Bounded operators and isomorphisms of Cartesian products of Fréchet spaces
type: Item
instance of: scholarly article
title: Bounded operators and isomorphisms of Cartesian products of Fréchet spaces
zbMATH Open document ID: 0973.46004
author: M. Yurdakul
author: Plamen Djakov
author: A. Tosun Terz{i}oğlu
author: Vyacheslav Zakharyuta
published in: Michigan Mathematical Journal
publication date: 2001-01-01T00:00:00Z
full work available at URL: https://hdl.handle.net/11511/30189
review text: 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.
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46A63
zbMATH DE Number: 1548283
zbMATH Keywords: isomorphic classification of Cartesian
zbMATH Keywords: products of Fréchet spaces
zbMATH Keywords: non-Montel Köthe echelon spaces
zbMATH Keywords: condition (DN) of D. Vogt
zbMATH Keywords: isomorphic classification of Cartesian products of an infinite-type and a finite-type power series space
zbMATH Keywords: stability of automorphisms under bounded perturbations
reviewed by: José Bonet
MaRDI profile type: Publication
OpenAlex ID: W2166601955
DOI: 10.1307/MMJ/1030132302
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description: scientific article; zbMATH DE number 1548283