Bounded operators and isomorphisms of Cartesian products of Fréchet spaces.

Abstract

where Lc(X, Y ) denotes the subspace of all compact operators. This phenomenon was studied later by many authors (see e.g. [1; 5; 11; 12; 13; 14; 15; 20; 21]); of prime importance are Vogt’s results [24] giving a generally complete description of the relations (1) for the general case of Frechet spaces (for further generalizations see also [3; 4]). Originally, the main goal in [25; 26] was the isomorphism of Cartesian products (and, consequently, the quasi-equivalence property for those spaces). The papers made use of the fact that, due to Fredholm operators theory, an isomorphism of spaces X × Y ' X1× Y1 (X,X1∈X , Y, Y1∈Y ) that satisfies (2) also implies an isomorphism of Cartesian factors “up to some finite-dimensional subspace”. In the present paper we generalize this approach onto classes X × Y of products that satisfy (1) instead of (2). Although Fredholm operators theory fails, we have established that—in the case of Kothe spaces—the stability of an automorphism under a bounded perturbation still takes place, but in a weakened form: “up to some basic Banach space”. In particular, we get a positive answer to Question 2 in [7]: Is it possible to modify somehow the method developed in [25; 26] in order to obtain isomorphic classification of the spaces E0(a) × E∞(b) in terms of sequences a, b if ai 6→ ∞ and bi 6→ ∞? Some of our results are announced without proofs in [9].