A generalization of Banach’s lemma and its applications to perturbations of bounded linear operators

Let X be a Banach space and let P : X → X be a bounded linear operator. Using an algebraic inequality on the spectrum of P , we give a new sufficient condition that guarantees the existence of ( I – P ) −1 as a bounded linear operator on X , and a bound on its spectral radius is also obtained. This...

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Bibliographic Details
Published inApplied Mathematics-A Journal of Chinese Universities Vol. 39; no. 2; pp. 363 - 369
Main Authors Wang, Zi, Ding, Jiu, Wang, Yu-wen
Format Journal Article
LanguageEnglish
Published Singapore Springer Nature Singapore 01.06.2024
Springer Nature B.V
Subjects
Applications of Mathematics
Banach spaces
Linear operators
Mathematics
Mathematics and Statistics
Operators (mathematics)
Perturbation methods
Banach lemma
perturbation analysis
65F20
47A55
generalized inverse
spectral radius
15A09
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Summary:Let X be a Banach space and let P : X → X be a bounded linear operator. Using an algebraic inequality on the spectrum of P , we give a new sufficient condition that guarantees the existence of ( I – P ) −1 as a bounded linear operator on X , and a bound on its spectral radius is also obtained. This generalizes the classic Banach lemma. We apply the result to the perturbation analysis of general bounded linear operators on X with commutative perturbations.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:1005-1031
1993-0445
DOI:10.1007/s11766-024-4872-3