A problem on the structure of Fréchet spaces
type: Item
instance of: scholarly article
title: A problem on the structure of Fréchet spaces
zbMATH Open document ID: 1262.46001
DOI: 10.5052/RACSAM.2010.26
author: José Bonet
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2011-07-18T00:00:00Z
review text: \textit{S. A. Argyros} and \textit{R. G. Haydon} [Acta Math. 206, No.~1, 1--54 (2011; Zbl 1223.46007)] have recently shown that there exists a complex Banach space \(X\) such that each operator \(T\in L(X)\) is a perturbation of a multiple of the identity by a compact operator, i.e., for each \(T\in L(X)\) there exist \(\lambda\in \mathbb{C}\) and \(K\in K(X)\) such that \(T=\lambda I+K\). The author states the following problem that arises naturally in the structure of Fréchet spaces: {Problem}. Is there a non-normable Fréchet space such that every continuous linear operator \(T\in L(E)\) has the form \(T=\lambda I+S\), where \(\lambda\in\mathbb{C}\) and \(S\) is a bounded operator, i.e., there exists a 0-neighbourhood \(U\) in \(E\) such that \(T(U)\) is bounded. The author shows that a positive solution of this problem would have consequences in several open problems about the existence of hypercyclic, chaotic and topologizable operators in Fréchet spaces (we refer to the paper for the definitions). More precisely, if there exists a non-normable Fréchet space \(E\) which is a positive solution to the problem, then the following results hold: (a) If \(E\) can be selected Montel (every closed and bounded subset of \(E\) is compact) then \(E\) does not admit a chaotic operator. (b) There exists \(T\in L(E)\) hypercyclic and \(\lambda>0\) such that \(\lambda T\) is not hypercyclic. (c) There exists \(T\in L(E)\) which is not topologizable.
reviewed by: Enrique Jordá
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A11
Mathematics Subject Classification ID: 46H35
Mathematics Subject Classification ID: 47A16
zbMATH DE Number: 5925100
zbMATH Keywords: Fréchet spaces
zbMATH Keywords: bounded operators
zbMATH Keywords: hypercyclic operators
zbMATH Keywords: chaotic operators
MaRDI profile type: Publication
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full work available at URL: racsam.2010.26
OpenAlex ID: W372365014
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description: scientific article; zbMATH DE number 5925100