W - - VA W C Y it does not follow, in general, that W C V if. i. V = L=(R), W - - Lo~(R) -- the

o,

class of functions in Lm(lg) vanishing outside of a compact). However, i f we change the norm on V to the norm {2.1), the relation W C V follows from the

c

above theorem.

[2.I¥]. RE:~IARK - If [V, TkV] is a compatible couple of Banach spaces and W - - VA W C V then W C V ~by immediate application of closed

c

graph theorem).

i2.V]. REHAI~K. - It is welt known and easily proved that if [17, W] is a compatible couple of BA~ACH spaces, then V A W and V q - W with the norms (2.1} are BA~cl-I spaces.

[2.VI] REM:ARK;- The compatibil i ty of a l inear couple of normed spaces is clearly equivalent to the fact that the identificatiou mapping is

closed.

§ 3. Completely compatible couples. - B e f o r e passing to the main considerations of this section, we make a few remarks.

A completion V of a normed space V is a BA~AC~ space containing V as a dense subspace and such that L I v I ] f f : l l v :v for v e V. The classical

abstract completion ~a (formed by equivalence classes of CAUCI~¥ Sequences in V) is not the only useful one. if V is a subspace of a HAUSDORF~ vector space E, there exists sometimes a completion of V in E, that is, a completion which is C E. If a completion in E exists, it is unique and

will be denoted by I ~E. Any two completions are canonically isometrically

isomorphic. The question of existence of l ~E is solved by the following:

[3.1] TI{EOREM.- ]If a normed space V is a subspace of a Hausdorffvector space E, then ~7 E exists i f and only if: a) V c E : b) every Cauchy sequence

in V converges in E; c) when a Cauchy sequence in V converges in E to O, it also converges in V to O.

N. ARo~sz.~J~ - E. GAGLt2*RDO: Interpolation, spaces, etc. 59

The proof is e lementary but not very short. We omit it since our interest will be centered in the case when E is a BANAC~[ space and this case will be settled as a simple corollary of our for thcoming theorem.

[3.I[]. RE~AR]~. - If E is a BA~AC~ space, the condit ion b) in the above theorem is clearly impl ied by a). However c) is not impl ied by a) (example: on the interval l - - [ - - 1 - - < t ~ 1] take E = L ~ ( I ) , ]tu]IE=!]UI/L,.

v = c ( I ) , II v = t lv I% + I v(O} ?). The main problem of this section is to extend a compatible couple

[ ~ W] of normed spaces to a compatible couple IV, W] of t h e i r completions. If such a couple exists, it is called a completion of IV, W].

[3.III] REMARK. - ~f V and W are subspaces of a HAUSDORFF vector

space E and ~E and ~V ~ exist, then [~z, I~V El is a complet ion of IV, PV].

[3.IV] REMARK.- Two different complet ions are not necessar i ly iso.

morphic : if they are denoted by [l~, 17V] and [~', ~7,], then the canonica l

i somorphisms F , ( -~ V' and I~T( ~ 17V' which leave respect ively V and W

invar iant do not t ransform necessar i ly ~7 (~ ~-TV onto ~7'A I7V'.

Suppose that there exists a complet ion [ ~ VV] of the compatible couple

IV, W]. By the canonical i somorphisms V < - > YV ", W<-> I/V ~ we then t ransfer

the ident i f icat ion mapping ~: of IV, ~V] to ~'V a and ~V¢ a and obtain

thus an ident if icat ion mapp ing z' and a l inear couple ~~' "~' [V~,, W~,] which is a complet ion of [V, W]. Since z' is an extension of the ident i f icat ion

mapp ing -: of [I/, W] and z' is a closed mapping from ]~a to ~d a, the

m a pp ing -~ mus t be closable in I 7" and -I/~d a. A direct t ranslat ion of the last fact is the following proper ty :

(8.1) I f }u~}C V(h V~ and lu,~} is Cauchy in V and W~ then

i[ u, ! t v ~ 0 is equivalent to l] u~ ttw--~ O.

[3.V] DEHlgITIO1,;.- A. compa t ib l e~coap le lIT, W] is called completely compatible (v-compatible) if propert, y (3.1) holds.

[3.VI] REIVIARK.- Obviously, for a l inear couple of BA~xc~ spaces compat ibi l i ty is equiva len t to complete compatibi l i ty.

[3.VII] TI-IEOREM. - Let IV, W] be a compatible couple and ~: its identification mapping, a} All the completions of [V, W], up to isomorphisms, are given by [V~,, I~,] where is any closed identification mapping from V~

60 ~ . AR0:NSZASN - E . GAGLIARDO: [~te~'poIatio~$ spaces, etc.

to W ~ which is an extension of z outside of [V, W]. b) Such mappings ~' exist i f and only i f [V, W] is completely compatible, c) I f [V, W] is c-compatible, then among all ~' satisfying the conditions of a) there exists a

minimal xo-the closure of x in I?~ and ~ ; the identification ~o can be directly described as follows:

(3.2) For v~ V~. w e ~VV a, ~°v= w means that for some sequence

( u,} C V (~ W which is Cauchy in V and W, Ilu, ,~vi] f f ' ~ O

and [I u,~-- w I1¢¢. ~ O.

The proof of this theorem results immediately from the preceding considerations. We note that the extension xo of x is outside of [V~ W] since is closed in V and W.

[3.VIII]. DEFInitIOn. - For a completely compatible couple [17, W] the

completion [V¢o, VtT~0] where z ° is described by (3.2) will be called the minimal completion of IV, W].

[3.IX]. COROLLARY. - Let V be a normed space and E a Banach space such that V C E. The completion of V in E exists i f and only i f the linear

couple [V~ E] ~is c-compatible.

PROOF. - If ITE exists, then [l~ E, El is a completion of [17, Ej; hence [V~ El is c-compatible. On the other hand, if IV, El is c-compatible, we

~Cb consider the minimal completion [V~o, E~o] of [V, El . Since V c E , the

c

identif ication mapping z of IV, E~ is a continuous mapping of V in E. Its

closure ~o in ~'a and E therefore maps ~(a continuously in E, hence identifies l ~" with a subspace of E which is ~E.

[3.X]. RE~ARK. - In some previous papers the first author introduced a notion similar to complete compatibility. The notion was called (( compatibility z and was introduced for two norms defined on the same vector space. We had then two, iu general different, normed vector spaces forming a l inear couple IV, WJ with V : V N W ~ W. The norms were called compatible if the condition (3.i) was satisfied. I t is immediate ly verified that in the present ease~ condition (3.i) alone is equivalent to complete compatibil i ty of [V~ W].

§ 4. Changing compatible couples into completely compatible. - Let us suppose now that the couple [V, W] is compatible, but not necessari ly completely compatible. Our problem will be to change the norms in V, W, f inding two new norms (not necessari ly equivalent to the previous) in order

N . A R O N S Z A J N - E . G A G L I A R D 0 : Interpolation spaces, etc. 6 1

to have a comple t e ly compa t ib l e couple . Th is can be done in m a n y d i f f e ren t ways, bu t a m o n g all poss ib le ways there is a canon ica l one wi th spec ia l p rope r t i e s :

[4.I]. THEOREM. - Let IV, W] be compatible. Let ~ = { II II ~, l[ II ~v} be the class of all norms on V a n d W which make IV, W] completely compa. tibIe a n d such that

(4.1) /I v I 1 ~ II v i1~, II w ] ] ~ II w tlw,

The class C is not empty a n d has a m a x i m a l element; namely , i f we define (with v,~ a n d w,~ ~ V N W, n - - 1> 2, ...)

(4.2)

/ 1!v1111= inf lira I I v - - v , , l ] v for all

i l w I l ~ - - inf l im I~v,,l ' * - -~

1t w - - w . d w for a l l

Iv , } Cauchy in V

and II v,, ]t w - ~ 0,

l w,, l Cauchy in W

and II w. ! ] v ~ O,

we have [fl ]1% II ] ] ~ ] e ~ , and /'or every other [11 H%, II I h ~ ] ~ :

(4.3) 11 v I [v< II v llv, II re t l w ~ II w 11~.

Obvious ly II I]]z de f ined in (4.2) is a p s e u d o n o r m ; we have to prove tha t a c tua l l y it is a norm. If v e V , v = ~ 0 , and for every e > 0 there is

a s e q u e n c e IvY} such tha t :

tl v~,,lt~---.-o, lira I I v - v ~ i ! , , < ~ ,

it is obv ious ly poss ible to c ons t ruc t a s e q u e n c e v,,} such tha t :

II v,~ li w ~ o , lira ll v - v., v = O

but, s inoe we suppose [V, W] compat ib le , this canno t h a p p e n for a v ~ 0. i0 , F r o m the de f in i t i on {4.2} of the new n o r m s I[ il~v, II iw; by t ak ing

vn----0, w, , - - -0 , we see tha t

0

62 N. ARONSZAJN - E. GXGL~Am)O: Interpolatio¢~ spaces, etc.

To prove that [l[ il°v, [] [ t ~ ] e ~ we have fu r t imrmore to prove that by tak ing these norms [V, W] becomes a comple te ly compat ib le couple.

Firs~ of all the compat ib i l i ty (see [2.I]) can be proved as fol lows:

Let I[ un - - v l]°v --,-0, ]l u,~-- w lj]~ . . . . . O, (u, e V A W).

For every s > 0 we can find.:n~, and accord ing to Def in i t ion (4.2), we can f ind v~, w ~ e V A W in such a way tha t :

I I

Consequent ly , it ( u , - - v~-- 'w~) - - v i v < 2~ and I{ ( u , - - v~--w~)-- n, It ~v< 2~.

S ince IV, W], wi th norms ]l [Iv, [I liw, is supposed to be compat ible , v - - w e V A W .

W e pass now to the proof of the comple te compat ibi l i ty .

Le t l u , l C [? N W be a CAucI-[Y sequence in the norms I[ H]z, [I I]~

such that ]J u , , ] ] ~ 0 . ~re have to prove that N u , li~ ~ 0 . Wi thou t loss of gene ra l i ty (by replac ing , if necessary , {u, l by a sui table subsequence) we may as sume

1 1

Accord ing to def in i t ion (4.2) for all posi t ive in tegers n, m ~ e can f ind ~v~,,, and v,, in V O W in such a way t h a t :

1 1 ilu,,~--v~[iv<m,

1 1 lLwn, y . , . - -~v. . .~ , , l tw<~-~n for m ' , m " > m, J]v~}lw"~ m '

1 m2,~ •

F r o m these inequal i t ies it follows, for m ~ n :

,~-1 1 1

~---1 2 II (u , . - - v ~ + " ~v~. ~ ) ]lv < - ,

N. ARONSZAJN - E , GAGLIARDO: Interpolation spaces, etc. 63

and for m " > m ' > m > n :

II ( u . . , - v . . + Y, ~vk. . , , ) - - ( u ~ , , - - v . , , , + ~ wk . .,,,, }! w

% tl v,., }1- + II v~ . ;lw + II Uk - - U~:+I -- W/~,m, IW

3 I < m + ~-~ •

1 Hence by definit ion (4.2) II u,~ tt~ <_ 2~-,-

To complete the proof of Theorem [4.I] we have to prove (4.3).

Let v e V , and ivy} be any sequence in VA W such that.

l im tl v~ - - v,., ~'v = O, II v . ttw - ~ O.

By (4.1)

II s , , iii~ ~ 0 .

Hence, by complete compatibility, II v,~ I!~-~ 0.

Consequently, by (4.1),

P t r : t I l v i w = l i m I l v ~ v , , l l v ~ lira l l v ~ v , , Iv,

which obviously implies the first inequali ty in [4.3). The second is obtained similarly.

[4.[I]. COROLLARY. - Under the hypotheses of Theorem [4.I] i f W is a

BA:NACt]: space, then II w tl~----- ll w IIw for w ~ W.

The proof is immediate since now, in the second part of definition (4.21, there exists w '~ W with I[ ~v~-- w' t iw-- ' -0 and by compatibility of [V, W~, ~ d - - 0 and lira t l w - - w , , ! I w = l t W l l w .

64 -N. ARONSZAJ.n - E. GAGLL~,RDO: Interpolation spaees~ etc.

CHAPTER U .

I n t e r p o l a t i o n s p a c e s b e t w e e n a c o u p l e

o f c o m p a t i b l e B a n a c h s p a c e s .

§ 5. The la t t ice of normalized Banach subspaces o f a B a n a c h s p a c e . - Con. sider at first an abstract vector space E without topology. The class of all subsparces of E is part ial ly ordered by inclusion. In this part ial order the class of subspaces forms a latt ice: for two subspaces Y and IV, the meet is given by V F~ W and the joint by V + ]~. This lat t ice is complete: for any class of subspaces V~, i e I, the meet is given by (3 V~ and

the jo int by E l~ which is the set of all vectors ~2 v~ with v~e F~ and

v~ ~ 0 for at most a finite number of indices i.

From now on in this section, E wilt denote a ]~A~AOH space.

[5.I] DEFINITION.- A subspace V of E is called a BT-subspaee (Banach- topologival subsp~ce) if it is provided with a BANAOH space topology such that V C E.

In a B T - s u b s p a c e the norm is not fixed. However, if for a subspace V there exists a topology making it into a B T- subspace this topology is unique i~).

The ordering by inclusion makes the class of all BT-subspaees of E into a lattice. One checks immediately that for two BT-subspaees, V and W, if we choose two corresponding norms, then the meet and the ,joint are given by V f3 W and V-{- W with topologies de f ined by the norms (2.1). It is therefore a sublat t ice of the lattice of all subspaces. However , as we shall show in Remark [5.V], for inf ini te-dimensional E this lat t ice is not complete, not even ~-complete. Since the fact that V is a B T - s u b s p a c e of E depends only on the topology of E (.but not on the choice of the norm in E) we can speak of BT-subspaces of BT-subspaees . The closed graph theorem gives immediately the following s tatement:

[5.II]. - I f V is BT-subspace of E and W is a subspace of V then W is ce BT-subspace of E i f and only i f it is a BT-subspace of V.

('~} This follows by ar~ immediate application of the closed graph theorem.

~ . ARONSZAhN - E . GAGLIARD0: Interpolation s p a c e s , e tc . 65

[5.III] RE~AI~K.- If {Vk} is an increasing sequence of BT-subspaces,

then U V~: is a BT- subspace if and only if the sequence is stat ionary (~).

In fact, if U V~ were a BT-subspaee the Vk's would be of first category in it unless they were equal to it.

[5.IV]. R E ~ A R K . - If {Vk} is a decreasing sequence of BT-subspaces ,

V ~- N Vk is a BT- subspace if the sequence t l?~} is stationary, where V k is the closure of V in Vk. The proof here is more sophisticated. Wi th any

choice of corresponding norms on ~k, V becomes, in a natural way, a

FRECttET space, and by canonical identification the conjugate spaces of V k form an increasing sequence whose union is the dual of V. If V were to be a B T - s u b s p a c e its ~RECgET topology would be a BA~AC~ topology and we could then apply the previous remark to the conjugate spaces.

[5.V] R E ~ A m ~ . - We can show now that for an inf ini te-dimensional space E the lattice of BT-subspaces is not a-complete. To this effect we need only a strictly decreasing sequence of BT-subspaces , Xk, such that (~ Xk is dense in each Xk. If there were a BT-subspace X which was the greatest lower bound of the X~'s it would have to be C N Xk and, on the other hand, would have to contain all the one-dimensional subspaces generated by elements of C~Xk. It would follow that X - - A X~ - - in cont rad ic t ion to the preceding remark.

It remains to construct a suitable sequence t Xk}.

Take a sequel~ce of l inearly independent normalized elements u~,, II u , l l E : 1, n - - 1, 2 . . . . . Denote by c,, the largest positive number su~ch

that c, ~ I ~ k l ~ ] l Z ~UklIE for all systems of n complex numbers ~,. k --I /¢~1

Clearly, 1 ~c~c~+~. For a sequence of positive numbers A.~ 1, consider the subspace X of all vectors v representable as a series Z~,u,

convergent in E with z--A" l ~ I ~ o c . It is easy to show that for a ~n

vector v there exists at most one such representat ion. Therefore we can

put Il v l l z - - ~, A, . c-~[~ [. Thus X becomes a BT-subspace of E. By choosing,

for a positive integer k, A n - - 2 '~i '~ ' ,~) we obtain the subspace Xk. Clearly, the subspaces Xk form a strictly decreasing sequence. Furthermore, since each u,,~ NXk this intersect ion is dense in every Xk. Thus we have constructed the desired sequence {Xk}.

(~) I .e . , for some k0, V k ~ Vko for k ~ k 0.

Annalt di MaSematiea

66 ~N. ARONSZA;IN - E . GAGLIARD0: Interpoh~tion spaces, etc.

[5.YI] DEFINITION. - A subspace V of E is called a Banach subspace of E if it is a B~N_~cH space cont inuously contained in E.

It follows that a BANAC~ subspace is obtained from a BT-subspace by fixing a corresponding norm in it. A B T - s u b s p a c e with two different corresponding norms gives rise to two different B~NAc~ subspaces.

For BANACH spaces we introduce the following part ial ordering.

[5.VII] D]~F1NI~IO~.- W e write W C V if V and W are B ~ - A c ~ spaces, W i s a B~N~c~r subspace of V and for w o W , []W!!w~]]W[!v. If W ~ V and W ~ IT, then we wilt wri te W ~ V; in this case V and W are equal as well as their norms.

Wi th this ordering, the class of BAN~C~ subspaces of E forms a latt ice: the meet and the jo in t of V and W are given by V N W and V - { - W with the norms descr ibed by (2.1}. W e denote this meet and jo int by

(5.~) vN w, v ~ w .

[5.VIII] DEFINITION.- A BANACIt subspace V is called normalized if V ~ E . A system of BANAC]Z subspaces V~, iE I , is called normalized if each V~ is normalized and IT/C Vj implies V ~ V i.

/£ simple inspection of the norms (2.1) gives the s ta tement :

[5XX]. - I f two Banach subspaces V and W form a normalized system, then the four subspaces V, W, F~J W, and V(~ W also form one.

[5.X] LE~I~A. - I f the system IVy}, k == 1, 2, ..., n, is normalized and V~÷~ is a BT-subspace, there exists a corresponding norm in V,+~ such

that the system IV t:}, k - - 1 , 2, ..., n, n -~ 1, is also normalized.

PRoof . - In fact, we can choose to start with any corresponding norm in Vn+l making it into a BA~ACtt subspace. Consider all the indices k ' 1 < k ' ~ < . . ~ n for which V~,D V,+I~ and also all the indices k i '~

k~'~ . . . ~ n for which Ve, C V,~+I. Then it is immediately verified that

fhe formula

(5.2)

gives the subspace G,+l provided with a norm for which our s tatement

is true.

The last statement, by induction, leads to the~fol lowing proposition.

1~. ARONSZAJ'N - E . GAGLIARD0: Interpolation spaces, etc. 67

[5 .XI ] . - For every countable system of BT-.subspaees we can find cor. responding norms for its subspaees which make the system normalized.

We come now to the main theorem of this section.

[5.XII]. T~rEORE~. - Let t Vi }, i e I, be an arbitrary system of Banavh subspaces

a) there exists a lways the meet (~ V~;

b) i f there exists a Banach subspace W such that V~ ~ W for every i, then there exists the jo in t ~ ~ .

~ I

P ~ o o s . - a) W e define

(5.3) ve (~ V~ if y e n V~ and l tvII (~ -~ sup lJVllv.<c~.

It is obvious that the set ~ V~ so defined is a subspaee, and that ] lvl l~v. is a norm on this subspace. To prove the completeness we notice that a CAUCEY sequence {vkl in this norm is necessari ly CAUC~[Y in every V~ and hence in every V~ converges to the same element v for which we then get

I ] v - v k ] l ( ~ v ~ - - s u p ! [ v - - v k l l v i - - sup lim H v m ~ v k l ] v i

__~ sup s u p l l v , ~ - - v ~ l l v / - - sup I I v ~ - - v ~ ] l ~ v i.

Thus (~ V~ is a BANACg space and since it is clearly a BAbTACK subspace of each V~ it is also one of E. It is immediately checked that the definition (5.3) gives, in fact, the greatest lower bound of the system {V~}.

b) W e define the jo in i as follows

(5.4) ~) ~ is the set of all v e E such that there exists an admissible sum

Zv~ wi th the properties v i e V~, ~A [lv~livi<cx~ and that V-v~ con-

verges to v in E.

(5,4') II v !Iv v,. = inf ~ If v~ lJvi, the in f imum being taken over all admissible

8~A~n8.

~rom the propert ies of an admissible sum ~,v~ it follows that at most an enumerable number of v(s differ from zero. Therefore the sum is, de

68 ~ . ARONSZA3N - E . GAGLIARDO: In terpo la t io~ s p a c e s , e t c .

facto, an enumerable sum. Since I J v ~ I I ~ IJ v~i lw it follows that the series E v~ converges in W, and by compatibi l i ty converges in W to v. From (5.4') we then get

(5.5) I] v 1 1 ~ £ 11 v I1~ 5 "

It is clear that (~V~ is a subspaee of E and that t t v t l ~ is a norm on this subspace. To prove that this norm makes UV~ a complete space, consider a CActi-ix sequence (u (~) } in it. Wi thout toss of general i ty we may assume that this sequence satisfies the condit ions:

oo

u ( ° ) - - O , ~ II u (~) ~ u(k-~) l lu v~. < ~ . k ~ Z

There exist admissible sums Ev~ k), k ~--1, 2, ..., such that i

v~ ~) -" u (~) - - u (k-~) i n E a n d ~, I1 v! ~) ttv~, < II u k - - u (~-~) lIu v + 2 - k . i i

It follows that ¢2(3

~. :~ tl v~ k) Jlv~ < ~ . (5.6) t E I k~z

Therefore for each i, Y~v~ ~) converges to some element v'~ in V~ and k

]L v'~llv~ ~ c~. Hence Ev'~ converges in W to an element u and thus also

converges to u in E (since W C/~}. Therefore u e (~ i~ and we have, ff

k

II u - - u(~) iIu v, = it u - - :~ (u(t) - - u(~-~) )Nv v.

k

~ It v ' ~ - ~ v 7) Ilv, 5=1

cx~

By (5.6) we then get I l u - - u (k)]]Uv~.--~0 for k - - ~ . Hence ~V~

:N. ARo~szxJ~ - E. GAGLIARD0: Interpolation spaces, etc. 69

is complete. By (5.5} we have ~ W and hence UJ i~ is a BANAC]~ subspaee of E. It remains to show that it is the smallest upper bound for the system / V~!.

That V~,C U)Fi for each index i' is shown by assigning to any element v ~')eV~, the admissible sum Ev~ where v ~ = 0 for i ~ i ' and v~--v(~') for i : i ' . Suppose now that V ~ V~ for every i. Then, in the, same way as we proved that W ~ U) V~, we prove also that V ~ U) I~.

Our theorem proves that the latt ice of all BA~AcH subspaces of E is boundedly complete. That it is not complete is shown by the trivial example of B~CAOl=[ subspaces E~ which are equal to E but have for norms s I[ II ~. It is quite clear that this class does "not have an upper bound.

W e will denote by £(E} the class of all normalized BA~AC~r subspaces of E. Our theorem gives

[5.XIII] COROLLARY. - ~(E} is a complete tattice.

[5.IX] R E ~ A R K , - W e could proceed to define the joint U)V~ in the

following way. Consider first the algebraic sum E Vi composed of all vectors v - - E v ~ where only a finite number of v~'s ~= O. For all such sums one

*2V~ considers then the inf imum of EII v*l!r~ and takes it as [] v ll'zv, is then a normed space cont inuously contained in E. In general, this space will not be completely compat ible with E. By applying Corollary [4.II] we obtain then, canonically, a new ~orm Ilvjl~v + on E Vi which makes it completely compatible with E. If we then form the completion of Y,V~ in E we obtain exact ly U V~ (~).

By an example we will now show that the subspace v l~ with the norm }[vtl'~v~ int roduced above ,actually may not be completely compatible with

E. To this effect consider an infinite dimensional BA~cAcg space E, and in this space a sequence of l inearly independent elements u~ such that I] u,~ []z -- 2-'*, n - - -1 , 2 . . . . . Form the one-dimensional subspaees Vi as follows: V~ is generated b$ u~ and V~ is generated by u i - -u~_~ for i > 2.

On ~ take the norm of E; thus V~ is a normalized BA~ACK subspace of E. The elements in EV~ are of the, form v - - E v i with finite number of non-zero terms. Since the u~ are l inearly independent for each such v the representa t ion Ev~ is unique. Hence l] v !]'zy i = E 1[ viH~. It follows that for

(7) In the first arrangement of the proofs of the theorems in Chapter ] I L we were using this procedure to define ~I~.. This led to the considerations and theorems of Section 4.

70 N. ARONSZAJN - E. GAGLIARDO: I'nterpol¢tior~ spaces, etc.

the elements u~ = u~-~ ~ {u~-- u~_~) we get ilu,~ i l '~ --][u~ lIE Jr ~llu~--u~-~llE; / : 2 i = 2

hence these norms increase with n. On the other hand, for n > m,

i • _ _

i ~ m @ : [

hence lug} is a C~go~Y sequence in the norm ]l lt'~v~ which converges in E to ~ero, whereas lira 1[ un II'z¢i ~ 0.

[5.XV] REg£RK. - In proposition [5.XI] we proved that any enumerable system of BT-subspaces can be normalized. This is not true for an arbi t rary system of BT-subspaces ; it is not t rue for the system of alI BT-subspaces . In fact, if we could normalize the whole system of BT-subspaces of E they would clearly form a complete lattice - - in contradiction to Remark [5.V].

[5.XVI] DEFINITION.- Let A be a Ba~cAo~ space and ~ a positive number. We will denote by ~A the space A with the norm 11 v I[pA = a II V[[A. Thus 0A is a BANA0]t space and ~ A ~ A , ~ A , or ~D A, depending on whether ~ > 1, = 1 , or < 1 .

The following proposition is obvious.

[5 .XVII ] . - a) p '(~A)= (f~)A;

b) f~AU,o'A --(min(,~, p')tA, ~At'~I f A - - ( m a x (~, ~'))A;

c) i f A c B , then ~ A ~ 2 B ;

d) for any family of Banach subspaces of E, t V~l, i e I,

(~ V~-'(~ ~V~, and i f UV~ exists, then ~ ) V~= ~J~V~.

[5 .XVII I ] . - Let {~}, i - - 1 , 2, ..., n, be a finite system of Banach subspaces of E such that they are mutually distinct as subspaces. There exist positiw numbers, ~1, pa, ..., p,~ st~oh that the system {~V~} is normalized.

\ .

Pnoo•. - We prove this by induction on n. I f n - - 1 , since V1 C E , the

identi ty mapping V~--,-E is bounded with bound c > 0 and we can then take ~, ~-e. Suppose that the proposition is already proved for some n ~ 1, so that we have the positive numbers ?'~, ..., ~',~ making the system {~'iVi}, i----1 . . . . . n normalized. To prove our s ta tement for n - ~ 1, we proceed as in the proof oi Lemma [5.X] by considering the indices k', ~ k'2 < . . . ~ n

~ . ARONSZAJN - E . GAGLIARDO: Interpolation spaces, etc. 71

for which Vk, D V.+~ and the indices k'~' < k~' ~ ... :~ n such that Vk,, C V,+~. If E is not among the V,'s, with i ~ n , put p'oVo~E so that 0 is among the indices k'. Denote by ck, the bound of the identi ty mapping V,~+~'k,V~, and by ck,, the bound of the identity mapping ~'~,,Va,,--~ T~+~. We choose then ~; = ~'~ for i ~ n and i not among the indices k"; ,~ .+~--max c~,; for

k' i among the k", ~ ~ ~'~ (max c~,)(max c~,,). It is then immediately checked

that the system ! ~ , V~}, i ~ n - ~ 1 is normalized.

[5.XIX] t~ES~AtlK.- The normalization procedure described in the preceding proof may be considered as simpler than that described in the proof of Lemma [5.X]. However, the procedure above does not lead to a normalization of an arbi t rary enumerable system, since in each step of the indue- tion we h,~ve to change norms established in preceding steps.

§ 6. The Banaeh algebra ~[V, W], in te rmedia te and interpolat ion spaces for a compatible Banach couple IV, W]. - In this section IV, W] will denote a compatible couple of BA~'ACH spaces (~). With the norms (2.1), V-~- W and V(% W are BA~AC~ spaces and V, W and VA W are BA~ACH subspaces of V-l- W such that V ~ W ~ lz-4- W and V ~ W ~ V A W .

For a l inear continuous operator T on a BA~AC]t space A we write I TIA for the bound of T on A, i .e. , the smallest constant c with ]1 Tu !IA ~ c[I u ]]A for every u e A. We will wr i te ~'~A for the BANACtt algebra of all such operators. We recall that a mapping T of V O W into V(2 W is called a l inear continuous operator on the couple IV, W] if it t ransforms l inearly and continuously V into V and W into W. Such a mapping has a canonical extension to a l inear mapping of V + W into V + W; also it l inearly t ransforms VC~ W into VN W.

[6.I] - I f T is a linear continuous operator on [~/, W], then

(6.1) [TIv+w~max (IT[v, [ TIw), I T r v n w < m a x ( IT v, I TIw).

PROOF. - For u e V-4- W, we have, by (2.1}, II Tullv+w~ inf (ltTvlivd-

+ l l T w l t w ) £ inf ( tT]v l iV] ]vd- [ T twllWllw) which proves the first

part of (6.1). If u e VNW, then [] Tullyaw-- max (H Tullv, l] Tullw) S~ ~_ max (I TIy I1 u [iv, ] T Iw [I u []w), which proves the second part.

(s) W e r e m i n d the r eade r that for BANACH spaces, ,,compatibles) is equ iva len t to comple te ly compatible ,,.

72 N. ARONSZA;IN - E . GAGLMaDO: Interpoh~tion s p a c e s , e tc .

[6.II]. DEFIni t ion. - We denote by ~[V, W] the class of all l inear continuous operators on [V, W]. For T e ~ [ V , W] we put I Tl~;iv, w ] - - max( tTlv, ITIw).

In the present chapter the couple [I~ W] will be fixed and we will write, briefly, ~ and 1 T I ~ for ~;[ V, W] and I T l~;[v, w] respectively.

The proofs of the next two propositions are immediate and we omit them

[6.III] - ~ ~[V, W] is a Banach algebra with the natural operations aS~--~T, ST (~, ~ scalars, and S, T i~ ~). and with the norm I T t~ . (9).

~; is a sub-algebra of the Banaeh algebra of all bounded linear operators on V 2 7 W in which it is continuously contained.

[6.IV]. - Consider the direct sum of the Banach algebras ~ v and ~ w ,

~;v-~Cgw, with the norm I] iT~, T2} l l - - m a x (I T~Iv, I Tz[w). For each Te¢~, define T~ and T2 as restrictions of T to V and W. The mapping so established,

of ~ into ~ z - ~ ¢gw is an isomelric isomorphism of Banach algebras. The

range of this mapping is the set of all co~ples t1'1, T2} e ~ v ~ - ~ w i satisfying the condition

(6.2) For u e VQW, T , u = T~uE VQW.

[6.V] D~FINITIO~. - An intermediate space between V and W is a BAtCAe~ subspace of V 27 W containing V Q ~ . A is a normalized inter. n~ediate space between V and W if V27 W ~ A D V Q W.

[6.VI] - a) Each intermediate space A can be provided with an equivalent norm which normalizes it; for instance, one can choose the norm of [AC~(V + W)]U(~Q W~.

b) All normalized intermediate spaces between V and W form a complete lattice under the ordering C .

The proofs follow immediate ly from Lemma [5.X] and Theorem [5.XII] respectively. ~Notiee that if VV~ W ~ 10), and we put only the restr ict ion V + W~DA on the in termediate spaces A, the result ing latt ice would not

be complete (not even ~-complete).

[6.VII] DEf in i t ion . - An intermediate space A between V and W is called an interpolation space (between V and W, or in [V, W]) if T ( A ) c A

(~) I n p a r t i c u l a r we h a v e f S T I ~; ~ I S I ~ ! T I ~7"

N . ARONSZAJN - E . GAGLIARD0: Interpolation spaces, e tc . 73

for every T e ~ . An interpolation space A is called normalized if it is a normalized intermediate space and satisfies the condition

(6.3~ t T I A < i TI~ , for all T e E .

[6.VIII] REMAnK.- We could try to introduce a more general notion of an interpolation space A by requiring, only, that it be a BAl~ACI~ subspace of V + fi/ with the property T ( A ) C A for every T e E . However, under these conditions, except where A - - ( 0 ) , A must be an intermediate space. In fact, if 0 @ a e A C V + l~; there exists a continuous l inear funct ional f(~c) on V + W such that f(a) = 1. If u is an arbi t rary element of g ( ~ W, the i inear operator T x = f ( x ) u clearly belongs to E. Hence u = T a e A .

By Proposition [6.1] we have

[6.IX] - V + W and V (~ W are normalized interpolation spaces between V and W.

[6.X] LE~t~[A. - Let E be a Banach Space,and A, B two Banach subspaces of E. I f T e ~ E and T ( A } C B then T transforms A contiuuously into B.

PROOF. - By the closed graph theorem it is enough to show that the restr ict ion TA of T to A is a closed mapping of A into B. Suppose that a~ ~ a in A and Ta,--~ b fin B. Then, since A C E and B C E , it follows

c

that a , - + a in E and Ta, . - -*b in E. Therefore b - - T a , which finishes the proof.

[6.XI] THEORE~L - Let A be an interpolation space between V and W. There exists a constant c > O such that t T~]A <~C I TI~; for every T e E ,

where TA is the restriction of T to A.

PROOF. - Since, by Proposition [6I], T ~ v + ~ , the preceding lemma gives T.4~ ~A. Thus the theorem states that the l inear mapping T---~TA of

into EA is continuous. Again, by the c]ose~d graph theorem, it is enough to show that if T(k)-~ T in ~ and T(Ak)--.-S in ~ , then S - - T A . To this effect consider any element a e A . Then clearly T( k) a--~ Ta in V + W a n d , T < k ) a ~ S a in ,4, and since A C V + W we get S a = Ta. Hence S = T A .

[6.XII] - I f A and B are interpolation spaces between V and ~¢¢ so are also A ~ B and A(~B. Thus the class of all interpolation spaces between V and W is a lattice; it is actually a sublattice of the lattice of all inter. mediate spaces.

Annal i di Matemat iva 10

74 N. ARONSZA~ - E. GAGLIARDO: Interpolatio~ spc~ces, etc.

The proof is immediate .

[6.XIII] TI-IEOI~E~L - All normalized interpolation spaces between V and W form a complete lattice which is a sublattice of the lattice of all normalized intermediate spaces.

P]~ooF. - 0n ly the first par t of the theorem requires a proof. Let tA~!, i ~ I , be an arbi t rary family of normal ized in terpola t ion spaces. We have to show that ~ Ai and UA~ are normal ized in terpola t ion spaces. Since they are normal ized in te rmedia te spaces, we need only prove that T(f~Ai)C(~ As and T(UJA~)C UJA~, and that I T [ ( ~ A . ~ [ T[ v and [ TIUA <~ [ TIV.

Suppose a e r i A l . Then a e A i , i e I , and [ [ a l l ( ~ , ~ i : s u p []a[[A{<C~. i

Hence TaeA{ and [[ Tat[Ai< = [ T I v i l a I t A ~ and thus TaE(~A~ and

]i Ta[[(~A~ [ T I v [[aH~A i.

Suppose now a e UAi. Then for any e > 0, there exists an admissible sum ~ai wi th a - - ~ a ~ in V + W and ~ l l a~ l [ . 4 , .< [ [ a l [uA¢+~ . I t follows

that Ta -- Z Tai in V + W and Z il Tai ttA~ ~ ! T tV Z H ai tt 4, < t T I V(I[ a HV.~, + ~). Therefore Tae ~JA, and II T a i i v j . ~ l Ti~;(llaltvJ~+*). Since • is arbi-

trary, the proof is f inished.

[6.XIV] - Let A be an interpolation space satisfying the condition (6.3) (x0).

a) For every positive ~, ~A satisfies (6.3).

b) I f B is a normalized interpolation space, then AUJB and Af~B also satisfy (6.3).

c) I f B and C are normalized interpolation spaces then (A (~ B)U)C is a normalized interpolation space.

P R o o f . - a) is obvious, b) follows from Proposi t ion [6.I] if we replace [V, W] by the couple [A, B] and notice that for Tecg[V, W] we have I T [ A ~ [ 7'[V and ] TtB~I TIV. To prove c), we r emark that the

BAI~AC~t subspace D-----(A(~B)~C satisfies (6.3) by vir tue of par t b). On the other hand, since V + W ~ B ~ VN W and V + W ~ C ~ VNW, then also D satisfies these inequal i t ies .

[6.XV] - Let A be an interpolation space. Define for a e A,

(6.4) II all'A---sllp [ITalIA, the supremum taken over all T e ~ ~vith l TI v <_~ l. 11 a]t'a is a norm equivalent to the original [I aliA. The space A,

provided with the norm (6.4) satisfies condition (6.3).

(i0) A may or may not be a normalized intermediate space.

N. ARONSZAJN - E. GAGLIAI~DO: Interpolation spaces, etc. 75

PROOF.- For a ~ A , denote by a the linear transformation of ~; into

A defined by a T - - T a . Clearly II a Il'A --" [ a[a,~; (x~). It follows that ilaH'a

is, in any case, a pseudo-norm. On the other hand, by applying a to the

identity mapping we get [alA,~ ~ [I alia and by using Theorem [6.XI] and

(6.4) we get [I,a 1t'.4 ~ c ][ a[[A. Hence (6.4) gives a norm equivalent to I[ a Ila. Finally, to prove condition (6.3) it is enough to consider it for opera-

tors SeCg with I S I ~ = I " We have then

I! 8 a ll' = I1TSa liA < sup it T ' a lIA = I1 a II'A. I TiV~_t IT ' iV_~

[6.XVI] COROLLARY.- Let A be an interpolation space between V and W. We write A' for the space A provided with the norm {6.4). Then the space [A'(~ IV-}- W}]UI V(~ W) is e~ual to A and h~s a norm making it into a normalized interpolation space.

The proof follows immediately from [6.XV], [6.IX], and [6.~IV, c)].

[6.XVII] THeOREm. - a) I f A and B are interpolation spaces between V and W and C is an interpolation space between A ancl B, then C is an interpolation space between V and W. b) I f A and B axe normalized interpolat ion spaces between V and W and C is a normalized interpolation space between A and B, then C is a normalized interpolation space between V and W.

PROOF.- It is clear that [A, B] is a compatible couple of BA~,~CH spaces and that for TEcC[V, W] the restriction of T to A-[ -B belongs to CC[A, B]. From this remark a) follows immedialely. Under the conditions of b) it is clear that C is a normalized intermediate space between V and W. Furthermore, I T I c ~ max (I T]A, I TIB) ~ ] Ti,~;'.

[6.XVIII] RE~ARK.- If A, B, C are interpolation spaces between V and W and C is an intermediate space between A and B, it does not follow, in general, that C is an interpolation space between A and B.

§ 7. The ease when V N W is not dense in V + W . - For a subset C of a space A the closure in A will be denoted in general by C -A. For brevity, the closure in the space V-{-W will be denoted C. As in the preceding section IV, W] will be a [ixed compatible couple of BA~ACH spaces.

(ii) ]~or two ]SANAOH spaces X, Y, and a linear mapping S of X into Y we denote the bound of S by 1SIY, X .

76 :N. ARONSZMN - E. GAGL:[ARDO: Interpolation spaces, etc.

[7.I] LE~MA. - Put

(7.1) V~ = ( v n w) v,

Then the follotving relations hold.

W~ - - ( v n W) W.

(7.2) F = v + w: , = w + v:,

(7.3) vTn = ( V n W ) =

PaooF. - In the canonical mapping of V + W onto V + W (see (1.4)) the

inverse image of V is clearly V~- (VN W). The closure of the last set in

V ~ W is therefore the inverse image of V. Hence, we get for this inverse

image the subspace V + W~ which leads to the first formula in (7.2)-; the second formula is obtained similarly.

Since the topologies in V and W are f iner than in V-~ W, we have

V N W D ( V N W) D V ~ + W ~ . On the other hand, by (7.2), V N W -- ( V + W~) n (v~-{- w) -- v ~ + w~, which gives (7.3).

The notation (7.1) will be mainta ined through this section. V~ and W~ are closed subspaces of V and W respectively and will be provided with

the norms of V and W. V, W, and V N W ~ - V~+ W~ are closed subspaces of V + W and will be provided with the norm of V + W.

[7.II] - 111, W~ V, W, and V ~ - W ~ are normalized interpolation spaces

between V and W.

The proof is obvious.

[7.III] L E P T A . - Let A be an interpolation space behveen V and W.

a) I f A C V then A D W ; b) i f A C W then A D V.

PROOF.- It is enough, obviously, to prove a). If A~: ~ there exist

a and % such that: 0 ~ a e A , a~ V, ~ is a bounded l inear functional on

V--}- W vanishing on l/, and %0(a)-- 1. Let w be an arbi t rary fixed element of W. The l inear mapping T u = ~(u)w beloflgs clearly to ~C. Hence w = 7 'ae A which finishes the proof.

[7.IV] THEOREm.- I f A is an interpolation space bet~veen V and W,

~*. ARONSZAJ'N - E . GAGLIARDO: Interpolatio~ spaees~ etc. 77

it must satisfy one of the four conditions:

1 o) A-~ V + W , 2 o ) F C A C ~ ?,

3 o ) W c A c i V , 40 ) V N W C A C V~+ W,.

PROOF. There are four mutual ly exclusive possibilities for A.

1 o) A@~" and Ac~ W; 20 ) A ~ l~ and A C :~Y; 3 °) A@ V and A C I ~ ;

40 ) A C V and A C W. By the preceding lemma these four possible situa- tions lead to the four corresponding cases of our theorem.

An immediate corollary of this theorem is the following.

[7.V] Coao~L~t~Y.- The

which are closed subspaces of

W + V~ -- ~I -z, and V -[- W.

only interpolation spaces between V and W

V + W are V~+ W I - - ( V N W), V + W ~ - ~ V,

[7.VI] R E ~ A R K . - a) Suppose that V ~ V and W~ ~ W, i .e . , that VC1W is nei ther dense in V nor in W. The four cases of Theorem [7.ZV] are mutual ly exclusive and represent disjoint classes of interpolation spaces between V and W, exhaust ing all of them. bt Suppose now V~- -V and W~ ~ W (or, symmetrically. V ~ V. ~t~\ : W), i .e : , that v~n w is dense in V but not in W. The classes of interpolation spaces represented by 3 o ) and 4 0 ) are disjoint and contain those represented by 1 °) and 2 0 ) respec. tively, c) Suppose finally that V~ : V and W~'= W, i.e., VN W is dense in both V and W (and then also in V ~ W ) . The classes of interpolation spaces corresponding to 1°), 2°}, and 3 ~} are contained in the one corresponding to 4o).

[7.VII] TgEORE~.- a) I f AI is an interpolation space between V~ and W~, then V-b A~, W ~ A I and AI are il~terpolation spaces between V and W corresponding to cases 2o), 3o), and 4 °) of Theorem [7.1¥] respectively.

b) Assume that the following property holds.

(7.4) Each mapping from ~vl and ~dw~ has an extension to a mapping in ~dv or ~ w respectively.

Then for any interpolation space A between V and W, AI-~A (~ (VI+ W1) is an interpolation space between V1 and W1 and i f A belongs to the case 2 °) or 3 °} or 40 ) of Theorem [7.IV], we have A : V - I - A 1 or A -~ W-i-A~ or A ----At, respectively.

78 ~N. ARONSZAJI'¢ - E . GAGLIAI~DO: Interpolation sp,aces~ etc.

P R o o f . - For part a) we have only to refer to Theorem [6.XVII], Proposition [6.X[I] and formula ~7.2).

Assuming the hypotheses of part b), we notice that for every T e ~:[V:, W:] there exist extensions T~CCv and T2+CCw of the restr ict ions Tv, and Two. Since V~N W1-----VN W and 7'v~ and Tw~ satisfy property (6.2) relative to V: and W~ (see [6.IV]), this property also holds for 7'1 and T2 relative to V and W. It follows by [6.IV] that T~ and T2 are restrict ions of an operator 7"erC[V, IV]. A~ being an interpolation space between V and W, it follows that T(Ax)--T'(A~)C A~. The remain ing statements of part b) result immediately from formula (7.2).

[7.VIII] REMARK. - We d~o not know if, property (7.4) is necessary for the validity of part b) of the last theorem. The property <<each T e ~ [ V x , W1] can be extended to a T e~C[V, W])>, which is probably weaker - but also more complicated to c h e c k - would suffice, i more customary proper ty which implies t7.4), and is presumably stronger, is the fol lowing:

(7.4't There exist continuous linear projections of V onto V~ and of W onto W~.

This proper~y holds, for instance, when V and W are I-~I:LBERT spaces. Another example where this property holds is when V - L*(R~), W - - t h e class of all totally finite signed BOREL measures in R"; we have here v n W = L~(R'~)NLI(R% V~ -- V and W1 -- L~(R ~)

[7.IX] REMARK. - The interest of Theorem [7.VII] lies in the fact that when it is valid it allows us to reduce the study of interpolation spaces to the case when VN W is dense in V and W (hence also in V + W). Almost all interpolation methods in the l i tera ture produce classes of interpolat ion spaces depending on one or several continuous parameters. These interpolation spaces are in .the case 4 o ) of Theorem [7.IV], and for most of them it can be proved that they are interpolat ion spaces between ]71 and W1.

[7.X] RE:~[ilC, K . - We finish this section by a simple il lustrative example. Suppose that V N W is a closed subspace of V as well as of W. It follows then by (7.2) that V, W and v n W are closed subspaces of V + W. To each of the cases of Theorem L7.IV] there corresponds only one interpolation space: V + W , V, W and VN IV, respectively, Consequently, between V + W and V N W there are only two interpolation spaces: V + W and V N W . Hence, in this case, V and W are not interpolation spaces between V + W and V N W . This is a special case of a general resul t {see Theorem

[lO.XVI].

N. ARONSZaaN - E. GAGLIARDO: Interpolation spares, etc. 79

§ 8. Conjugate couples. - As in the preceding section IV, W] is a f ixed compatible BA~AC:~ couple. We shall investigate the conjugate spaces V*, W*, especially in cases when they form canonically a compatible couple.

We recapi tulate first the way in which the BANACI~ spaces V N W and

V-~-W were formed in Sections 1 and 2. We form V q - W and its closed subspace, Z - - [{v, w ! : w - - - - v ] . Put t ing the norm II {v, w} fl --

max { I] v]Iv, ]] w]'w) on V4- W we get a canonical isometric isomorphism of -VQW onto Z: u ~ I u , - - u l . Choosing the norm L[~v, wlH=!lv~fvq-llwllw we obtain a canonical isometric isomorphism of Vq--W onto ( V 4 - W ) / Z : u = v ÷ ~ v ~ { v , wt + Z.

We will use the following well known theorems. 1 o) The conjugate space of a direct sum is the direct sum of conjugate spaces. If the chosen norm is max (I] v ltv, il w lIw), then the conjugate norm is II v* t~v*l~ -k I] w* l!w*, and if the chosen norm is !l v l[v q-llwlIw, the conjugate norm is max(ll v*tlv, , Hw*!lw.). 20 ) If A is a closed subspace of a BA~rAO~t space E then A * ~ E * / A I , and ( E / A ) * ~ A I , where Al is the the subspace of funct ionals from E* vanishing on A.

It will be convenient for us to define the pair ing between V4-W and

V*4-W* by the scalar product,

(8.1) < { v , w}, {v*, w ' i > - - < v , v * > v - - < w , w*.>w;

this deviation from the usual definition by sum obviously does not change any of the previous statements. One checks then immediately that

(8.2) Z l - - [ { v * , w*}: w - - - - v for all u e V A W ] .

The preceding remarks lead immediate ly to:

[8.1] - a) The conjugate space of V+W is canonically isometrically isomorphic with ZI , with norm ]l {v*, w*} I [ - - m a x (I]v*][v*, IIw*[]w*), the scalar product being the one induced by {8.1).

b) The conjugate space ot YN W is canonically isometrically isomor. phic to ( v * q - w * } / z l with the norm induced by the norm llv*IIv*q- q-[] w* J[w* on v* q - w * and scalar product induced by (8.1).

The comparison of the definit ion of Zl by (8.2} with the definit ion of

80 ~-~. AI¢0NSZAJN - E . GAGLIARD0: Interpolo~tio~ spaces, etc.

Z leads in a natural way to the following identif icat ion of elements in V* and W*: the l inear funet ionals v*eV* and w * e W * will be considered equal if they have the same restr ict ion to V N W . However, this identification is only acceptable if it does not identify different elements of V* (or W*) with the same element of W* (or V*). 0ne sees immediately that this condition is satisfied if and only if V N W is dense in both V and W. We are thus led to the following definition.

[8.11] D E F I : N I T I O N . - I f

(8.3} V(5 W is dense in both V and W,

V* and W* form a l inear couple IV*, W * ] - the conjugate c o u p l e - with the identif ication mapping defined as follows"

(8.4) v*~-w* i f and only i f < u , v * ~ v - - ~ u , w * ~ w for all u e V ( ~ W.

When we speak about the conjugate couple or use the symbol [V*, W*] it will be assumed that condition (8.3} is satisfied.

From Proposition [8.I], by vir tue of the last definition and formula (8.2), we obtain

[8.III] T K E O R E ~ . - I f the conjugate couple IV*, W*] exists, then (V + W ) * ~ V*N W*, (VN W)*~ V* + W*. The corresponding scalar products

are those induced by the scalar product (8.t} on the direct sums V-~ W

and V* ~- W*~ i.e.~

(8.5) ~ v + ~ v , u * ) v + w - ~ ~ v , u * " .~.~ + < w, U*>w

< u, v* + w* ~ vn w - - < u, v* -~ v + w .

An immediate consequence of this theorem is:

[8.IV] COROI~L~R¥. - 1[ IV*, W*] exists then : a) I f W C V {i.e., W is a Banach subspace of Y( then Y* C W*. b) I f W C V , the~ V*~ W*. et V~V~W* is dense in V*, W*, and Y* + W* in their respective weak*-lopologies (12). d) I f V and W are reflexive, then V*NW* is dense in V*, W* and 17. + W* {in their normed topologies}, and IV**, W**] exists and is equal to [V, W] (in the usual identifications).

(tz) Bu t not necessar i ly in their normed topologies.

N. ARO~'SZAJ~ - E. GAGLIARDO: I nterpolatio~v spaees~ etc. 81

[8.V] REI~ARK.- If V and W are not both reflexive, the density condition may or may not be preserved in the conjugate couple. Simple illustra- tive examples arc the following: 1 °) V--L~(O, ~) and W- 'L~(0 ,1 ) ; here the densi ty is preserved. 2 e) V : L 2 ( 0 , 1 ) a n d W-----L~(0~I); here the density is not preserved. As long as the density is preserved we may form the conse- cutive conjugate couples [V*, W*], [V**, W**], etc. We do not know of an example of a non-ref lexive couple IV, W] for which the fourth conjugate exists. That the third conjugate may exist is seen by taking

1

W - - [ t ~ , ! : n l ~ , ~ l ~ 0 ] with I l t ~ t [ I w - - - m a x ln~,~[ .

We have here W C V .

Consider an intermediate space A between V and W. A continuous l inear functional on A is, a f o r t i o r i , a continuous l inear functional on VOW. This gives a natura l identification of A* with a subspace of (VOW)* on condition that two distinct elements in A* are not identified with the same element. It is obvious that this res t r ic t ion is equivalent to the condition

(8.6) V• W is dense in A.

If this condition is satisfied and [V*, W*] exists, the identification is valid and A* becomes a subspace of V * ~ W* containing V*C~ W*.

The following proposition is then immediate.

[8.VI] - I f A is an intermediate space between V and W satisfying (8.6), and i f [V*, W*] exists, then a) A* is an intermediate space between V* and W*. b) I f A is normalized so is A*.

Consider now an operator TecC~_~[V, W]. Consider fur ther the restrictions of T, T v E ~ v , T w e ~ w and their adjoints T~.erCv,, T~v~'~w.. For u e V O W and u*e~7*OW * we have by (8.4),

~ u, T~zu* ~ v - - ~ Tvu, u* ~ v - - - ~ Twu, u* ~ w - - ~ u, T~vu*~w.

Hence by Definit ion [8.II] and Proposition [6.IV], T~. and T~v are restrictions of a unique operator T*~75[V ~, W*]. T* will be called the ~-adjoint of T and the class of all such adjoint operators will be denoted by ~C*.

AnnaU di Matemat ica 11

82 N. ARONSZAJN - E . GACLIARDO: I~terpolatio~ spaees~ etc.

[8.VII]. TKEORE~ - a ) The class ~* of all ~-adjoint operators is a Banach algebra; it is a subalgebra (with the same norm) of ~ [ V ~, W*].

b) The mapping T--* T ~ is an isometric anti-isomorphism (~) between the Banach algeb~:as ~ and ~ .

c) The operator T ~ on V ~ ~- W ~ is the adjoint of the restriction Tr'NW ; the restriction T~ ,nw . is the adjoint of the operator T on V ~ W.

d) In order that an operator L e ~ [ V ~, W ~] belong to ~ it is necessary that it be weak* continuous on each of the four spaces V*, W*, V* (3 W* and V* -[- W* ; it is sufficient that it be weak* continuous on V* ~ W*.

e) ~* = ~ [ V * , W*] i f and only i f V (3 W is reflexive.

f) I f V A W is reflexive, then [V*, W*] satisfies (8.3)and [V**, W**] exists.

PROOF. - Par ts a) and b) are immediately checked by the use of elemen- tary propert ies of adjoint operators (such as I TIA-----! T*]~., ete).

To prove c) take v* -Jr- w* e V* ~ W*, u e V N W, u* G V* ('1 W*, v -1- w ~ V -k W and use (8.5). This gives

< Tu, v*-bw* > v Nw~-<Tu, v* >;7 --k < Tu, w*> w-- < u, T*v ~ >v"b <u, T~w * > w

---- < u, T*(v* Jr- w*) > v n w ,

< T(v + w), u* > r ' + w = < Tv, u* >v 4- < Tw, u* > w - - < v A- w, T'u* >v+w"

The first part of d) follows immediately from part c) s ince an operator on a conjugate space is an adjoint if and only if it is weak* continuous. For the second part assume that L is weak* continuous in V* "4- W*. It is then the adjoint of some operator H e ~ r , nw. Hence, for any u ~ V A W and v* ~ V* we have < t tu , v* >rnw---- < u, Lv* > r a w , and, since Lv* ~ V*, <Hu , v* > r - ~ r . Choosing v* so that ilv* lit.-= 1, and <Hu, v* > v - - [IHul[r, we get IIHu[[v < i lul lvlLl~. . It follows that H is a bounded operator in V, defined on V (3 W which is dense in V. There exists, therefore, a

unique extension of H to Hr- e ~r'. Similarly, there exists an extension Hw E ~w,

and I by Proposit ion [6.IV], Hr" and T/w form an operator T e ~ [ V ~ W]. The ~-ad jo in t of T must then coincide with L since L is the adjoint of H : TVNW.

(13) Since , in th i s paper , we use b i l i n e a r sca la r p roduc t s ( and no t h e r m i t i a n as in HILBERT spaces), the m a p p i n g T ~ T* is l i n e a r ( and not an t i - l i nea r ) . The p r e f i x • an t i of ~ a n t i - i s o m o r p h i s m b e t w e e n BANACK a lgeb ras ~ r e f e r s to the p r o p e r t y (ST)*~ T'S*.

~. ARONSZAJN - E. GAGLIARD0: Interpolatio~ spaces, etc. 83

To prove e) assume first that V (3 W is reflexive. Then V*-}= W* (V f~ W)* is reflexive too and every bouncied operator in V* ~- W* is weak* continuous. I-Ience, by part d), our assertion follows. Assume next that VCI W is not reflexive. The same is true then for V* "4-W* and there exists a bounded l inear functional f(~v) on V* "4- VP* which is not weak* continuous. Taking any u* ~ V* (3 W*, u* ~ O, and putt ing L x ~ f(a~)u*, we get a l inear operator in ~[V* , W*] which is not, in ~* by part d).

Finally, we obtain f) by noting that since V* n w * is weak* dense in V* -{- W* (see [8.ZV, v)]) if V* -{- W* is reflexive, V* n W* must be dense in it in the weak topology and therefore also in the normed topology.

[8.VIII] RE~IARK - We give here a simple example of IV, W] such that IV*, W*] exists, V N W is reflexive and each of the spaces ]?, W, V ~ W is non-reflexive. Take two disjoint bounded intervals /1 and /2 on the real line and consider functions f defined on I~ U [2. Put V-----If: f e L ~ on /1 and f e L ~ on h], W----If : f ~ L ~" on 1~ and f ~ L ~ on I~]. Then V n W -~ L2(h O I2) and V ~ W - - L ~ ( I ~ U I 2 ) . If we now replace V by the space V - ~ [ f : f e L P on /1, 1 < p < 2 , and f : L ~onI~], we get V N W - ~ L ~ ( I ~ N I 2 ) andV- [ - W [ f : f e L p on /1 and f e L ~ on /2]. Hence V and V n w are reflexive whereas W and V-{- W are non-reflexive. We may add that V and W are reflexive if and only if V n W and V ~ W are reflexive (~).

[8.IX] RE~A~: = As a complement to Remark [8.V] we notice that if one of the spaces V, W, V ( 3 W is reflexive, whereas V-~W is non-reflexive, there can exist at most two conjugate couples (otherwise V ~ W would be dense in (V ~ W)** in the normed~topology). It follows that if V ~ W is reflexive and one of V, W, V N W is non-reflexive (hence V ('1 W non- reflexive) there can exist at most three conjugate couples; the maximal number may be at tained as in the last example of Remark [8.V].

[8.X] - Suppose that [V*, W*] exists, A is an intermediate space between V and W satisfying (8.6) and T e ~ [V, W] satisfies T (A)C A. Then the restriction TA belongs to ~A and the restriction T 'A, is the adjoint of TA.

PROOF.- The first assertion follows from Lemma [6.X]. To prove second assertion, we write for u e V (~ W and a* c A * ,

< TAU, a* >A Z < Tu, a* >vnw---- v n w .

the

(~4) To see this we apply the theorem : If E is a BANAC~ space and A a closed subspace of E, then E is reflexive if and only if A and E/A are reflexive. We then put E-----V-~ W and A~--Z~--- [Iv, w}:~v~--v] .

84 N. ARONSZA:IN - E . GAGLIARDO: I~vterpol, ation spaves~ etc.

This expression is a bounded l inear functional in u with the norm [lu[lA (since T A ~ A ) , hence, by (8.6) is uniquely extendable to A, i.e. T ' a * c A * and < T,~u, a* >A--= < U, T ' a * >~. Again by (8.6) it follows then < T.~a, a* >A---- < a, T ' a * >A [or all a EA which finishes the proof.

[8 .XI] TIIEOREI~I - I f [V*, W * ] exist8, ]7 N W i8 reflexive and A is an interpolation space between ]7 and W satisfyi+~g (8.6) then A* is an interpolation space belweel, V* and W*. I f A is normalized, then so is A%

The proof is clear by virtue of [8.X], [8.VI] and Theorem [8.VIII. For use in the next theorem we introduce the following notations.

[8.XlI] DE~I~ITIO~¢ - ~ [ V , W] and ~[V, W] will denote the classes of intermediate or interpolation spaces (between V and IF) respectively which satisfy (8.6). The subscripts ~¢ n >>, ~ r >>, or << n r >> will mean that the classes are restr ic ted to << normalized >>, << reflexive >>, or ~< normalized and reflexiv e ~> spaces, respectively.

I t is clear that all these classes C[, ~ , , , . . . , ~ , with the order relation C are latt ices (~). The preceding theorems and propositions lead immediately to the

[8.XIII] DUALITY ~HEOm~I - Let V and W be reflexive and V A W dense in V q- W. Then: a) the same is true for the couple IV*, W*] and IV**, W * * ] : [V, W]. b) The mapping A ~ A* is one-one and order reversi~g on each of

the classes ~ , IV, W], 5 ,~ [V, W], ~, IV, W], ~,,.[V, W] and transforms them V* onto the corresponding classes formed for [ , W*] for each of these classes

the inverse mapping is given by A* --* A* * ~ A.

§ 9. Some propert ies of the Banach algebras. ~[V, W] and ~;a[V, W]. Consider an operator T e ~ - = ~[V, W]. If A is an interpolation space

then the restriction, TAe ¢GA. We may then consider the inverse (if it exists) of T in ~ and TA in ~A.

[9.I] TFLEOREI~I - Let T e ¢G, then: a) I f T -~ exists, then Tdl exists for every interpolation space A and is the restriction of T-1 to A.

b) I f T-~ 1, T~) and T - ~ W (or T ~ v ~ ) exists, then T -1 exists.

c) I f T - ~ W and T-~r¢ exist, then T -~ exists.

(l~) Which in general are not complete,

N. ARONSZAJN - E . GAGLIARDO: Interpolation spaces, etc. 85

Pnoo t~ . - a) If T -~ exists, then (T-~)ae ¢gd and clearly equals (T~) -~.

b) If T~ ~ and Tw~ exist, we have to check if they form a transforma-

tion in ~ . By [6.IV], it is enough to check that if u e V Q W, then T-~u-----

Tw*U~ V ~ W. If, in addition, we assume that T ' ~ W exists, then u =TvnwU, for

some u~ ~ V VI W and hence T ~ u --~ T-~Tv~ ~u~ ~- T~,~T~u~ --- u~ and similarly,

Tw~U----u~. We check then immediately that the combined transformation,

T-~(V-3cW } : T ~ V 2 V T-~lw is , in fact, the inverse of T in ~ .

If, instead of the existence of -~ T~NW, we assume that of Tv+-~ ~, then for

any u e V N W, we have T v + w ( T - ~ u ) - - T v ( T - ~ u ) - - - - u and, similarly,

Tv+~7(T-~ ~u) = u ; hence T-~u -- T'v~wu -- Tw~U.

c) If T ~ and Tv-~w exist, then clearly, T ~ is the restr ict ion of

T ~ v to V (5 W. It is enough to show that T~w(V) C V and T ' ~ w ( W ) C W. In fact, for v e V let T ~ v -- v~ + w~, v~ e V, w~ ~ W. Then v = Tvv~ -~- Tww~.

Hence, Twwz ~ V N W. It follows that T-~hwTww~ = T - ~ w T z + ~ w ~ .~ w~ and

thus w~ e V C~ W and T ~ w V e V. The proof of T-{~w(W)C W is similar.

[9.1I) RE~[ARK- In general the existence of T -~ does not follow from the existence of T~ ~ and Two. For example, consider the L~-space on the

c i rcumference I~l "- 1. As V and W take the closed subspace of L~[]~I - - 1] which vanish for Re~ < 0, or Re~ > 0 respectively. We introduce an iden. tification mapping x between V and W as follows: w - - x v if there exists in I~1 < l an analytic funct ion f(~)eI12 such that f ( ~ ) = v(~)for t~t = 1, R e ~ > O and f ( ~ ) = - - w ( ~ ) for I~1- -1 , R e ~ < O . We obtain thus a l inear couple [V~, W~] of HILI~ERT spaces. I t is easy to check that this couple is compatible, that V~. Q W~ is canonical ly topologically isomorphic to H ~ and is dense in V and W. Taking then Tvv -- ~v, TwW -" ~w, one sees immediately that for v - - w , TgV =-Tww, and hence Tv and Tw combine to a l inear operator T on [VT, W~]. On the other hand Tv and Tw are uni ta ry operators in V and W respectively and thus T e ~ , / ' ~ and Tw~ exist whereas T -~

does not exist ( T ~ does not exist since Tvnw is the shift operator on //2).

[9.III] DEFIniTION - For T e ~ we will denote by ~(T) the spectrum of T in the BA~AC~ algebra ~ . If A is an interpolation space between V and W, we will denote by ~A(T) the spectrum of TA in the BA~Ac~ algebra ~;A-

[9.IV] T~EORE~- Let T e ¢g. Then: a) for any interpolation space A, •A(T) C a(T). b) v(T) ---- ~v(T) t) ~w (T) U ~vn w (T) -- +v (T) tJ ~w (T) L9 ~v+w(T) - - a vnw(T) tJ ~v+w(T). v) C being the complex plane, the following statement is

86 N, ~RONSZAJN - E , GAGLIARDO: l nterpo~a,tion spaces, etc.

true : Each component of C - - ~(T) is equal to a component of (C -- ~v(T)) N ( C - ~w(~)).

PnOOF.- a) and b) fotlow immediately from Theorem [9.I] applied to T - - ~I (I being the identity).

To prove c) we denote by R(T; ~) the resolvent operator of T in ~ , and by RA(T; ~) the resolvent operator of TA in ~A (for an interpolation space A). These are analyt ic opera tor-valued functions of ~, regular in the open sets C--<;(T) and C - ~A(T) respectively. Since, by a) of our theorem, C--~(T)C(C- -~v (T) )A(C- -~w(T) ) , what we have to prove is that if G is a component of C - - <;(T) and G' is the component of (C-- ¢;v(T)) n (C- - aw(T)) containing G, then G - - G'. In fact, if this were not true, there would exist a ~ 'e G', such that ~' belongs to the boundary ~G. There would then exist a sequence i~,, !C G with ~,, -~ ~'. For each ~,,, Bv(T; ~,) and Rw(T; ~.) combine together to form R(T; ~,~). Since ~'e G', it belongs to the resolve~nt set of Tv as well as T~, and the operators By(T; ~;,) and Rw(T; ~, ,) form CAUCI~Y sequences in the respective aigebras '~v and ~w- Hence the operators R(T; ~,) form a CAucHY sequence in the algebra ~ . This, however, is impossible since by a well known property of resolvent operators, if ~' is

1 in ~(T~, then IR(T; ~,,)[V~_ ~ i ~ , - - ~ ' l "

[9.V] COI~OLL~RY- I f T e ¢G, A is an interpolation space between V and W, u e ( V -5 W ) - - A and T u : = k u - S a with a e A , then l II <-~tT[~.

PnooF. - If ), were > I T ]~ then (T - - kl)- 1 would exist ; hence (T - - ),I)~ 1 would exist too, and would be the restr ict ion to A of ( T - - X I ) -1. It would follow that u - - ( T - kI) la = ( T - k l J A ' a e A against our assumption.

[9 .VII DEFINITION - Let A be an interpolation space between V and W, We denote by ~A_-- ~A[V ' W] the closure in ~ of the set of all T e '~ with T(V~- W) C A . As norm in ~A we take the norm of ~ .

[9.VII] - I f for an interpolation space A, T e ~A, u ~ ( V + W ) - A and T u = k u ~-a with a e A, then k - - O .

PROOF - By definition, there exist T, ~ ~ with T,(V -5 W) C A and !T - - T, I~7 ~ 0. I t follows that (T -- T , )u -- ku -5 (a -- T,u). Since a - T ,u e A, we get,. by Corollary [9.V], i kl ~ I T - - T. l~.

~ . ARONSZAJN - ~E. GAGLIARDO; Interpola, tion spaces, etc. 87

[9.VI]I] THEOREM - Let A be an interpolation space between V and W, strictly smaller than V ~ W. There ¢~A is a prolJer closed ideal in the Banach algebra ~ .

PROOF.- From the definition it follows that c~,4 is a closed linear subspace of ~ and that the set of all T e ~ with T ( V - [ - W ) C A form a

dense subspace ~A of ~A. If S~ and $2 are any two operators in ~ then

obviously T~ ~A implies S ~ T S ~ ~ A hence, by cont inui ty, T ~ ~A implies S:TS2 e cG n. Therefore ~A is a closed ideal. To show that this ideal is proper we first notice that for any q ~ ( V - k W)* and u ~ V A W C A , the operator

Tx ~ ~(x~)u belongs to ~A, and hence ~A:~:(0). On the other hand I ~ ~ n since for u ~ (V "t- W) - - A, I u - - u q- 0 -- in contradiction to [9.VII],

§ 10. Relative completion. - The notion of relative completion was introduced recently by E. GAC~LIARDO in his study of special interpolation methods [4].

[10.I] DEI~II~ITION - Let /~ be a B~NAC~ space and A a BA~AC~ subspace

of E. The completion of A rel. E, denoted by A(~), is the set of all elements

~eeE which lie in the closure in E of a bounded sphere in A, i.e., 2~( E ) -

t2 SA (R)E. For x ~ ~t(E) we put R>o

(10.1) li x ItX(E) : i n f R for all R wi th x e SA(R) E .

[10.II] REMARK - ,~(E) will not be confused with the completion of A in E (denoted in § 3 by ,~E) which is obviously equal A.

[10.III] - A being a Bana('h subspace of the Banach space E, we have

a) ~(E~ is a subspace of E and with norm (10.l) is a Banach subspace of E.

M

b) A (~) ~ A.

c) ~(E) ~ (completion of A tel. AE).

d) The space ~t(E) and its topology depend only on the topologies of A and 1~ and not on the choice of their norms.

e) The norm of : (E) depends only on the norm in A and the topology of E (and not on the choice of norm in E).

88 N . ARONSZA;~N - E . GAGL~ARDO: Interpolation spa,ces, etc.

f) I f B is another Banach subspace of E and A C B then A(~) C A(~)

and .~(~) C I~ (~) ; i f A ~ B, then ~4 (~) ~ [~(~).

g) The space 71(~) and its norm are not changed i f in their definition

the strong closure S~(R) ~ is replaced by the closure of SA(R) in the weak

topology of E~ i.e., by SA(R) ~'~.

h) (Completion of A¢~) tel. E ) ~ A(E).

PRoo~ ~. - a) That .~(E) is a subspace and ]]xIIx~E ) is homogeneous and satisfies ~I~KOwSKI'S inequal i ty follows immediately from the observation

that if x ~ S A ( R ) E and yeS~(R~) E, then ~x-~ ~yeSA(R]~t-~ RII~[) E. Further- more, since there exists a constant c such that ! lu! lE~ cllul!.4 for u e A , it

follows that II~c!tE ~-- cll~cIiX(E)- Hence, (10.1) defines a norm and A~) C E. It

remains to show that _~<E) is complete. Let {x , l be a CAVCH¥ sequence

in t](E). It is, a f o r t i o r i , CAuc~IY in E with limit x. For any e > 0 we can

choose an n~ such that for m , n > n ~ , x ~ - - ~ . e S A ( e ) E . Hence alsox,~--x

e SA(~-) ~ and thus x is the limit of ( x , f in .4(E).

b) Obvious, since aeS~(Hal~) ~ C S~(tlaIIA) ~ for any a e A .

The properties c), d), e), and f) are also obvious consequences of the

definit ion of .4(~).

g) is proved by noting that SA(R) being a convex set, its weak and strong closures are the same.

h) Denote by A1 the completion of ~(E) rel. E. Since by b), AI~-~<E>

it is enough to show that 2~( E)~D A1. To this effect, take x~ A1. Txaen there

exists a sequence Ix .}CA(E) with IIw. I!X(E)~IxlA~ and x . ~ x in E. For each x , there exists a sequence (a(k')lCA such that IIa(k~)IIA-~ l[~v. l iZ(~)= [I~][A, and such that a ~ ' ) ~ x , in E. By diagonal procedure we obtain, therefore, a sequence { a , } C A , a , , ~ w in E. und l]a,,rl[A~-]lx]tA,, which gives

+ 2<K) and }] -< I] lIAr.

[10.IV] - /t(E) -- E i f and only i f A -~ E.

PROOF.- It is enough to show that A ~ E implies z] ~E) =~ E. In this case the identi ty mapping I : A ~ E is a continuous isomorphism o[ A into E (and not onto). A well known property of such mappings is that a sphere

~T. ARONSZAJN - E. GAGLIAItDO: Interpolation spaces, etc. 89

GO

in A is t ransformed into a nowhere dense set in E. Since ~(E)- - U SA(n) E

our conclusion follows. W e introduce the following notation to be used in the next theorem. If X

is a subspace of a BANACI-I space B, the polar set of X, i.e., the set of all e lements u * e B * such that < x , u * > = 0 for all x e X will be denoted by X AB (in usual notation B is omitted).

[10V] TI~IEOI~E~[- Let A be a dense Banavh subspace of the Banach space E. Then with the usual canonical identifications we have

~I( E> = E (5 (A** /(E*)±A*)

and the norm in ~4 (E~ is equal to the norm in A** /(E*)~A*.

PROOF - We first give some explanations regarding the formula in the theorem. Since A is dense in E the couple [E, A] has a conjugate couple [E*, A*] with E* C A*. The second conjugate couple may not exist since E*

is not necessar i ly dense in A*. If we consider E *A*, which is a closed subspace of A*, its conjugate (with conjugate norm)is given by A**/(E*)±A*.

Now, the couple [E*, E *A*] has a conjugate couple which is [E* *, A**/(E*)±A*]. Obviously, E** D A**/(E*)~-A'). E being a closed subspace of E**, the inter.

¢

section E (5 (A**/(E*) .~-A*) is a closed subspace of the second term and hence, provided with the norm of the second term, is a BANACI~ subspaee of E.

To prove our formula, assume first that x eA(E). Then there exists a sequence I a , } C A such that ]i a,, []A -- ]] X, l].~(~ ) and [ ] a , , - -~c ] t z - -0 . We have then < a , , u * >E--~ < X, U* >E for every u * e E * C A * . On the other hand I < a , I , U * > E ] = [ < a . , U * > A I ~ []Xl].~(E)[[U*I]A*. Hence I <0e, u* >,[_~<

[]x~lT~(z) l]u* ][A*, and therefore x is identified with an element of (E---TA*) * -- A * * / ( E * ) ± : . Thus we have x e E ( 5 ( A * * / ( E * ) ± A * ) _ and ]lxl].~(,)~ I]xll~-,~.4.)..

Suppose now that oeeEV~(E*A*) *. Then for u * e E * , < x , u * > z - - < x , u* > ~ A * - - - < X**, U* >A* for every x** belonging to the equivalence class i~ A * * , ~ A* / (E )- corresponding to x. Pu t I1 x II(~A*). -- inf It :c* * tin'* ---- R. ~,, Since, by a well known theorem o ,D,,A** 'JA*n*~J --S.4(R') ~:A**, it follows that for

R ' > R there exists x** e SA(R') ~:A**, This means that for every finite system a*~,. . . , a* , in A* and any s > 0 ihere exists an a ~ A with Jlal]A < R' s u c h t h a t

J < ~ * * - - a , a ~ >.4* l < ~, k ---- 1, ... , -n

In particular, when the al* are in E*, we get I < a ~ * * - - a , a* >A*I----

I < X -- a, a*k >~:/ < ~ . This means that x ~ SA(R') ~vz, and by [10.III, g)] x e A qE)

Anna~i eli Matematica 12

90 ~ . ARONSZAJ~N - E . GAGLIARDO: I~terpolation spaces, etc.

and ]I x ItA(E) < R'. Hence, choosing R' arbi t rar i ly near to R : tlx l!~*)" we I ~ < get II ::c ~.~(E) _-- iI x/!~A*), which finishes the proof.

[IO.VI] COROLLARY - I f A is reflexive, then .~(E) --_ A.

PROO~ ~. - By [10.III, c)] we can replace E by ~E and thus assume that A is dense in E. Then E ' C A * and E* is dense in A* (since E* is a total subspace of A* and A is reflexive). It follows that (E*) ±A* ---A*±A*--(0)

and ~(R) _~ E G (A* */(0)) - - E N A = A.

[ t0 .VII ] REI~IARK - By [10.III, h)] the repeti t ion of the operation << completion rel. E>> does not lead to any new spaces. However, a different kind of i teration of the relative completion may lead to new spaces. Put A o - - E

and for n ~ 0 put A,_~-----A(~). A priori, this sequence can be extended into a transfinite sequence by defining for ordinals ~ of second kind A~---- ~ A~.

However, this t ransfini te sequence must become stationary and the last A~ different from the preceding must be equal A. At present we do not even know if this last index of << non-s ta t ionar i ty )~ may be infinite. Actually we know only of examples where this index is ~ 2.

Such an example, with index 2, is constl 'ucted as follows: We take A---- l~ (bounded sequences converging to zero). Then A * * - - l °~. For any prime number p consider the element uv = {~,,}eA** defined as follows. For n : p ~ , l ~ k ~ p , ~ - - 1 . For n - - p * and k > p , ~ , = - l / p . For all other indices n, ~ , - - 0 . Denote by B the closed subspace of A** generated by elements u~; then A N B - - - ( 0 ) and A--}-B is not a closed subspace of A** . Consider now the HILBERT space H of all sequences {a,, } such that

Vn-2[~,,[ ~ < ~ . Clearly H D A * * D A . We define E : H / I ~ H and identify elements of A with the equivalence classes containing them. (Clearly A N/~H --(0)). Using Theorem [10.V] we prove that the closure of A in A~ is equal

to (A-b B)a*'/B, and the last space is strictly greater than A. However, it can be proved that A2 = A, A simple example with index 2 is given by E----L~[0, 1] and A - - C[0, 1]. Then A ~ - - L ~ [ 0 , 1] and A ~ - - A . Here, however, A is a lready a closed subspace of A,.

[10.VIII] THEOREM - Let [V, W] be a compatible Banach couple. We have

a) (V n W) (v+W~ -- (V N W)(V) + (V n W) (w).

b) ~(v+v) = V + (V N W) (W), "W (v+r<) "=-- W + (V n ' w ) (v).

e) ?(v+~> ~ ~v<v+~v> = (V N-~V) (~'+W) ¢%

Q~) I n g e n e r a l t h e • ~ ,) c a n n o t b e r e p l a c e d b y , Z : , .

2(. ARONSZAJN - E . GAGL1ARD0: Interpolation spaces, etc. 91

PROOF. - a) By [!0.III, f)] it is clear that we have the inclusion (~ D >~;

hence it remains to prove ((C>>. Take x e ( V N W ) ( V ~ -W). There exists a sequence Iu , i C V N W such that Hu, , i lva)v<R and i l x - u o i t ~ + w - o . Let x = x ~ + x 2 , x~eV, x2~W.. It follows that x ~ + x 2 - - u , , - - ' v , . + w , , with [ ; v , l ] v ~ 0 , ]]w,,[iw ~ 0 . The last equali ty can be writ ten ( x ~ - - v , ) ) + ( x 2 - - w . ) - - u , showing that ( x ~ - - % ) and ( x 2 - - n ' , ) belong to VI'3W. An immediate evaluation shows that

II x~ - ~ . H v a w ~ m a x [( I! x , ! , + ~:t v . iI.), (1I x~ lt~ + !l'~v. II,~ + R)].

Hence x~ e (V (3 W)(7) and similarly, xz e (V/"3 W) (W).

b) Here again the inclusions ((D ~> are obvious, and we prove (< C ~.

Suppose xe~Z(v+Y). Hence there exists a sequence { v , } C V with liv, l . v < R and I t x - - v - IIv+w ~ 0. Put t ing x = x~ + x2, x ~ V, x 2 e W, we get x ~ + x 2 - - v,, - - v,,' + w , with Ilv,'[[v ~ 0 , l l w , [ I w ~ O . Hence x 2 - w , ~ v N w and the sequence { x 2 - - w , , } is bounded in V N W and converges to x~ in W. The proof of the second equation proceeds in the same way.

c) is an immediate consequence of b) and a).

[10.IX] THEOttEM - Let [V, W] be a compatible Banach couple and A

and B two interpolation spaces behveen V and W and A c B . Then ~t(B) is also

an interpolation space between V and W. I f A is normalized so is ~(B).

P R O O F . - Let T~ ~;[V, W] and ace ~t(B). Then there exists ( a , } C A such that tl a . ]!~ = II x I/_~'(.> a n d ]i ~ - - a,, l l - - - O . It f o l l o w s that li Ta,, I1~ --<1 T j., 1t x IIZ(.)

and li T x - - T a,, ))B ~ O. Therefore Zx Z ~B>, fl TX i'i£(B) < I TI.~ ti x fIX(B), and ] T I z ( , ) ~ ]TI~. If A is a normalized interpolation space, then with the above notations, ]Ix -- a,]lv+jr ~ 0, and hence ] ] x [ ] r + w - lim]la~,]lv+l~-<

i[xll£(,) . Thus ~(B) C V + W. On the other hand ~(S) D A ~ VI'3W and I T]g(B> <~ I Tla < I TtV, which proves the second assertion of the theorem.

[10.X] LE~IMA - Let IV, W] be a compatible Banach couple, h ~ V + W, and A an intermediate space between V and W.

a) Suppose that A satisfies one the following two properties:

(lO.2a) The topology o[ A is not coarser than the one of W when restricted to Sva w(R), 0 < R < oo.

(10.2a') The topology of A is not coarser than the one of V when restricted to Svnw(R), 0 < R < oz.

92 N. ARONSZAJN - E . GAGLIAI¢DO: lnterpolatiol~ spaces, etc.

Then i f h~ ~zr+ w) or h~ f¥(17+w) respectively, there exists T e ¢gva w[V, W] such that Th ~ A.

b) Suppose that A satisfies property

(10.2b) The topology of A is not coarser than the one of V - b W when restricted to Sva w(R), 0 < R < ~ .

The~ i f h ~. V(V+w)U l~ r(v+W) there exists T e ¢gvn ~[V, W ] such that Th ~ A.

PROOF.- a) By symmetry in V and W it is clear.y enough to prove

our thesis if (10,2a) holds and h g ~,(17+w). We show first that (10.2a) is equi valent to the following proper ty :

~ 1, (10.3a) There exists a sequence I u ~ ! C ~ T A W such that l!u~F!vnw-- N u~ [~w ~ O and ]I u~ ll.~ ~ ~ for some ~ > 0.

Obviously if (10.3a) holds, then for any R, 0 < R < o% ~ Ru~ C Svnw(R)

1 and ~Ru,* converges to 0 in W but not in A. Suppose now that (10.2a) holds.

Then there exists }x~ } C Svnw(R) such that x , converges to some x ' ~ Svnw(R) in W but not in A: Because of compatibili ty of W and A, {x~} is a CAtchY sequence ia W but not in A. I t follows that for some sequences of integers converging to ~ , ( m~ } and I n~ }, we will have, if we put Yk = x , ~ - - x , k, that I/Yh - o, i[ y , it~ --> ~' > 0 and [I Yh Hvn~7 ~ ~ ~- 2R. We cannot have k = 0 since then H YhlIA would ~ 0 (the topology of A being coarser than the one of V N W). Hence u,*--Yh/l lYkIlVNW satisfies (10.3a). We now pass to the proof of a).

If for all T e c6vaw T h e A , then Th would be a l inear mapping of ~VNW into A. By a simple application of the closed graph theorem it is

seen that this mapping would have to be bounded. Therefore, in order to prove the statement of the lemma, it is enough to exhibit a sequence IT,,} C ¢G yaW with 1T,*l,g uniformly b o u n d e d and ]IT~httA ~ . We will construct 7,* as an operator of rank 1, T a x - - < x , f~>z,+, f ~ e ( V ÷ W ) % z . e v n w, <h, f,*> = i and IIz**iiA - - ~ .

By Theorem* [8.I, a)] the space ( V + W) + is canonical ly isometrical ly isomorphic with the space of all couples (f ' , f " l such that . f ' e V * , f " e W* and l T - - - - w for every u e V N W . To fE(V-~- W)*, there corresponds If ' , f"} with f ' -- P ' f - - restr ict ion of f to V, and f " --~-- P " f w h e r e , P " is the restr ict ion of f to W. Fur thermore , Ilftl<17+~r)* -- max [Ill'Ill7°,

iI f" Consider first the case h~. V v+W. We put f~----fo where fo is a l inear

functional in (V-t--W)* vanishing on ~17+w and such that < h, / o > - - 1 .

~-~. ARONSZAJN - E . GAGLIARD0: Interpolation spaces, ctc. 93

Using the sequence {u~}, which exists by (10.3a), we define zn~--[tu, ll~/~u,. Then -P'fo ~ O. Hence for the operator T n x - - < x, fo > z** we have ITn I v - - 0 ,

i [ ¢ ' f

IT.[w - IIP"fo ]w'llz,*i!w -- tl P folIw, and IIT,~hll~= Hz,,H~ >_wHu,,[i~ ~ - o z . Suppose now ~hat h e ~v+~Y. Consider the subspace B C (V-I- W)* of all

f ' s satisfying < h, f > -- 0. B is a closed subspace of (V -[- W)* of co-dimension 1. P'(B) is therefore a subspace of P'((V-{- W)*) of co-dimension at most i. W e prove now that

(*) P'(B) is not a closed subspace of P'CV + w)*) in the topology induced by V*.

In fact, the l inear functional ~(f'), f ' e P'((V q- W)*), defined as < h, f>v+,+ for any f with P ' f = f ' is well defined (since for P ' f - - O , < h, f > - 0 in

view of hE VV+w). Since ~(f') is not identically 0 and vanishes on P'(B) it follows that P'(B) is the nul lspace of ¢p(f').

I f (*) were not true, the nullspace o f ~ would be closed in the topology of V*, hence in this topology ~ would be continuous. By HAH~-BANAC~ theorem~ ~ could be extended to a l inear functional h** E V**. It follows that h** is in the closure of Sv(R) in the weak* topology of V** for R - - I I h * * fly**. We will show that as a consequence, h would have to be in the closure of S~(R) in the weak topology of ~V-t-W. In fact, for any finite system l f(k)l C ( V - t - W ) * and any ~> 0 we can find an element, v~ST(R) such that ] < h * * - - v , P ' f ( k ) > v * r < s for every k. By definition of h** as an extension of the funct ional ~ we have therefore, ]< h, f ( k )>v+w- - <v, f(k)>v+w I < s, which proves our assertion, But this would mean that

h e ~(v+w) against our hypothesis. Since P'(B) is a non-closed subspace of co-dimension 1 of P'((V+ W)*)

in the topology induced by V*, then in this topology it must be dense in P' ( (V+ W)*). Consider any element fo e (v -[- W)* with <h, f o > v + w - - 1 . Then there exist elements b k e B such that ]IP'(f o - bk)ll~.. ~ i /k, k -~ 1, 2, .... If we consider b as running over the polygonal line formed by segments joining b k with b~+ ~, it is clear that the quotient

Q(b) - - [I [o __ b [l(v+w)" [1 p ' ( f o _ b)llv"

varies cont inuously with b and since < h, [o __ b >v+T+- -- t, we have "tl fo __ b ii~'+w)* > ~1 h IIv+w, and the quotient Q(b), for varying b on the polygonal line, takes all the values between Q(b 1) and ~:x~. W e consider now the sequence {u~ i exist ing by (10.3a), and pick up an no such that Ilu,,l~!~? ~ max [2, Q(bl)] for n > no. Then, for n - - 1, 2,. . . , we choose bn so that Q(b,,)--Ilu~o+~It-~. We put [~ -- f~ - - b,,, and z,~---]]P'(fo~b~,)tl'f:u,~.+,,. Since Q(b,,) ~ 2 and ]] r e _ b, II(v+w)* = max [ii p , ( f o _ b,)Hv*, ll P"(f° - - b,)]',,~,], we get ]i P"(f°-- b,)]]w*

94 N. ARONSZAJN - E . GAGLIARDO: InterpolatioJ~ spaces, etc.

- - t ] f ° - b,,lI¢v+r~),. Also, since 1 = II u~,o+,,livnm = max tllu~+-ilT*, Ilu~o+,~ll,] and [iU,o+,H~v~ < 1/2, we get Hu,0+, / lv= 1. It follows, therefore, that for the operator T~x --- < w, f~, > z, , IT~!v- - i T,)t~,-- 1. On the other hand, since Q ( b , ) - * ~ , the b,'s cannot be confined to a finite number of segments joining b ~ with bk+ 1, and it follows that I IP ' ( f° - -b , ) l iy • -+0. Hence Iiz,~i[,~

It p , ( f o _ b,)IIv~. ~ co, and our proof of a) is finished.

b) To start, we notice the equivalence of (10.2b) to the property

(10.3b) There exists a sequence {u,} C V N W such that ilu~Ilvn~.-- 1, ilu,,]]v+~ ~ 0 and ilu~ll~ ~ for some ~ > O.

The proof is completely similar to the one of equivalence between 10.2a) and (10.3a).

Next we show that

(10.4) Properly (t0.2b) holds i f and only i f one of the properties (10.2a) and (10.2a') holds.

In fact, if ei ther (10.2a)or (10.2a')holds, then, a f o r t i o r i , (10.2b)holds since the topolog} ~ of V + W, restr ic ted to V A W, is coarser than ei ther of the topologies of V or W. On the other hand, if (10.2b) holds, then (10.3b) holds and we can use the sequence {u~} from (10.3b). The condition

• t ! Ilu,,liv+s. ~ 0 means that u ~ - - u , [ d - u ~ " , u',~ and u+, in V N W, I iu,( t tv---0 and 'alu,,"Hr~-~O. Since llu,;ll. +lluo"ll, >llu,,il >, at least one of the sequences {u,~'} and {u,~"} satisfies lim sup Hu,' Ha ~ ~ / 2 or lim sup H u-" li* ~ ~/2 respectively. If it is the first sequence then (10.2a') holds, and if it is the second, then (I0.2a) holds.

The thesis of b) follows now from (10.4) and a).

[10.XI] REMARK- The interpolation spaces between V and W which satisfy properties (10.2a) or (10.2a') are of ra ther special kind ; those introduced in t h e l i tera ture in connection with interpolat ion methods almost never satisfy these propert ies (i7). As examples of in termediate spaces satisfying property (10.2a) we mention here that if V A W is not closed in W then any intermediate s p a c e C V satisfies (10.2a). (Clearly if V V~ W is closed in W there are no in termedia te subspaces with property (lO.2a)).

The proof of Lemma [10.X] actual ly gives a stronger resnl t : namely, that the operator T of the lemma can be chosen in the BA~CAClg sub-algebra ~o[V, WJ of ~;vnw[v , W] formed by closure in ~ of the class of all ope-

('~) The usual in terpola t ion spaces have the property that/[uIIA--<--cllu[l~ - 0 Ilu h~- wi th constants c and 0 independen t of u, 0 ~ 0 C 1.

~*. ARONSZAJN - E. GAGLIARDO: Interpolation spaces, etc. 95

rators of finite rank with range contained in V A W. ~o is also an ideal of ~ and is formed by operators which are completely cont inuous in every interpolat ion space between V and W.

Passing' to the applications of the lemma we note first an immediate corollary.

[S0.XII] CO~tOLLAR¥- Under the conditions of Lemma [S0.X] (part a) or part b)) if, in addition, h e A, then A is not an interpolation space.

For a compatible BANAC~ couple [y, W] there usual ly exist infinitely muay couples [V', W'] with V'(3 W ' - - - V ( 3 W , and V'-{- W ' - ' V ~ W (all the spaces being mutual ly compatible) For each such couple the class of intermediate spaces is the same. A natural question ar ises : are the inter- polatiou spaces also the same. The next theorem gives an essential ly negative answer to this problem.

[t0.XIII] TttEORE~f - Let [V, W] and [V', W'] be two compatible Banach couples with V-~ W : V ~ W', V (3 W - - V' (3 W', nei ther of them being identical to [V-~- W, V (3 W] or [V (3 W, V ~ W]. I f all the interpolation spaces belween V' and W' are interpolation spaces between V and W, then, possibly by interchanging V' and W', we have the following statements:

a) i f V (~ W is a non-closed subspace in V and W, then (V N W) (v) C

V' C ~7(v+w) and (V (3 W)(~ ) C IV' C(¥ (v+w~

b) I f V N W is closed in V or W, then (possibly with interchange of V' and W'), V' = V and W' : W.

:PRooF. - Since V' and W' are interpolat ion spaces between V' and W', by our assumption they must also be interpolation spaces between V and W. We consider the two cases.

a) Since V (3 W is not closed in iV there exists a sequence t un '}C v ( 3 w such that llu 'Ii n = S, and lin 'iiw - - 0 . We cannot have

U ! • ? and ][ , w ~ 0 since this would imply Ilu,, ]Ivnw 0. Hence we can assume that in one of the spaces V' and /o r W' the norms of u,,' are > ~ > 0 . If necessary, we can interchange V' and W' so as to have ]l u'~ ]Iv" > ~ > 0. Hence

by Corollary [S0XII] V C ~(v+~v). On the other hand, for any such sequence i~yLn't~g['~ W w i t h It un'IIVNW : 1 and :]u~'[[w --* 0 we must have ]lu~'[[~r, ~ 0.

Otherwise, for the same reason, W' would be C ~(v+w); hence V + W would

b e C ~ z(v+~), contrary to [S0.IV]. It follows that every CAUCHY sequence in W, bounded in V(3 W must also be a CAuc~¥ sequence in W', and thus

96 ~' . ARONSZASN - E . GAGLIARDO: Interpolation spaces, etc.

by compatibi l i ty we get W' D (]7 n w ) (Tv). Since also V n W is not closed in V,

the same argument shows that one of the spaces V', W' must b e C ]i r(v+W)

and the other D(V n W) (V). The first inclusion cannot be satisfied again

by V' since then the second inclusion would be satisfied by W' and we would

have, by using Theorem [IO.VIII], V' C ~(v+~) N ~¢(T~+~r) ___ (V' n W) (V+W) =

(V n W) (7) ~ ( V N W) (~)C W' , contrary to the hypothesis of our theorem.

Hence we must have W'c~V(V+~'~ and V'D(V N W) (V) as stated in a).

b) If v n W is closed both in V and in W, then it is closed i n V + W and hence closed also in V' and W ' (see Lemma [7.I]). As in Remark [7.X], we see then that ei ther V ' ~ V and W ' - - W or W ' - - V and V ' : W .

Suppose now that V N W is closed in V, but non-closed in W (the case when V(3 W is closed in W and non-closed in V is treated similarly by symmetry). We repeat now the argument from the beginning of the proof

of a) (possibly interchanging V' and W') and obtain V'C ~(v+T~). Also, as in

the proof o[ a) we get W ' D ( V N W ) (W). Since now V N W v - - V N W ~ - V ,

we have, by Lemma [7.I], W 7+n" : W and (V N Wy+~r ~_= (V n w ) ~r. v ' cannot

be C W : Wr'+r~ since otherwise V' would be C ( V ~ ( V N W ) (~:))N W "-

(V n W) (W~C W'. By Lemma [7.1Ii] it follows that V'D V and thus V C V 'C

V-4- (VN W) (w). If for some v e ( V N W) ( l r ) - ( V n W) we had v e V ' , then also we would have v E V ' N W ' = V A W which is impossible. Hence V ' - - V . Checking with Theorem [7.IV] we see that W' cannot be in case 1% nor in case 2 ° (since it would follow that (V'N W ' D V)); henc~e it must be in case

3 ° or 4 ° . In both these cases we would h~ve W ' C WV+~I~-- W and since V ' = V, we must have W ' : W (otherwise V ' A - W ' would be strictly smaller than V + W).

[10.X[V] COROLLARY - Under the hypotheses of Theorem [10.XIII] part a),

i f (V N W) (v+W) -- V n W (in particular, i f v , n w is reflexive) then V' -" V and W ' : W (or V ' : W and W ' : V).

PROO~ ~. - In the present case (VN W) V - - ( V N W) w- - V N W; hence

~-(v+n:)_-V and ]YV(V+w)=-W (see Theorem [10.VIII]). It follows by the preceding theorem that V ' C V and W ' C W (or V ' C W a n d W ' C V ) and the inclusions must actual ly be equali t ies because of V + W : V' ~ W' .

[i0.XV] REMARK - We do n o t ' k n o w of any example pertaining to the case a) of Theoren [10.XIII] where the couple IV', W'] is not identical with

[v, or [w, v].

~N. AROZCSZAZY - E. GAGLIARDO: Interpolation spaces, etc. 97

[10 .XVI] T~EOR]~M - Suppose that neither of the two compatible Banach couples [V, W] and [W, V] is identical to [V N W, V + W].

a) I f V N W is a non-closed subspace in V and in W, then neither V nor W are interpolation spaces between V n W and V + W.

b) I f v n w is closed in V but not in W, then V is never an interpolation space between V n w and V + W whereas W is one, i f and only i f V n W is dense in W.

c) I f V N W is closed in both V and W, then neither V nor W are interpolation spaces between V n w and V ~- W.

P R O O F . - a) Suppose V is an interpolat ion space. There exists a sequence f u , , i c v n w such that Ilu. l l~n~=l , Ilu.ll -O and Ilu. llw<l. It follows that tlu.llv=l and Ilu.llg+ o. We can now apply 0orol lary [10.XII] to the couple [V N W, V q - W ] and the space A - V, and obtain

V C (V n W)(v+Tv) -- (v n W) v q- (V N W) w (see Theorem [10.VIII]). It follows

that V C(V N W) v - - in contradict ion to [10.IV]. The proof is similar for W.

b) Suppose first that V is an interpolation space between V ( ' /W and V-{- W. We can then repeat the argument at the beginning of the proof of a) and obtain again a contradiction to [10.IV]. Concerning W, in present c i rcumstances it is a closed snbspace of V-{ -W (see Lemma [7.I]). The only interpolat ion spaces for the couple IV N W, V q- W] which are closed sub-

spaces of V - b W are V + W and V n w v + w _ v N W W. Hence our assertion.

c) This case was settled in Remark [7.X].

§ I1. Minimal in terpola t ion spaces. - in this section [V, W] is a fixed compatible BAZ~ACH couple.

[l l .I] TItEOREM - Let A be any normalized intermediate space between V and W. Then

a) There exists a min imal (rel. the order relation C ) normalized interpolation space ~V(~ ~D A. It is formed by all elements u ~ V -b W representable by admissible sums E T~a such that T~ ~ ¢g with at most an enume-

a ~ 4

rable number of T~' s ~ O, the sum converges in V-k W to u, and ~ ITa I~g ]1 a li.~ < oo. a G A

The norm of u in this space is given by I1 u I I ~ inf Z Orc~ ~=AITal~llalJ'~ over all

admissible sums representing u.

A n n a l ~ d i M a t e m a t i e a 13

98 N. ARONSZAJN - E . GAGLIARDO: Interpolation spaces, e t c .

b) There exists a max ima l normalized interpolation space, z.r.r.r.r.r.r.r.r.rff~ ~ A. I t is formed by all u ~ A such that for every T ~ ~ , Tu@A. The norm in this space is given by JjuJje~q~- sup li TulJ~ for all Te~ +vith J Tj~g< l.

PROOF. - The existence of the maximal and minimal interpolation spaces spoken of in a) and b) follows immediately from the completeness of the latt ice of normalized interpolat ion spaces. It remains to prove the explicit representa t ions given in a) and b).

a) The proof that the space ~ , ~ described in a) is a B_A~ACH space is completely similar to the proof in part b) or Theorem [5.XII]. In par t icular it is a normalized BA~ACIt subspace of V ~ IV, since for any admissible sum /tuJlr%w < E il T~allv+w < E I T~l~g[lallA. If u e A , then the assignment

a ~ A a~ A

Tu -- [ (identity), and T a ---- 0 for a ~ u gives an admissible sum for u which show-s that /luj]~!~LA <--]lUi'A. Hence 917~D A D V N W. Furthermore, if T e ~ ,

then for any admissible sum u = E Tan we get Tu --- E TTaa and since EJTT al~gllaHa~jTj~gy,ITalcgljal l / it follows that ~'6.~ is a normalized interpolation space. Finally~ if B ~ A is a normalized interpolat ion space and u e~th:~, then for any admissible sum u - - Z T a a we have ~. H T a a N B ~ v i Ta ]~ H a HB ~ Y, IT a J~d I] a [j.~. Hence u E B and ]] u lIB ~ Jl U J!~ihA, which finishes

the proof of a).

b) In proving that ~ A is the maximal normalized interpolat ion space ~ A , we begin with a prel iminary remark.

Every u e~TqL:~ gives rise to a mapping of ~ into A given by Tu. This mapping is linear, and by using the closed graph theorem one shows immediately that it is bounded. Thus we have a correspondence between elements

in ~ A and ~;A,~. This correspondence is obviously linear, one-one, and

with the norm given in ! ~ A it is an isometry. One checks also that in this

correspondence the image of ~ A is a closed subspace of ~A,~;. Hence ~ = > = II-ll, + a n d is a BANACI~ space. Since Ilu![g~; l sup JlTu =llull:4 >

II .ll n <_ll li n , we see that ~[L~ since for u E V (3 W, Iluli~'i5 A < sup [ T:'~'~_~ 1

is a normalized intermediate space between A and V ('I W. On the other

hand, for I T']~ l we have ]lT'u I ~,~ -- J,~Ic~-- sup I!TT'uJ]:~ ~ IIu:]~A and thus

! ~ is a normalized interpolat ion space. It remains to show that if B is a

normalized interpolat ion space with B ~ A , then B ~ ~CA. In fact, if u ~ B , then T u e B C A for all T e ~ and H uJ]+=-E~ -~ sup ]1Tu [ l ~

snp il Tn <- II u il .

~ . ARONSZAiIN - E . GAGLIARD0: I~tcrpolation spaces, etc. 99

[ t l . I I] I)EFI~ITION -- For any u s V-b W we denote by ~(u) the set of all ~ ~ Tn with T s ~ . The space '~(u) will be considered with the norm ]Ixll,~ --- inf ITI~, the inf imum being extended to all T e ~ with Tu - - ~c.

[ l i . I I I ] - For each u s V + W, u ~ 0, we have

a) ~(u) is an interpolation space between V and W . ~ ( u ) is a nor. realized Banach subspace of V "b W for p _> [t u ]ll~+~v.

b) For every T' ~ ~ , [ T' I~'(u) <-~ J T' ]~g.

c) V A W A ~(u) for every u e V + W .

d) i l U i ! u = l , i f u ~ V ( 5 W then i l u - - y ! i u >-I for every y e V U W .

PROO;~ ~. - a) For fixed u, Tu is a l inear mapping of "-~ into V-i- W and C~(u), as defined, is the range of this mapping. One checks immediate ly that this mapping is continuous. If ~ is the nullspace of this mapping, then by definition of Ilw]Iu t h e m a p p i n g becomes an isometric isomorphism of ~/~VLu onto ~(u). Hence ¢~(u)is, a BAleAClt space. Since for each x - - T u ' w i t h T s ¢~, t!xlIv+w<=tTl~;]]utiv+w it is clear that ~(u) is a BA~AClX subspace of V + W and that ~ ( u ) is a normalized BANACE subspace of V-}- W for p ~_> ilultl++~r. That ~(u) is an interpolation space is obvious. Fur thermore , if x - - ~ ( u ) and T ' e ~ , then IIT'xllu--

inf [TI~- ~ inf [T'TJ~ <_ i T']~;liW[[u. Thus parts a) and b) are proved. T~=T'x Tu=z

c) Follows from a).

d) If Tu ~- u then I Ti~'=> 1. Since Iu = u and J I[~---- 1 we have t !Ui lu-=l , If ug. V A W we have by Corollary [9.V] that for y e V N I V and every T e ~ with u - y - - Tu, [ TI%-~1. Hence the second part of d).

[II.IV] THEOREM.- Let B ~ (9[u])~A where A is a normalized interpolation space, [u] is the one-dimensional space get, crated by u e V-+- W, u g: O, with the norm of V + W, and p > 1.

b) I f u s At • A, A1 is a normalized interpolation space, and ~ :,lu ~llv+w => ii u l lA~, then ~DThB ~ A~.

c) 1[ u A, then Ilu;Ie ,= ilujl +, and Fru -ahem,-> il lIv+, for every a e A.

100 N. ARo~szaJ~x - E. GAGLIARDO: Interpolation spaces, etc.

PROOF. - a) We prove first that 01~' = (pllU]lv+w ¢g(u)) ~ A is a normalized in terpola t ion space. We ~pply Prop. [6.XIV, e)]. This is possible in view of [ l l . I I I , a), b)], since [1 l . III , a)] impl ies ~ II u IIr+w ¢g(u) ~ (~ [] u I'7+w ¢g(u)) ~ (V-{- W) for ~ _~ 1. By [ l l . I I I , d)] ~Nul[v+wCg(u)~[u]. Hence ~D~'d'~B. If we now prove s ta tement b) of oar theorem, with ~]~Cs replaced by ~e)iU we will have at the same t ime proved both a) and b). To this effect we notice first that under the hypothes is of b), Cg(u)CA~ and for w = Tu and T ~ ~ , It w IIA~ --~ I T tog I: u tla~ ~--~ I T I"/7 ~iI u I!v+w. Hence ~II u !lv+w Cg(u) ~ At which proves the desi red s ta tement .

c) The proof is analogous to that of [ l l . I I I , d)).

[ l l .V] COROLlaRY- I f A~ and A~ are normalized interpolation spaces, A~ ~A~ , and A~ not a closed subspace of A~, then there exist interpolation spaces strictly between A~ and A~; for instance, the spaces ~D~f'Cn where B -

(~[u])~A~, u e A ~ ~ - Az and ~ --Ilutt~,/ l lulI~+w.

PROOf. - Under present conditions, ~[u]CA~; hence B ~ A 2 , and A~ ~['5BCA~. Obviously O]'CB @ A~ since u ee.~)~. On the other hand, ~ff'dB cannot be = A~ since, in the topology of ~ B , U is not in the closure of A~ (by Theorem [II . IV, c)]) whereas in the topology of A~, u is in the closure of A~.

For brev i ty ' s sake, we will in t roduce the notat ion

(lZ.1) 9rc~,~=grc,~=(P~(u))vJ(vnw),l[ IIo.,=11 IlercB,

where B = Ilullv+ [u] u ( v n w) and 9 ~

By Proposi t ion [ l l . I I I , c)], ~ , ~ , -- ~ (u) but the norms on the two spaces are in general different .

[ l l .VI] THEOREM - Consider an arbitrary set ~ C V -~ W and a function ~(u)___~ llullr+~7 defined for u ~ . Then the space ~ ~)rCe(~),~ is a normalized

ue$ interpolation space between V and W and all such spaces can be obtained

in this way.

PROOF.- That U!~FC~(,,),~, is a normal ized in terpola t ion space follows f rom Theorem [b.XIII]. Suppose now that A is a normal ized in terpola t ion space. P u t ~ - - A - - ( 0 ) and ~(u)-- l lulIA for u e A . By Theorem [ l l . IV, b)], ~]'Cg~), ~ C A for every u ~ $. Hence t.~ ~'Cpc,,), ~ C A, On the other hand by [II . IV, c)] tlullp(,),~,-'--llull~. Therefore, AC~J~)YCpO,),, a n d the norm in U ~]7~(~), ~ _~ the norm in A, i.e;, A C ~Ce(~), , .

N . ARO~-SZ~_JN - E . GAGLIARDO:~ Interpolatio~ spaces~ etc. 101

§ 12, The s t ruc tu re of the spaces ~(u). - As in the preceding section [V, W] will be a fixed compatible BA~AC~ couple. We will speak about interpolation spaces (meaning interpolation spaces between V and W). The last theorem of section 11 showed the role played by the subspaces Ca(u) (or, more precisely, their normalized versions ~ p , ~ ) i n the construction of interpolation spaces between V and W. In this section we will collect some more properties of the spaces '~(u).

It will be convenient to use the following definition.

[12.I] DEFINITION- In V + W we introduce a partial order relation denoted by < . We write u <~ u~ (or ux ~- u) if u e ¢g(u~), i.e., if ~(u) C ~(ux). I f u - < u ~ and u x < u we wr i t e u--,-u~: this means that Ca(u)= ~(ux). All u's satisfying u-,.-u~ form an equivalence class (of ux).

It is obvious that the relations <: and ~,~ introduced above are actually a partial order relation and ari equivalence relation, respectively, and that V + W is thus decomposed in mutual ly disjoint equivalence classes.

[12.II] - a) The following sels are equivalence classes : (0), (V G W) - - (0),

( V - - V N W V ) + ( W - - V N Ww) (~8), v - V l ' ~ w v , w - - ] z n w w . All the other equivalence classes are contained in one of the five mutual ly disjoint sets :

( V - - V N W v) + (V N W ~ - ( V N W)), (V N W T - - ( V N W)) + ( W - V N W,V),

(V n W v - - (V N W)), (V n w w -- (v N W)), (V n W W - (V N W)) + (V n W " ~ - (V n w)~.

b) Every interpolation space is a union of equivalence classes. Hence

i f A and B are interpolation spaces, A C B, then B - - A, i7. B ~ A, and B ~ ~B are unions of equivalences classes.

PROOf. - a) The proof of a) is immediate if we use Lemmas [7,1] and [7.III] and Theorem [7.IV]. We have ~(0) -- 0 ; for u ~ (V fl W) - - (0), '~(u) --

VFI ~V; for u e ( V - - V A W v ) - ~ - ( W - V A W W ) , ~ ( u ) - - V + W; for u ~ V - -

V N W v, ~ ( u ) : V ; for u e W ~ V fi W w, ~ ( u ) - - W.

b) The proof is obvious since under the hypothesis of part b) ~ s is an interpolation space.

[12.III] - a) I f T ~ a n d T -~ exists in "~ then Tu~x .u for all u.

b) I f x e ~(u) and I ] x - u H u < 1, then x - , - u .

c) The equivalence class of u forms an open set in Ca(u).

[is) For two subsets A and B of a vector space, A + B means the set of ail vectors of the form a-~ b with a ~ A and b q B. If one of the sets A and 13 is empty, then A-l-B is also empty.

102 X A~o~sz.~z~" - E. GAGLIARD0: Interpolatio~ sp,aces~ etc.

PROOf. - a) is obvious. To prove b) we notice that if i i x - - u l I ~ < I, i.e., there exists T e ~ such that I T l ' ~ < l and x - ~ u + T u : ( I ~ T ) u . It follows that I - } - T has an inverse in ~ , and part b ) fo l lows from a). Finally, to prove c) we remark that if u~ ~ u the spaces ¢~(u) and ¢~(u~) are the same as topological spaces (even that their norms are in general different). There- fore b) implies c).

[12.IV] - a) Each interpolation space A C ~ ( u ) - (u) lies outside of the open un i t sphere S~(u, 1) with center u.

b) I f u~ ~ ~(u) and u~ 4 u, then there exists a max imal interpolation space A~: , (in general not unique) with u~ e Am~:, C ~(u) - - (u). Am~x is always a closed subspace of ~(u) contained in ~ ( u ) - S~(u, 1).

c) For every interpolation space A C ~ ( u ) - - ( u ) there exists a l inear funct ional c?(x) on "~(u) with bound 1 such that ¢?(u)- 1 and ¢?(A)~-O. The subspace ~ ( x ) ~ 0 is a supporting hyperplane for S~(u, 1).

PROOF. - a) If u ~ A then A does not contain any point of the equivalence class of u hence [12.III, b)] implies A f l S~(u, 1 ) - 0.

b) There exists at least one interpolat ion space, namely ~(u~) satisfying

u~ ~ A C '~(u) -- (u). By a), all such spaces lie in ~(u) -- Su(u, 1). Hence .~;~u) is also an interpolat ion space satisfying this condition. For any class of interpolat ion spaces ordered increasingly by inclusion such that each space is C ~(u) -- Su(u, 1), the closure (in ~(u)) of their union is again an interpo. lation space C ~ ( u ) - Su(u, 1). Hence the conclusion of b).

c) It is enough to construct the funct ional ~ for the closure 2~ "~tu) C ~ ( u ) - - S u ( u , 1). W e define it first on the closed snbspaee A ~(u) ~ [u] by put t ing ~(a -}- ~u)--~ where a e ~: (u) . Since a ~ u, we h~ve for ~ ~ 0 , II~ T ~u Ilu----

I~I II~ -~a + Ul[u ~ t~l and thus q~ satisfies our requirements on A~(u)~q --In]. Using HAHz~-BA~cACrI Theorem we extend q~ to the whole of ~(u) according

to the requirements .

[12.V] REMARK - There is an interest ing algebraic interpretat ion of the last proposition. Consider u ~ 0 and denote by ~ , the set of T e c~ with Tu = 0 (the nut lspace of the mapping T ~ Tu). Then ~ , is a closed lef t-sided ideal of cG and there is a one-one correspondence between interpolat ion spaces A C ~(u) and lef t -s ided ideals ~9~,,A satisfying ~ C ~ 7 ~ , ~ C which are BA~AC~ subspaces of ~ . This correspondence is given by ~ , , , ~ - - [Te ~ : T u e A ] . Par t b) of [12.IV] shows that there are such maximal proper ideals and that they are necessari ly closed in ¢g, The functional ~ of part c)

N. ARONSZAJN - E. GAGLIARDO: Interpolation spaces, etc. t03

gives rise to a bounded l inear funct ional ~(Tu) on ~ which, on the subat- gebra [I] + ~Y'5,,,_~ is multiplicative. An especially interest ing case is when A is a closed subspace of ~(u) of eodimension 1. Then ~ , , , A is a two-sided maximal proper ideal of "-~ and ¢p(Tu) is multiplicative on the whole of ~ .

In the remainder of this section we wil] investigate the possibility of the ext reme case mentioned at the end of the preceding remark. We actually study the more general case where

(1,,.1) ¢g(u)C [u] + A, A being an interpolation space not containing u.

If (17.1) holds, then obviously C~(u)--~ [u]-[-(A N ~(u)) which presents the ease mentioned at the end of Remark [12.V].

[12.VI] - Let A be an interpolation space satisfying (10.2a) or (10.2a') or

(10.2b). Then, in order that relation (12.1) hold, it is necessary that u e ~V (v+~r)

or, u ~ ~z(v+w), or, u ~ (Z(v+v) U ~¢(v+~) respectivelj.

The proof is immediate by Lemma [IO.X] (the space A in the lemma being replaced by [u] -k- A).

[12.VII] - I f the relation (17.1) holds, then:

a) I f A satisfies (t0.2a) and (102a'), then u e V f l W(V+w).

b) I f u ~ V fl W (v+W) then A satisfies (10.2b).

PRooF. - a) By the preceding proposition uE ~(v+w) N VV(v+w)"-V N W(V+w) (see Theorem [10.VIII, c)].

b) If A did not have property (10.2b) the topology of V + W would be f iner than that of A on every sphere Svow(R), 0 < R < c ~ . Since

u e VO W(v+ ~) there exists a sequence t u , } C S v n w ( R ) for some finite R such that the sequence {u,} is CAtTCg¥ in V-{-W, hence CAUCH~¢ in A. Therefore iu,} would converge to the same point u in V ~ - W andA. This would imply l l u , , - u ll.~ ~ 0 which is impossible since u ~ A .

[12.VIII] COROLLARY - Suppose V fi W is non-closed in V and W. Then the relation

(12.1') ~(u) - - [u] + (V fl W),

is possible only i f u ~ V fl W(V+ W)

u ~ V N W

104 N . AROIqSzAhR ~ - ]~. GAGLIAltDO: Interpolation spaces, etc.

PROOF. - In the present case there exists a sequence t u,~t C V N W with llUnII~ow--t and I l u . l t w ~ 0 . Hence VN W has proper ty (10.2a)(also with V and W interchanged). We then apply the preceding Proposi t ion [12.VII, a)].

[12.IX] REMARK- The ease when (12.i') holds represents the smallest spaces "g(u) after the space (0) corresponding to u - - 0, and V N W corresponding to u~(VN W ) - - ( 0 ) . If, in the corollary~ we did not suppose that VN W is non-closed in V and W, the only addit ional possible cases of (12.1') would be the trivial ones when V -" [u] -~ (V N W) or W - - [u]- t-(VN W). When (12.1') is valid we can also completely determine the equivalence class of u: it is the set of all elements of the form ) . u + ~ where x e V N W and ) , ~ 0 (since V N W is then the only maximal interpolation space C ~ ( u ) - (u)).

As a counterpar t to the corollary we construct an example where

u e VN li '(r+W) and where the relation (12.1') actual ly holds. Let V be an infinite dimensional BA~ACH space and let v,, ~ u ~ 0

in V. W e may and will assume that the elements u and vn, n - -1 , are l inearly independent and that l tu l l{-~ l t v . I v - - 1 . We construct the elements uh - -v ,~ , k - - 1 , 2,..., by induction, choosing u~--v~. If ~ all u~...ua--v,~ are already defined~ we consider the smallest distance ~h from u to the l inear subspace [u~,.. . , uk] generated by u~,. . . , uk. W e denote then by nk+~ the smallest integer > n ~ such that Iiv,b+ --ull~-~2-a~a. The sequence {uh!

OO

being so defined, we consider the subspaee W ~ (~ fuji and the l inear

couple IV, W]. We notice first that u ~ W. Otherwise we would have u - - Z ~kua with Z ]~h] ]IuhIIv < ~ . Since IlukNr-~ I[uIIT ~ - 1, we may find N such that

Z l ~ h t < l / 2 " The equat ion for u then gives Z ~ ( u k - - u ) ~ u ( t - - ~. ~h)-- N-~I N ~ N~z N Z ~u~. The norm of the le f t -hand member is then ~ 2--~"-~N, whereas the

1

norm of the r ight-hand is > 1/29~. Hence a contradiction. W e show next

that VN W ~ + ~ ) - - W~) - - [u] -{- W. To this effect suppose that iI~c, - - x l l r ~ 0

and that ilx.]t,~<M. Hence x , = Z ~'~u~, Z l~")l < M . We may assume k=l k=l

cO

(if necessary, by choosing a subsequence) that lim ~ ) - - ~a and lira Z ~(~") ----- ~.

It follows that Z I~al ~ M and I al ~ M. If we put x ' - - Z ~ua Jr (~¢ -- }2~a)u, i 1 1

then x ' : x . In fact, x , -- ~ ' - - Z ( ~ ' ) - - i ~ ) u ~ + Z ( ~ ' ) - - i a ) ( u a - - u ) + k = l k = N + l

N u[ Z ( ia-- i (~ " ) ) + ( Z ~(~'~) - - :¢)]. When n - - - ~ and N remains fixed we get

N . ARONSZA;IN - E . GAGLIARDO: I n t e r p o l a t i o n spaces, etc. 105

lim sup ] I x , , - ~c'll ~ ~ 2M2--N~N. Since we can take 1V--~ ~x~, this shows that

x , ~ w ' in V; hence w ' = x . It follows that ~7(v) C [ u ] + W . 0n the other

hand, I lu ,] lw~ 1 and u, ~ u in V; hence u ~ !~V (v), and our proof is finished.

C~AV~E~ I I I

Interpo la t ion Methods.

§ 13. In te rpola t ion theorems. - In this chapter << couples >> will mean BA~AOE couples (unless otherwise stated). Different couples which we consider will be dist inguished by lower indices such as [Vo, Wo], [Vx, Wx], IV,, W~].

We already introduced the notation ~S ,a to denote the BANAC~ space of bounded l inear t ransformations of A into B with norm --ITIB,,~ for any such transformation T. for two couples IVy, Wi] and [Vj, W j] we can consider the mappings of V~ U Wi into Vj U Wj which t ransform V, into V~ and W~ into Wj l inearly and boundedly. As we know from Chapter I, they can be extended in a unique way to a bounded l inear mapping of V, + W~ into V j + Wj. This class of mappings, called bounded linear mappings of couple ~> into couple <<j >> will be denoted by ~ , , . In particular, ~,,i is identical to the previously introduced class, ¢g[V~, W~]. For T e '~/,i we introduce as norm, the expression

(13.1) IT] i ,~ - max[ITtvj , v, ' tTIwj, wi].

If T e '~2~ and T' ~ ¢g82 we can compose these t ranformations and obtain T'T ~ ¢Gsl. We note first a few obvious properties of the above introduced ~ot ions .

[13.1] - a) ~i, i is a Banach space with ]TI~,i as norm

b) I f T e ~o,1 and T' ~ ~l,e then TT' e ~o,2 and ITT'1o,2 ~ IT[o,1 IT'11,2.

c) The space ~ , i can be canonically identified with a closed subspace of

¢gv i, v~ -~ ~ y i , w i composed of couples i T', T'} satisfying T'u i :- T"u i e V~ ~ W~ for all uj e Vj N Wj . This identification preserves norms if, on the direct sum, we take the maw-norm.

Clearly, part c) above is an extension of [6.1V]. As an immediate exten. sion of [6.I] we obtain

[ 1 3 . I I ] - For T e ~i,i , [TEv;n~,,vjn,~j < [Tki, [Tlv+~ < I T k j .

Annali di Matematiea ~

106 N'. ARONSZASN - E. GAGLIARDO: Interpolation spaces, etc.

[13.III] DEFINITION - [ f in a couple [ r i , Wi] we choose an in t e rmed ia t e space A~, the sys tem IVy, W~; Ai] will be cal led a triple, and if no confus ion arises, it will be d i s t ingu i shed by the same index as the couple (~9). I f the intermediate space is normalized, we will call the t r iple normalized. I f for two t r iples • i >> and (<j ~ we have the property that for every T e ~i,~, T(A~) C A i, we say that the (i ~ j ) - i n t e r p o l a t i o n theorem holds. The (i ---~ j ) a n d (] ~ / ) - i n t e r - pola t ion theorems are cal led unilateral . I f both un i l a t e ra l in te rpola t ion theorems are valid, we say tha t .a bilateral interpolation theorem between the t r iples <) and (<j)) is va l id ; we denote it as (i <->j)- interpolat ion theorem.

If the two t r ip les are the same, then obviously the ( i - ~ j ) - i n t e r p o l a t i o n theorem holds if and only if A~ is an in te rpo la t ion space be tween V~ and W~. I n ana logy to L e m m a [6.X] and Th e o r e m [6.XI] we obtain here, by s imi la r proofs, the fol lowing s ta tement .

[13.IV] TEEOREM - a) I f for two triples <<i)) and <(j ~ and a mapping T e ~i,~ we have T(A~)C Aj , then TA~e ~.~,_% where T~4~ is the restriction of T to A~.

b) I f the (i ~ j ) - i n t e r p o l a t i o n theorem holds, then there exists a con- s tant c such that I T tA i, ~ ~ C t T {j,~.

Consider a t r iple IV1, W i , Ai]. Since V i N Wj c A i c V i + W i, there exist cons tants mj ~ m1(A ~) and m i' ~ mj(Ai) such that

(13.2)

(13.2')

]la]]~j ~ mtl]allv~+w j for a e A i ,

l lull~ <= m/liutiv~n,:~ for u e v i n w i

W e will a s sume tha t these cons tan t s m i and m i' are the best possible in i nequa~ t i e s (13.2) and (13.2') respect ively .

Consider f u r t h e r m o r e a couple [V~, W~]. W e will use the fol lowing nota t ions .

(13.3) For any T~,ie ~ , i and ueTi,~(Ai), Hu [r ,s(Ap-= inf[ allA j , the infi" m u m extended over all a e A i wi th T , , i a - - u .

(13.4) For any Ti ,~e cgi,~ and u e T~,~(Ai) (so), IlU[!T/~(,tj ) __ m a x [mi'[luli~ +w+,

11Tj,'uHAj]"

(i~) If for the same couple two differont intermediate spaces are chosen, the resulting triples are different and will have to be indicated fully~ say [V~, ~7t, Al] and [Vi, W~, Bi].

(~o) T~,~(A]) is the subspace of all elements u e V¢+W~ such that Ti,iu e Aj.

N . ARONSZAJN - E . GAGLIARDO: Interpolations, spaces, etc. 107

[13.V] L E M ~ A - a) For each T~,¢e ~;~,i with ITi,¢li, i ~ l, the space Ti,i(Ai) with norm (13.3) is a Banach subspace of Vi ~ Wi salisfyin9 Ti,i(Ai) mi(V~ + Wd.

b) For every T i , ~ ~ ,~ with I T~,~li,i ~ 1, T~(Ai) with the norm (13.4) is an intermediate space between V~ and W~, and we have m / ( V ~ W~)~ • ~,~(A~) ~mi'(V~ + W~).

PnOOF. - a) That T~,i(AI) is a BA~AC~ space is clear since it is canonically isometrically isomorphic to the quotient space A¢/N where N is the intersect ion of the nullspace of T~,i with A1 (N is closed in A¢). For u e TLi(Ai) and a ~ A i , with T~,ia-- u we have, by using [13.II], II u J I ~ + ~ IT~,II~,i

1 • l[ a lvj+w~ _--< lla !l~j+% =< n~ )l a lI~. I~ follows therefore that T,,~(A~) ~mi(Vi + W~).

b) To see that T~,~(Ai) is a BA~AC~ space we consider the direct sum

m/(V~ + W~)-~ At, with the max . -norm and notice .that the correspondence u ~ {u, Ti,,u! is an isometric isomorphism of T~(A/) onto a closed subspace of the direct sum. Thus the inclusion T~,~(Ai)~m/(V~-4- W~) follows immediately. To prove the other inclusion, consider u e V~ f3 W~. Then Ti,~u e V i ~ W~ and II T],iutl,~y < m/ [I TI,iUHV]f~Wj ~---- m/[Ti,~[¢,~ [I Ultvin ~ ~- m/ N ullr'~nw, (using [13.II]). This implies T~(Ai) ~ m/ (V~ ~ W~).

[13.VIi LEMMA- a) For each u eVi(~ W~ and for each ~ > O, there exists a transformation T~,]e ~i,~ with IT~,~l~,i--1 of the form T~,ix~--

1 <x, f>~%.+w~ --u~ where f ~ ( V ~ + Wi)* and :¢>0 , such that IlulI~,i(_~i)~<

(mi + ~)II u II, ,n~:.

b) For each u~ Vi + Wi and for each ~ >0, there exists a transformation 1 u'

T j ,~e~ i , i with bound ITj, i[i , i--1 of the form Ti , ix ,= <~,f>v¢+w~

where f ~ ( V ~ + Wi)*, u' e Vj(] W~ and ~ > O, such that llTl,~ull.~j>~_ (m~ - - ~) )) u )!v,+~,.

PRoo)'. - a) Since the constant m i in (13.2) is the best possible, we can find an aoeA i with I[ao]lAj <(mi+~) l lao] lv j+wj=mi - ] - s . We choose f ~ ( V i + Wj)* so that [I fll(vj+wj)* -- 1 and < no, f>vj+w~ -- 1. Since I T~,i[~,i-- 1

--a max[ H ftIvj* tt utIv,, IJ fllwj* tt ull)~], by choosing ~.---- m a x [ tl fJI%" tl uHv,, II flJ%" H u ti~] we get I T~,i [~,i - - i. It follows that T~,i(aao ) - u and tluilTi,i(.~i) _~< [taaoii~tj < @m i + ~) ~-- (m i + e) il u tiv, flw, since :¢ --< max ( II u iiv,, t[ u ltw~)" max ( [] f l[v,', I[ f Hw.¢*) = tlu [Iv, n ~ II f II<~/+wj>* = II u liven ~', (we use here [8.I]).

108 N. AaoNszna~ - E. GAGLIARD0: Interpolation space% etc.

b) Simi lar ly as in a), we choose f e ( V i + W~)* with tl f ll(v~+~i)" = 1 and < u, f >vi+w = ]] Utlv~+w i. We choose u' ~ ViA W i so as to have [] u' I!,~i> ( m / - - ~)]] u'[[~-in~[.. By choosing then ~ = max [!l/live* I] u' II~j, ]lf I[w." II u' iiwj] we

1 obtain ! Ti,i Ii,i = 1, t[ T~,iu 11~ - - II u tl u ~+ u~ - ~ N u' t l ~ >= (mi' - - ~) ]] u liv ~+w~ .

[13.VIII TKEORE~I - a) The space

(13.5)

is an interpolation space between V~ and Wi with property (6.3) (i.e., I TBtit "<: ITli,~ for T~ 'g~,~), and satisfies

(13.5') mi(V, + W,) ~ B,~ ~ m i (V~ fl W~).

b) The space

(13.6) C~ =-- C~i (A i) (~ T ~ (Ai)

is an interpolation space between V~ and W ~ with property (6.3) and satisfies

(13.6')

b') C~i(Ai) is the space of all u ~ V~ + Wi such that for every Tj , ie ~i,~, Tj,~ u E Aj and such that sup I[ Ti,i u I IA~ < co. For u ~ C~i (Aj) we have also

I1 u llcj(~ i) = sup tl ri,,u ll.~i" I ~ j , i li,~ ~

PROOF. - a) By Lemma [13.V, a)] and Theorem [5.XII, b)], B~i, as given by (13.5), exists and satisfies B ~ C m j ( V ~ + Wi). The second inclusion in (13.5') follows from Lemma [13.VI, a)]. To prove that B~i is an interpolation space with property (6.3), consider any u eBb1 and Ye ¢g~,i with iTl~,~<=l. Then by definit ion of ~ , u - - Z T~,}a,, the sum converging in V~q--W~, T(.") I~,j < 1, a, ~ A i anti ~. II an ]l~i < ~ . Then Tu = E T T ~ a, ; clearly, the

¢71 = sum converges in V~+ Wi and ITT~:}I~,i~-1. It follows that TueB~t and ]l Tu ttB~ =< tl u tlmi which proves property (6.3).

b) That C~i is an intermediate space and satisfies (13.6')is a direct consequence of Lemma [13.V, b] and Theorem [5.XII~ a)]. That it is an inter-

N . A R O N S Z A 3 N - E . G A G L I A R D O : D~terpolation spaces, e t c . 109

potat ion space wi th p rope r ty (6.3) is based on the def in i t ion of i~. In fact, for u~C~1 we have Ti ,~u~A¢for all T ~ , i ~ y , ~ and IIullc~i - sup m a x -

[m/l[ u I[v,+~, [] Ti,~u ]]~j]. H e n c e for T e '~,~ with ]T[i,~ ~ 1, we get Ti,~TuEA i and [I Tu [[cii : s u p max Ira/ II Tu I[v¢+n~, I[ Ti , ,Tu ]l:lj] < ]l u ]Ic,1. W e use

IT i , i i j , i<= z

he re the fact tha t V~--}- W ~ sat isf ies p rope r ty (6.3).

b') By def in i t ion of ~ (see (5.3)), u E Cii(Aj) if for eve ry Ti, i E '/;i,~, Ti,iu E A] and 1] u [[c~i(.~j) "-- sup m a x [m/ [[ u []~¢+w¢, [[ Ti,~u [!aft < c~. S ince

I T i , i l j , i ~ - - < ~

by L e m m a _[13.VI, b)] there exists a T~,, with ]T;,,ti,,---1 and llT~,~ull.~i>_~- ( m / - - ~) l] u !l~,+w, for any a > 0, it fol lows that sup m a x [m/It u ttv,+w,,

H Ti,,ut]~j]-" sup t[ T~,~u ]l~j which f in ishes the proof.

[13.VIII] - Suppose that in the triple [ V i, Wj ; A~] we replace A i by another intermediate space Ai ~ PAi, for some p > O. Then.

a) The constants m i and m~' corresponding to ~4 i in (13.2) and (13.2')

satisfy m t <-~ pmj and m i' ~ pml .

b) B~i(Aj)~pB~i(Ai) and CiJ(Ai) D pB~Y(Ai).

PROOF. - a) F o r any a E A i we have l} all,4/>- 1/p II a II~j ~ ~j/~ lI a iivs+~: + and for any u E V i N W i we have [ [ u t l X j ~ P ] ] u H ~ j ~ P m i'l[u[Ivifl.~. Since

m] and mY' are supposed to be the best possible cons tan ts in the i r respect ive

inequal i t ies , our conclus ion follows.

b) W e c h e c k i m m e d i a t e l y that TCi(3.i)~P(Tci(Ai) ) for every T~,jE '~i , i

and also T~,~(A~)~p(T~,~(Ai) ) for every TI,iE ¢gi,i" For the last inc lus ion we

use the fact proved in par t a) of this proposi t ion that m / _ < pmi ' . The con-

c lusion of par t b) follows then immedia te ly .

[13.IX] - The following relations always hold:

B~ ( C~i (A~)) G A i G Cj ~ (B/ (Ai) ) .

PROOF. - Suppose f irst that u E BI~(Gii(Ai)). By (13.5), u is r ep resen tab le by an admiss ib le sum in V i + Wj : u = E T('~)x with [Y(.").li,~ < 1, x,,6 C~J(Aj) y , i n 1 , ~ :

and [[ u [[Bi~(C~J(Ai) ) - - inf E If X,~[[c~i(Aj), the i n f i m u m being ex t ended over all such admiss ib le sums. S ince x,,E Cd(A~) it fol lows that- T(~)x E A i and i , i n

110 ~ . ARONSZAJN - E . C{AGLIARDO: Interpolatio~ spaces, etc.

ttx,,~llc~(~[i) >= [I T}~,lx,~ II,~" i~lence for each admissible sum, E ~(~)x,,, ~ converges - - T(n)x~ ll~j < ~' [I x~ which proves the first in A i to' u and [Iu lIAr. -<- Z II i,~ = [Ic~i(~./),

inclusion of our proposition. To prove the second inclusion suppose u E A i. Then for every T~,iE '~i,~

with ~ Ti,iIi,i -< 1 we have T~,iu E Ti,i(Ai) cB~i(Ai). Hence u E T~,~(Bj(Ai) ) and, by Theorem [13.VII, b')], ]tu I]cii(~i¢%>) = sup l] T~, i u t!Bii(Aj) ~-~ sup i] T~,i u II Tt, j(Aj) tfUtI~¢ which finishes the proof.

In the next theorem we will use the following definition.

[13.X] DEF~I~IO~¢ - The best constant c in the inequali ty of Theorem [1B.IV, b)] will be called the (i ~ i)-interpolation constant and denoted by y~,i.

[13.XI] Ti~EOREM - Let [V~, Wi; Ad and [Vi, Wi ; Ai] be two triples with all the spaces fixed except for Ai.

a) In order that the (j ~ i)-interpolatio~ theorem hold it is necessary and sufficient that A ~ B~(AI). I f the theorem holds and Ti,i is the ( j -~ i)- interpolation constant then A~ ~ y ~ , i B~i ( A1).

b) In order that the (i--~j)-interpolation theorem hold it is necessary and sufficient that Ai ~ C~i(Ai). I f the theorem holds and T1,~ is the corresponding constant then Ti,~ A~ C C~I (At).

c) In order that there ex~ist an Ai such that the (i .e-v, j) interpolation theorem hold, it is necessary and sufficient that B j ( A j ) c C,J(Aj). I f this condition is satisfied, then the bilateral interpolation theorem holds i f and Only i f B~i (Ai) c Ai ~ Cii (Ai). I f the bilateral theorem holds, then Yi,i BiI(Ai) C Ai

1 C~i ( A ~). Yi,~

PROOF. - a) If A, DB,i(Aj) then Ti , j (Ai)c BiI(Aj)c At. Hence the ( } ~ i ) - interpolation holds. Assume now that the (j ~ i) interpolation holds. Then I T*,il~,,.~i < Yi,i! T*,JI~,i" It follows for u E Ti,j(Aj) and tTi,/li,j <~ 1 that [l u ll.~,-- < Yi,~ [t a{t-~j for all a E Aj with T i , j a - u. Hence A ,D yi,j(T~,i(Aj)) and by (13.5) Ai~ ' ( i , IB j (A i ) which finishes the proof.

b) Again, A , c C~i(A i) implies the (i ~ j ) - i n t e rpo l a t i on theorem. Suppose now that the (i-- .- j)-interpolation theorem holds. Then for all Ti,~E ¢gj,~, ]T~,~I~j,A ~_~7i,~]T~,ili,~, and, for u EA~, Tj,~u E A i. Hence by Theorem [13.VII, b')], tl u [[C~J(Aj) ----- sup IE Ti,~ u ]]_~j < T~,~ II u Ii.~; which proves part b).

ITi,~Lt,~ <1

c) is obtained by combining a) and b).

N. Aao~szx3~ - E. GAGLIARDO: Interpolatlou spaces, etc. 1 i l

[13.XlI] DEFI~ITIO:~ - If in the t r iple IVy, W~; Ai] A~ is an interpolat ion space between V~ and W~, the tr iple will be called an interpolation triple. If A, is a normal ized in terpola t ion space, we have a normalized interpolation triple.

[13.XIII] DE~]~IO:~ - Let [V~, W~; As] and IV1, Wi; Ai] be two triples with (i -~ j ) - in te rpola t ion theorem. The in terpola t ion theorem is called optimal if we cannot strictly increase A~ nor str ict ly decrease A i without, losing the theorem.

[13.XIV] Tt:IEOREM - Suppose that the (j--+ i)-interpolation theorem holds.

a) I f the interpolation theorem is optimal, the triples (> and ~j must be interpolation triples.

b) In the triple (~ i f we replace Ai by B~(Aj) or B~i(Cj~(AI)); and in the triple ~j>> we replace A i by Ci~(B~i(Ai) ) or Ci~(Ai)respectively, then, between the resulting triples we get an optimal (j ~ i)-interpotation theorem.

c) If, in the triple ~i)>, A~ is replaced by ~_~cAi, and in ~j>> A i is

replaced by A1 ~ Ai and an optimal (] ~ i)-interpolation theorem is thus obtained, then Bii(Ai) c A~ c Bi~(Ci~(A~)), and Ci~(B~i(A~)) ~ zt~ ~ Ci~(A~).

PROOF. - a) By Theorem !13.X[,a), b)] it is clear that if the interpola- t ion theorem is opt imal then A~--B~I(Ai) and Aj--Cj~(Ai). By Theorem [13.VIII it follows that the tr iples are in terpolat ion triples.

b) By Theorem [13.XI] the (j ~ / ) - i n t e r p o l a t i o n theorem is main ta ined if I A~, Ail are successively replaced by I Bj(Ai) , Ai } and by { Bj(Ai) , Ci~(B~J(Ai)) !. Since A~DB~i(Ai)(by Theorem [I3.XI, a)] and A i c Cii(Bii(Ai))(by [13.IX]), one sees immedia te ly that in replac ing t Ai, Ajt by tB~J(Ai), Ci~(B~J(Ai))I an opt imal (j ~ / ) - i n t e r p o l a t i o n theorem is obtained. Similarly, when replac ing !Ai, Ajt by tAi, Ci~(A~)t and then by tB~i(Ci~(A~)) , C~(A~)!, we obtain again an opt imal in terpola t ion theorem.

c) In present condit ions we have a (j ~ / ) - i n t e r p o l a t i o n theorem when

replacing tA~,Ai} by tA~,Aj} or iA~,At}. I t follows by Theorem [13.XI]

that A_IDB~I(AI) and A-jc CI~(A~). The other inclus ions follow from Proposi- tion [13.IX].

[13.XV] REMARK- The opt imal in terpolat ion theorems which seem to be the most advantageous ones, were not, as far as we know, invest igated in the l i terature. On the other hand, it is known that some of the most used

112 N. ARONSZAJN - E. GAGLIARDO: Interpolation spaces, etc.

interpolation theorems, such as the M. RtEsz convexity theorem, are not optimal (~).

[13.XVI] T ~ E o m ~ - I f [VI, W i ; A]] is an interpolation triple and [V~, W~] an arbitrary couple, then we have

(13.7) ";i,~ B,~ (Ai) ~ C~(A~) .

I f the triple (<j ~) is a normalized interpolation triple, we have B j ( A i ~ C~i(Ai).

PROOF.- Consider any T~,iE ~i , j and Tj,~E ~j,~ with ]T~, i l~ , i~ l and IT i ,~ l j , i~ I . Take any uET~,j(A~) and any a ~ A i with u - - T ~ , ] a . Then II u IIT~,~(,~i)= inf ]1 a II,~" On the other hand, Ti,~u=Ti,iT~,ia. Since Ti,~T~,~ ~i,~,

and A~ is an interpolation space, we have T t , ~ ~ A i and sup It Ti,~u tt.~-- }Ti,iIi,~<-- - x

sup ]l Ti,iT~,ia I!A~ ~- "(i,1 [[ a fiat.. Hence I[ u Ile~i(~j) ~ V~,~ ][ u [iT~,~(.~j), C~(Ai) yi,i(T~,i(A])) and (13.7) follows.

If A i is a normalized interpolation space then ~],i-- 1, and our assertion follows from (13.7).

[13.XVII] Tt~[EORE51 - Consider two triples <~ i >> and c j ~>.

a) The optimal (] ~ i ) - in terpolat ion theorem holds i f and only i f Ai = Bii(A i) and A i -- Cj~(Ai).

b) I f the optimal (j ~ i)-interpolation theorem holds, then the (i ~ j ) - interpolation theorem is ~valid also.

PROOF. - The proof of a) follows immediately from Theorem [13.XI, a)b)]. For the proof of b), we notice that by a) A i = Ci~(B~Y(Ai)). Since B~i(AI) is an interpolation space between V~ and Wi, Theorem [13.XVI] applies, and BI~(Bj(Aj))cCi~(B~i(Aj)). Hence, by Theorem [13.XI, e)], the bilateral (i < -~ j ) - interpolation theorem holds.

In general the spaces B~J(Ai) and C~i(A]) are not normalized even when A i is a normalized interpolation space in [Vi, Wj]. We therefore introduce the following spaces:

(13.8)

(13.9)

~i(Aj) ~ [B~i(A~) (~ (V~ + W~)J U (V~ n W~),

C~J (Ai) ~ [ C~i (Ai) ~I ( V~ Jr W~)J ~J ( Vi N W~) .

(~t) These results follow, for instance, from E. GAGLtARDO~S investigation of special interpolation methods [5],

:N. ARONSZAJN - ]~. GAGLIARD0: Interpolatgon spaces~ etc. 113

Transla t ing the previously proved properties of B~i and C~i, we obtain the following list of propert ies for the spaces (13.8) and (13.9).

[13.XVIII] - a) B~i(At) -- B~i(Ai), ~i(Ai) -- C~i(Ai), ~l(al) and ~ (A i ) are normalized interpolation spaces.

b) I f A i is a normalized intermediate space, Bii(Ci~(Aj))C Ai ~ Ct~(B~i(A~)).

c) I f A i is an interpolation space, then (13.7) is valid when C and B

are replaced by C and B.

d) When A i is a normalized interpolation space then B~i(A~) and ~i(A~)

are normalized interpolation spaces in [ V~, W~] satisfying ~t (A]) ~ Bii(Aj).

All these statements are immediate consequences of preceding theorems. In some cases we use Proposition [6.XIV, c)]

§ 14. I n t e r p o l a t i o n methods .

[14.I] DEFI~ITIO~ - Consider a class ~ of compatible BANACH couples. If un intermediate space F[V, W] -- A~ is assigned to each couple [V, W] E ~ , we say that F is an interpolation method defined on g when the following condition holds: For any two couples [V1, W1], [V2, W2] in ~ , the (1 ~ 2)- interpolation theorem holds for the corresponding triples [V1, W1; A~] and IV2, W2; A2] (where Ai)- -F[V~, W~]). If all the interpolation constants are uniformly bounded for all choices of the couples, the interpolation method F will be cailed uniform. If all these constants are _--~1, the interpolation method F is called normalized.

As obvious consequences of the definit ion we have the following properties.

[14.II] - Let F be an interpolation method defined on a class of couples gg.

a) I f [V, W ] E ~ and A - F[V, W], then A is an interpolation space between V and W.

b) For any two couples in ~ the corresponding triples determined by F admit of a bilateral interpolation theorem.

PROOF. - For a) we take the two couples of the definition as -- [V, W], and for b) we interchange the couples of the definition.

Annali dt Matemattca 15

114 N . A~ONSZAJN - E . GA~L]'AI~DO: Interpolation spaces, e t c .

[14.III] RE~AI~K - Wo do not know of any interpolation methods in the l i terature which are not uni form; most of them are normalized. W e will show in the next theorem that any uniform method can be made normalized by a su~itaBle ch:ange :-of norm in the :spaces ;~F[V, W].: On the other hand we do not know if every interpolat ion metlmd can be rendered normalized in this way.

The importance of interpolation methods lies in the interpolat ion theorems they give rise to. One could imagine more general ways of obtaining interpolat ion theorems. For- ins tance, one might have, in two different classes of couples, ~ an d c~', two assignments of intermediate spaces, F[V, W ] - - A and F'[V' , W'] -- A' so that for any couple IV, W]~ df and [V'W'] ff OX' there would be an interpolation theorem between the corresponding triples. A number of p roper t ies could be obtained for such << generalized >> methods. However, past exper ience shows that all the interpolat ion theorems in use are obtained from interpolat ion methods as d e f i n e d in [14.I]. We will therefore confine otu'selves to this kind of methods.

[14.IV] THEOREM - Let F be a uniform interpolation method with all interpolation-constants bounded by ~ < ~x~. Consider any couple IVy, W ~] ~ and put, for u ~ A~,

(14. I ) II u r I:a~ ~ sup sup II Tj,~ u I]~tj where << j )> runs through all the ] ITj,~li,g<---~

couples in ~ : Then ]1 u }l-~ < [] u :]'~ <~ VII u ]!~i and the spaces A~ -- F[ V~, W~] provided

with the norm I] u []'~ make the method F normalized.

PROOF, - It is clear that (14.1) defines a norm on Ai. If we take (<j>> = <>, Ti,~ -- identity, we get [I u Its,. < H u It~. On the other hand, [] Tj,~u ]!.~j < Y~,~HulI~i <Y[ luN,4 i which gives I lu t l~ < 7 1 ] u ] [ ~ i . Hence ~he new norm is equ'ivalent to the original one. If we consider now any two couples ~< 1 >> and << 2 ~> belonging to gf and take u C A1 and T,.~ E ~2,~ with 1T~.,~ I~,~ < 1, then

sup sup II Ti:T::ul["/<~sup sup II Ti:ull +-- II U [[~:~1" 1 [ITI,zlIj,2 < 1 i I Tj,1 ]1:1 < 1

[14.V] REMARK -- If F is a normalized interpolat ion method we have, in part icular , that for every [l~, W~J E J~, A~--/J'[V~, W~ is an interpolat ion space between V~ and W~ with the corresponding interpolat ion-constant yi,~ ~ 1 (it must then be a c t u a l l y - - 1 ) . In general, these interpolat ion spaces will riot be n:ormalized, but if we wish, we may Teplace the norm in Ai by the transformation [A~(V~q-W~)]U)(V~N W~:). This t ransformation does not change the SPaCe but' only the norm, making A~ into a normalized interpolation space. It is immediately proved that with this change of norm F remains a normalized interpolation method.

N. kao~sz~z~ - E. G ~ a D o : Interpolation spaces, etc. 115

[14.VI] BI~I~ W~EO~E~ - Consider a no, malized interpolation method F

defined on a class of couples ~ and let ~ be a larger class of couples.

There are normalized interpolation methods F defined on ~ which are

extensions of F. Among these there are two extreme, F ' and F" such that for

any normalized extension [f of F and any couple [17o, Wo]E ~ we have

~'[ Vo, Wo] ~ ~[ Vo, Wo] ~ ~"[ Vo, Wo]. The two extreme extensions F ' and if" are given by the following formulas

(14.2') F'[Vo, Wo] ~ @ B*o(A~), $

(14.2") F"[ vo, Wo] ~ (~ C~(A,), i

where (~ i ~) runs through all the couples of the class c~ and A i - F[F, , W,].

PRoo:~. - We show first that for [Vo, Wo] E ~ and any two couples (<i~ and <<j~ in ~ we have.

(14.3) B~(A~) ~ C~ o (Ai).

In fact, take u E To,i(A~) for any To,~E ~o,~ with I To,~Io,~ 1. Consider any ai E Ai with To, ~ -- u ; then, II u ]lTo, i(~) -- inf ]1 a~ 11.4 t. On the other hand, eonsi-

a i

der any Tj, o E ~j ,o with lT],oli, o<_~ 1. Then, T¢,ou----T~,oTo, ia~ and I T/,oTo,~IA3.,~; I Ti, oTo,~li,~<~ 1 (since the method F is normalized). We have I] Ti, oU lIAr. ~--< ]] a~ ]IA; and thus II u ]]C0i(A~,) - - sup [[ T¢,oU !l~.~. ~ II a~ II.~ and finally, [Iu I[coJ(.Aj)

I~,o Ii,0 =<~ iI U Ilro,~(-~.)" Therefore Ci o (Ai) ~ To,~ (A~) and (14.3) follows.

F rom (14.3) we deduce

P'[Vo, Wo]~ F"[yo, Wo].

If ~ is a n y normalized extension of ix/', then by Theorem [13,X[, c)] we

have B~o(A,)c/~[Vo, W o ] ~ C~o(A~) for eve ry [V~, W ~ E c~. We obtain t h u s ,

by (14.2') and (14.2"), F ' [Vo, Wo]~F[Vo , W o ] ~ F " L V o , Wo]. It remains to show that F ' and F " are normalized extensions on ~ of the method F. We notice first that for [Vo, ~ ] e ~ we have. by Theorem [13.XI, c)], B~ o (A~) C Ao C C~o (A~) for every [V~, VV~] E c~. Therefore, since for an interpolat ion space Ao wi th cons tant To,o~--~l we have B°o(Ao)~_Ao~C°o(Ao), it follows by (14.2') and (14.2") that F'[Vo, Wo]~Ao ~F '~[V0, Wo].

116 N . ARONSZAJN - E . GAGLIARDO: Interpolation spaces, e t c .

Consider now any two couples [Vo, Wo] and [Vi, W~] in e~. Let T~,o E ~ , o with I T~,o i~,o -~ 1.

Consider first the method _~'. If u E F ' [ V o , Wo] this means that there exist (by (14.2') and by (13.5))admissible sums u - - E T("). a i with T('*! o~ <_1

0~$~, 01~ ~ ~ ~ -

and a~ E A~ = F I V e , Wi,~] such that ~ [I a~ ! i ~ < ~ " Further , I[ u [l~,~vo,wo] =

in f ' . ][ ai,~lla¢,~ for all such admissible sums. Consider now Tx,ou. For each

above defined admissible sum we have Tx,oU : ~ Tl,oT(o;~ai ~. Hence T~,ou is

represented by the corresponding admissible sum as element of F '[V~, W1]. It is now obvious that l] Tx,oU I!F'[r~, w~] ~ I] u [l~'[go, w0], which proves that

/~' is a normalized interpolation method.

Consider now the case of /~". Suppose u~F"[Vo, Wo]. By (14.2"), (13.6) and Theorem [13.VII, b')] we obtain that for every [Vi, Wi] E g , and every T~,oE'~,~ with tT~,ol~,o<~l,T~,ouEA~and l iu i{~ , , [Vo,%l :su p sup I]T~,oI!~..

i t Ti, o li.o_< ~ Consider then T~,oU for some T~,oE ~ ,o with [T~,o[i,o~___ ~ 1. For any T~,~ E ~ , ~ w i t h i T i , ~ ] ~ , ~ l , T#,~T~,ouEA,(since T~,~T~,oE~i,o)andsup sup IIT~,~T~,ouHAi~

i I Ti, 1 ll, i ~i

sup sup [1 Ti, ou [l,~ -- ]I u tim, Iv., %]. It follows that F " is also a normalized i ]iTi, o ll, o~_~

interpolation method.

[14.¥II] COROLLARY - Let [17o, Wo] be a compatible Banach couple and Ao an interpolation space between Vo and Wo with interpolation constant Yo,o -- 1. For any class of couples ~ containing IV o, Wo], there exist normalized interpolation methods F such .that 2 ' [ V o , W o ] - Ao. Among all these methods there are two e~ctreme, F ' and F" so that for any other such method F, and any couple [V~, W~]Ee~ we have F'[V~, W i ] ~ [ V ~ , W~]~tP"[Vi, W~]. The methods F ' and F" are given by

(i4.4) F'[V~, W~] -- B°(Ao) F " ~ v W,] -- CO(Ao)

The proof follows immediately from the preceding theorem when e~ is replaced by ~ , and J~ is replaced by the class composed of the single couple [Vo, Wo].

REMilCK (added in the proofs) - The following statement can be proved by using the axiom of choice and the we l l -o rde r ing of all BA~ACH couples: if F is a general interpolation method, then for each Banach couple IV, WJ we can choose for F[V, W] a norm (equivalent to the given one) so that the method becomes normalized.

N. ARONSZAJN - E. (~AGLIARDO: I n t e r p o l a t i o n spaces, etc. 117

BIBLIOGRAPHY

[ 1 ] N- ARONSZASN, Associated spaces interpolation theorems and the regularity of solutio~ls of dif[erential problems. Proc. of Symposia in Pure Matematies, Vol IV, 1961. AMS.

[ 2 ] n . P. CALDERON, Intermediate spaces and interpolation. Reports of the Conference on Funct ional Analysis , Warsaw, 1960.

[ 3 ] E G'AGLIARDO, Interpolation d'espaces de Banach et applications {:Notes I, I I , I I I ) , CR, Par is 248 {1959) pp. 1912-14, 3388-3390, 3517-3518.

[ 4 ] E. G-AGLIARDO~ A unified structure in various families of functional spaces. Compactness and closure theorems, Proc. in ternat ional Symp. on Linear Spaees~ Jerusalem (1960} pp. 237-241

[ 5 [ E. GAGLTARDO, A common structure in various families of functional spaces, Quasilinear interpolation spaces, {Part I, I I ) Ricerche di Mat. 10, {19~1) pp. 244-281 and 12 (19681 pp. 87-107.

[ 6 ] J. L. LIONS, Espaces intermddiaires entre espaces hilbertiens et applications, Bull, Math SSMPRPR. Bucharest 2 (50) 09581.

[ 7 ] J. L. LIoNs, Thdor~mes de trace et d'interpolation (I)-(V), (I), ( I I ) : Annal i Scuola Norm. Sup., Pisa, t. X I I I (1959), pp. 389-4037 t. X I V {1960)7 pp. 317-3317 "(III): J-ourn. de Liouvill% (IV}: Math. Annalen. (¥1: Summa Brasil. Mat.

[8 ] 3'. L. LIoNs, Une construction d'espaces d'interpolations. C, R. Ac. Sci., Paris , 251 {1961)7 pp. 1853-1855.

[ 9 ] J L. L~ONS and 3'. PEETRE, Sur une classe des espaces d'interpolation. To appear in Inst. des Hautes J~tudes Scientifiques, Publications IV[ath~matiques. Paris.

[10] S. G. KREIN~ The notion of normal stairs of spaces Dokl. Akad. ~Tauk, t. 132 {1960) pp. 510-513.

[1]-] 3'. ~/[~RC[NKIEWICZ, S~r l'i~terpolc~tior~ d'opdrate~rs. C. R. Ac. Sci. Paris, (1939) 208 pp. 127"2-1273.

[]2] 3'. PEETRE~ Nouvelles propridtds d'espaces d'interpolation. C. R. Ac. Sci. Paris, 256 09631 pp. 1424-1426.

[13] ~¢[ R~.sz, Stcr les maxima des formes bilindaires et sur les fonctionnelles lindaires. Acta Math. 49 (1926).

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Interpolation spaces and interpolation methods