Fredholm theory for pairs of closed subspaces of a Banach space
type: Item
instance of: scholarly article
title: Fredholm theory for pairs of closed subspaces of a Banach space
zbMATH Open document ID: 1071.47016
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-04-21T00:00:00Z
review text: \textit{T. Kato} in chapter IV of [``Perturbation theory for linear operators'' (Springer-Verlag Berlin) (1995; Zbl 0836.47009)] developed a perturbation theory for upper and lower semi-Fredholm pairs of subspaces of a Banach space. The present author introduces the notions of compact, strictly singular and strictly cosingular pairs of subspaces of a Banach space. He proves that a pair of subspaces \((L,M)\) is strictly singular if and only if \((L,N)\) is upper semi-Fredholm for every closed subspace \(N\) such that \((M,N)\) is upper semi-Fredhom. Moreover, he demonstrates that the upper semi-Fredholmness of \((L,M)\) and strict singularity of \((M,N)\) do not imply the upper semi-Fredholmness of \((L,N)\). Counterparts of these results for lower semi-Fredholmness and strict cosingularity are also obtained. Finally, the author proves that the classes of compact, strictly singular, and strictly cosingular pairs are transitive.
reviewed by: Oleksiy Karlovych
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A55
zbMATH DE Number: 2159772
zbMATH Keywords: compact pair of subspaces
zbMATH Keywords: strictly singular pair of subspaces
zbMATH Keywords: strictly cosingular pair of subspaces
zbMATH Keywords: semi-Fredholm operator
zbMATH Keywords: perturbation theory
zbMATH Keywords: Fredholm theory
zbMATH Keywords: strictly singular operators
zbMATH Keywords: perturbation classes problem
author: M. González
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.10.014
OpenAlex ID: W2042235682
cites work: Q3838080
cites work: Q5526484
cites work: The perturbation classes problem in Fredholm theory.
cites work: On incomparability of Banach spaces
cites work: Q5526485
cites work: The only continuous Volterra right inverses in $C_c[0,1]$ of the operator $d/dt$ are $ʃ^t_a$
DOI: 10.1016/J.JMAA.2004.10.014
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description: scientific article; zbMATH DE number 2159772