Bounded operators and isomorphisms of Cartesian products of Fréchet spaces
The main goal of the present article is to continue research of Zakharyuta from the early 70's and of \textit{P. B. Diakov}, \textit{S. Önal}, \textit{T. Terzioglu}, and \textit{M. Yurdakul} [Arch. Math. 70, No. 1, 57-65 (1998; Zbl 0919.46033)] on the isomorphic classification of Cartesian products of Fréchet spaces. As an example, the following result solves a question of the paper just mentioned. Let \(X_1,X_2,Y_1,Y_2\) be non-Montel Köthe echelon spaces of order \(p\in[1,\infty[\) such that \(X_1\times X_2\) is isomorphic to \(Y_1\times Y_2\). If \(X_1\) and \(Y_1\) satisfy the condition \((\text{d}_2)\) of Dragilev and \(X_2\) and \(Y_2\) satisfy the condition \((\text{d}_1)\) (or equivalently the condition (DN) of D. Vogt), then \(X_i\) is isomorphic to \(Y_i\) for \(i=1,2\). The isomorphic classification of Cartesian products of an infinite-type and a finite-type power series space is presented. The existence of pairs of Fréchet spaces \((X,Y)\) such that every continuous linear map from \(X\) to \(Y\) maps a neighbourhood of the origin in \(X\) into a bounded subset of \(Y\) (studied by Vogt and others) and the stability of automorphisms under bounded perturbations are essential tools to obtain the present results. Three open problems are stated.
isomorphic classification of Cartesian products of an infinite-type and a finite-type power series space
stability of automorphisms under bounded perturbations
scientific article; zbMATH DE number 1548283