The perturbation classes problem in Fredholm theory.
type: Item
instance of: scholarly article
title: The perturbation classes problem in Fredholm theory.
zbMATH Open document ID: 1036.47007
DOI: 10.1016/S0022-1236(02)00071-X
published in: Journal of Functional Analysis
publication date: 2003-05-25T00:00:00Z
review text: Let \(X,\) \(Y\) be complex Banach spaces, \(\Phi=\Phi (X)\) be the space of Fredholm operators on \(X,\) \(I_X\) be the identity operator on \(X,\) \(J_M\) be the inclusion of a closed sumspace \(M\) of \(X\) in \(X\) and \(Q_M\) be the quotient map onto \(X/M.\) An operator \(T\in {\mathcal L}(X,Y)\) is inessential if \(I_X-ST\in \Phi (X),\) for every \(S\in {\mathcal L}(Y,X);\) it is strictly singular if no restriction \(TJ_M\) of \(T\) to a closed infinite-dimensional subspace \(M\) of \(X\) is an isomorphism; and it is strictly cosingular if there is no closed infinite-codimensional subspace \(N\) of \(Y\) such that \(Q_N T\) is surjective. It is well-known that the perturbation class of \(\Phi \) is the set of inessential operators. The main purpose of the paper under review is to give a negative answer to the important perturbation class problems: the perturbation class of the upper semi-Fredholm operators contains properly the strictly singular operators, and the perturbation class of the lower semi-Fredholm operators contains properly the strictly cosingular operators.
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 1915620
zbMATH Keywords: semi-Fredholmness
zbMATH Keywords: perturbation class
zbMATH Keywords: strictly singular operator
author: M. González
reviewed by: V. M. Deundjak
MaRDI profile type: Publication
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full work available at URL: s0022-1236(02)00071-x
OpenAlex ID: W2069907583
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description: scientific article; zbMATH DE number 1915620