A problem on the structure of Fréchet spaces
\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.
scientific article; zbMATH DE number 5925100
scientific article