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Interpolation Spaces and Interpolation Methods

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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~). Introduction.- The beginning of what we now call interpolation methods between BANAC~ spaces was the convexity theorem of )][. RIESZ [13] in the late 1920's. This theorem gives an interpolation metlmd for couples [LP(di~), Lq(d~t)]. In the late 1930's, MAlCCINKIEWleZ [11] obtained an extension of the M. RIESz interpolation method to couples formed by weak LP-spaces. In the 1950's E. )I. S~EIN and G. WEISS [14] extended the method further by admitting couples [Lv(d~), Lq(dv)] with different measures l~ and v. Until the late 1950's, the research in interpolation methods between sp~ees remained essentially within the frame of couples of L ~ spaces. At the end of 1958, J.M. LIOZ,~s gave the first proof of the inter- polation theorem for quadratic interpolation between HILBER~' spaces t2). This work gave impetus to the research on interpolation methods for arbitrary couples of BA~c~tOK spaces. Since then several authors have introduced and developed a number of different interpolation methods for couples of general B,t~ACI-I spaces. We mention 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 many different interpolation methods it seemed timely to study the general structure of alt possible methods: to determine all of them and to analyze the properties which are common to all. This is essentially the subject of the present paper. The research was prompted by the following specific consideration. (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 applications it becomes apparent that it is not always of importance to develope a specific method, but rather to determine if a general interpolation method (~) exists which assigns to a given couple of B~Ac~ 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. VIIi. 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-Preliminaries, contains material familiar 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 usually supposed to be continuously 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 introduction of the complete lattice of normalized B-~NAC~ sub- spaces 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 interpolation methods. Chapter I[ deals with interpolation spaces. In § 5 we introduce the normalized BANAC]/[ subspaces of a given B~_NAc~ space. The main theorem is that the lattice of these subspaees is complete and we give the construction of the joint and meet for an arbitrary class of such subspaces. In § 6 we introduce the intermediate and interpolation 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 linear 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 restrictions which this hypothesis imposes on the interpolation 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. We then study the relationship between the interpolation 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*] if and only if VN W is reflexive. In § 9 we consider the properties of, and the relationships between the BANAC]t algebras ~[V, W], ~A (this is the algebra of bounded linear opera- tots on an 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 completion 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 interpolation spaces is Lemma [10. X]. By using this lemma we obtain the following- again, rather unexpected result: whereas V-J- ~, and VN W are always interpolation spaces between Vand W, V and W, with some trivial exceptions, are never interpolation space.s between V-~ W and V N IV. This result indicates t~aat it is not an ~unesseutiaI>> restriction when one considers only couples IV, W] with WC V. In § 11 we come back to interpolation spaces. We construct the minimal interpolation spaces containing a given intermediate space. In par- ticular, for any element ue V-{- W we construct tile minimal interpolation. space ~;(u) containing u: ~(u) is formed by all elements Tu for Te~[V, W] For u :~0, V N W is always contained in ~(u} but never dense in it {unless ueVN W when VA W--~;(u)). By using a normalized version ~,~ of ~;(~) we can construct all interpolation spaces between V and W (see Theorem [11. VII). In § ][2 we analzze the structure 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 statement in this section is to the effect that ~(u) -- [u] -{- VN W only in ver Z exceptional cases, namely when u~(vn W)(V+w). That this case can actually arise is shown in an example constructed in the last Remark of the section. 54 :N. ARONSZAJN - E. GAGL:[ARDO: Interpolatio~ spaces, etc. Chapter III deals with our main results concerning interpolation me- thods and interpolation theorems. In §13 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 bilateral interpolation 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 interpolation theorems between two triples and obtain, somewhat unexpectedly, that in any such optimal interpolation theorem the intermediate spaces in each triple must be, actually, interpolation spaces. In § 14 we pass to interpolation methods on a class of compatible BANACH couples. We define uniform and normalized interpolation methods. Each uniform method can be normalized. There are no known examples of methods which cannot be rendered uniform. An interpolation method assigns to each couple of B¢r~cAc~: spaces an intermediate space. This intermediate space must be an interpolation space. Corollary [14. VII] gives the result mentioned previously in this introduction, that for a given interpolation 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 conju- gate and self-conjugate interpolation methods, and also a study of existing interpolation methods with reference to the results of this paper.
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