{orders}ms/98424/cobos.3d -17.11.00 - 11:40 REAL INTERPOLATION OF COMPACT OPERATORS BETWEEN QUASI-BANACH SPACES

Let …A0;A1† and …B0;B1† be couples of quasi-Banach spaces and let T be a linear operator. We prove that if T : A0 ! B0 is compact and T : A1 ! B1 is bounded, then T : …A0;A1† ;q ! …B0;B1† ;q is also compact. Some results on the structure of minimal and maximal interpolation methods are also established. 0. Introduction. Assume that T is a linear operator such that T : Lp0 ! Lq0 compactly and T : Lp1 ! Lq1 boundedly. Here 1 po; p1; q1 1; 1 q0 <1. Let 0 < < 1 and put 1=p ˆ …1ÿ †=p0 ‡ =p1, 1=q ˆ …1ÿ †=q0 ‡ =q1. In 1960, Krasnoselskii [13] proved that under these assumptions T : Lp ! Lq is also compact. Krasnoselskii's theorem was motivated by certain compactness results for integral operators established by Kantorovich (see [14], p. 118) and it led to the study of interpolation properties of compact operators between abstract Banach spaces. These investigations has been done during two very di¡erent periods. A ¢rst one during the 60's, simultaneous to the foundation of abstract interpolation theory, and a second period developed during the last decade, where modern interpolation techniques have been successfully used to derive new compactness results. Contributions on this subject are due to many authors. We refer to [8] and [7] for a quite complete list of references and for historical remarks on the development during the two periods. Some more recent results can be found in the papers by Cobos [4], Cwikel and Kalton [9] and Mastylo [16]. Returning to Krasnoselskii's theorem, Zabreiko and Pustylnik [20] (see also [14], Thm. 3.11) proved in 1965 that the result is still true for the full MATH. SCAND. 82 (1998), 138^160 Supported in part by Ministerio de Asuntos Exteriores (Programa de Ayudas de Cooperacioè n Cientí |̈¢ca a Investigadores en Instituciones de Suecia durante 1995). Received January 15, 1996. {orders}ms/98424/cobos.3d -17.11.00 11:41 rank of parameters, that is for 0 < q0; q1 1. Note that the last couple is formed by quasi-Banach spaces when 0 < q0; q1 < 1. It arises then the question if a similar result holds true for abstract quasi-Banach couples, not only for Lq-couples. Accordingly, we establish in this paper such a result. We follow the approach developed in [8] and [7] (see also [5], [6]) based on the description of the real interpolation method as a maximal and a minimal interpolation method in the sense of Aronszajn-Gagliardo [1]. The main obstacle is then to ¢nd a useful extension of maximal methods for quasiBanach couples. The natural de¢nitions based on scalar sequence spaces, give nothing but the sum space when applying to a couple as …Lq0 ;Lq1† with 0 < q0; q1 < 1, because the spaces Lqj have trivial dual. We overcome this di¤culty by giving a maximal description of the real interpolation space in terms of vector valued sequence spaces involving the couple into consideration. We have then a description for each quasi-Banach couple, rather than a description for the real interpolation method. However, this will be su¤cient for our purposes. Such a description is given in Section 2, where we also derive the corresponding minimal characterization. Working in the category of Banach couples, any maximal or minimal method de¢ned by sequence spaces satisfying certain mild conditions, can be equivalently de¢ned by vector valued sequence spaces as we show in Section 1. This result, that we think has independent interest, is based on the HahnBanach theorem and applies not only to the real method, but also to the `` '' method (see [18], [10]) and Ovchinnikov's u-method [17]. In the ¢nal Section 3 we prove the announced interpolation theorem for compact operators in the quasi-Banach case. 1. Maximal and minimal methods in Banach spaces. Let us start by recalling the construction of the Aronszajn-Gagliardo maximal functor H‰…B0;B1†;BŠ… ; †; for Banach spaces (see [1]; see also [12], [3]). If A ˆ …A0;A1† and B ˆ …B0;B1† are Banach couples, we write T 2l… A; B† to mean that T is a linear operator from A0 ‡ A1 into B0 ‡ B1 such that the restriction of T to each Aj de¢nes a bounded operator from Aj into Bj …j ˆ 0; 1†. We write kTk A; B ˆ max jˆ0;1fkTkAj ;Bjg We say that a Banach space A is an intermediate space with respect to the couple A ˆ …A0;A1† if the following continuous embeddings hold … A† ˆ A0 \ A1 ,!A ,!A0 ‡ A1 ˆ … A†: If, in addition to the above property, whenever T 2l… A; A† if follows that real interpolation of compact operators... 139 {orders}ms/98424/cobos.3d -17.11.00 11:41 T : A! A is bounded, then A is called an interpolation space with respect to the couple A. Let B ˆ …B0;B1† be a ¢xed Banach couple and let B be a ¢xed intermediate space with respect to B. Given any Banach couple A ˆ …A0;A1†, the space H…A0;A1† ˆ H‰…B0;B1†;BŠ…A0;A1† is de¢ned as the collection of all those elements a 2 … A† such that Ta 2 B for all T 2l… A; B†. The norm in H‰…B0;B1†;BŠ…A0;A1† is given by kakH ˆ supfkTakB : kTk A; B 1g: In order to give some important examples of maximal methods denote by `q …1 q 1† and c0 the usual spaces of doubly in¢nite scalar sequences and, given any positive sequence …!m†, de¢ne `q…!m† by `q…!m† ˆ f m† : …!m m† 2 `qg: We give a similar meaning to c0…!m†. Example 1.1. Let 1 q 1 and 0 < < 1. If B ˆ …`1; `1…2ÿm†) and B ˆ `q…2ÿ m†, then the interpolation method generated by this choice is H‰…`1; `1…2ÿm††; `q…2ÿ †Š…A0;A1† ˆ …A0;A1† ;q the real interpolation method realized as a K-space (see [12]). Namely …A0;A1† ;q ˆ a 2 … A† : kak ;q ˆ X1 mˆÿ1 …2ÿ mK…2m; a†† !1=q <1 <: =; …1† where K…2m; a† ˆ inffka0kA0 ‡ 2ka1kA1 : a ˆ a0 ‡ a1; aj 2 Ajg: Example 1.2. If B ˆ …`1; `1…2ÿm†† and B ˆ `1…2ÿ m†, then we obtain H‰…`1; `1…2ÿm††; `1…2ÿ †Š…A0;A1† ˆ H1…A0;A1† Ovchinnikov's u-method (see [12] or [17]): Note that the sequence spaces X ˆ `q…!m† which arise in Examples 1.1 and 1.2 satisfy the following three conditions: a) Sequences having only a ¢nite number of coordinates di¡erent from zero are contained in X . b) k… m†kX ˆ sup n 0 k…. . . ; 0; 0; ÿn; ÿn‡1; . . . ; nÿ1; n; 0; 0; . . .†kX . c) If j mj j mj for each m 2 Z and … m† 2 X , then … m† 2 X and k… m†kX k… m†kX . 140 fernando cobos and lars-erik persson {orders}ms/98424/cobos.3d -17.11.00 11:42 All sequence spaces that we consider in the rest of this section are supposed to satisfy conditions a), b) and c). Next we shall show that the behaviour of a maximal method on a couple of sequence spaces can be shifted to couples of vector valued sequence spaces. Let s ˆ …s0; s1† be any Banach couple of scalar sequence spaces over Z and let s be any intermediate sequence space with respect to s. Given any sequence of Banach spaces …Fm† with Fm 6ˆ f0g for each m 2 Z, we put s…Fm† ˆ f…am† : am 2 Fm and k…am†ks…Fm† ˆ k…kamkFm†ks <1g and we de¢ne s0…Fm† and s1…Fm† similarly. Assumptions a), b) and c) on scalar sequence spaces guaranteee that the vector valued sequence spaces are Banach spaces. It is also clear that s…Fm† ˆ …s0…Fm†; s1…Fm†† is a Banach couple. We are now ready to establish the announced result. Theorem 1.3. Let H‰…B0;B1†;BŠ…:; :† be any maximal method and let …s0; s1†, s and …Fm† be as above. If H…s0; s1† = s then H…s0…Fm†; s1…Fm†† ˆ s…Fm†: Proof. Take any …am† 2 H…s0…Fm†; s1…Fm†† and let T 2l……s0; s1†, …B0;B1†† with kTk s; B 1. Using the Hahn-Banach theorem, for each m 2 Z we can ¢nd fm 2 F m such that kfmkF m ˆ 1 and fm…am† ˆ kamkFm : Next consider the operator R 2l……s0…Fm†; s1…Fm††; …B0;B1†† de¢ned by R…xm† ˆ T…fm…xm††: It is easy to see that kRk s…Fm†; B 1: Hence, according to the de¢nition of the maximal method, we have that R…am† ˆ T…kamkFm† 2 B and kT…kamkFm†kB ˆ kR…am†kB k…am†kH…s0…Fm†;s1…Fm††: Since this holds for any T 2l……s0; s1†; …B0;B1†† with kTk s; B 1, we conclude that …kamkFm† 2 H…s0; s1† ˆ s with real interpolation of compact operators... 141 {orders}ms/98424/cobos.3d -17.11.00 11:42 k…kamkFm†ks ˆ k…am†ks…Fm† k…am†kH…s0…Fm†;s1…Fm††: This proves that H…s0…Fm†; s1…Fm†† ,! s…Fm†: Conversely, let …am† 2 s…Fm†. Given any R 2l……s0…Fm†; s1…Fm††; …B0;B1†† with kRk s…Fm†; B 1 put T… m† ˆ R m kamkFm am !