On the existence of universal -nuclear Fréchet spaces.

Abstract

IntroductionSince its introduction by Grothendieck [4], the theory of nuclear spaces has acquired increasing importance in functional analysis. More recently, several generalizations and variations of this theory have been studied by many authors, culminating in the unifying theory of A-nuclearity presented in [3] (where numerous references to previous work can be found). In the theory of nuclearity a crucial role is played by the sequence space l 1 and, in essence, A-nuclearity is obtained through replacing l 1 by a sequence space A satisfying some suitable conditions. It is then natural to ask which properties of nuclear spaces go over to the more general setting of A-nuclear spaces. Given the enormous ränge of problems covered by such a question, in this paper we choose one such problem and give a complete solution to it, äs well äs present the form taken by the solution in the case of concrete sequence spaces encountered in the applications.We statt with the following observation. By the celebrated theorem of Kömura and Kömura [6], the Frechet space s of rapidly decreasing sequences is a universal generator for the variety of nuclear spaces and hence S N is a universal, nuclear Frechet space, in the sense that every nuclear Frechet space is isomorphic to a closed subspace of s". On the other band, in [10], Theorem 10 we proved that there is no universal ^-nuclear Frechet space and, so far, nobody has exhibited a Frechet space which is a universal generator for the variety of s-nuclear spaces (the reason will, of course, be apparent later).These considerations led us to consider the problem of characterising those sequence spaces (within a broad admissible class) admitting a universal, A-nuclear Frochet space. In this paper we solve this problem under minimal assumptions on and give some interesting applications. The results have been announced in [15].The paper is divided into two parts. Part I develops the general theory; we prefer to introduce the various hypothesis on the sequence space precisely at the point where they are needed. Starting with a preliminary section on sequence spaces, we proceed in § 2 with the definition and elementary properties of A-nuclear spaces. Our definition of A-nuclear spaces is slightly stronger than that adopted in [3] in the sense that our A-nuclear spaces are contained within the class of A-nuclear spaces äs defined in [3]. Furthermore, the two definitions coincide for a large class of sequence spaces A, while our definition Brought to you by | University of California Authenticated Download Date | 6/8/15 9:39 AM with (/") and (e n ) orthonormal Systems in E v and Å õ respectively, so that the map T vu is pseudo A-nuclear. Finally, if (H 4') is satisfied, (i) follows from (iv) by Lemma 2. Journal f r Mathematik. Band 301 2 Brought to you by | University of California Authenticated Download Date | 6/8/15 9:39 AM Ã ao Ú and form the space # = < { : Ó eJ{J < oo >. Arguing by contradiction, suppose that / 2 c # ; t «=i J then there exists c>0 such that oo / oo