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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| Published in | Applied Mathematics-A Journal of Chinese Universities Vol. 39; no. 2; pp. 363 - 369 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Singapore
Springer Nature Singapore
01.06.2024
Springer Nature B.V |
| Subjects | |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. 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. |
| Author | Wang, Zi Ding, Jiu Wang, Yu-wen |
| Author_xml | – sequence: 1 givenname: Zi surname: Wang fullname: Wang, Zi organization: School of Mathematics Sciences, Harbin Normal University – sequence: 2 givenname: Jiu surname: Ding fullname: Ding, Jiu organization: School of Mathematics and Natural Sciences, The University of Southern Mississippi – sequence: 3 givenname: Yu-wen surname: Wang fullname: Wang, Yu-wen email: wangyuwen1950@aliyun.com organization: School of Mathematics Sciences, Harbin Normal University, Academic Committee, Harbin Institute of Petroleum |
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| Cites_doi | 10.1007/BF02648883 10.1006/jmaa.1997.5482 10.1016/B978-0-12-514250-2.50013-5 10.1080/01630563.2020.1813164 10.1016/S0024-3795(02)00493-7 |
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| Keywords | Banach lemma perturbation analysis 65F20 47A55 generalized inverse spectral radius 15A09 |
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| References | M Z Nashed. Perturbations and approximations for generalized inverses and linear operator equations, Generalized Inverses and Applications, Academic Press, 1976, 325–396. CampbellS LMeyerC DGeneralized Inverses of Linear Transformations1979LondonPitman T Kato. Perturbation Theory for Linear Operators, Springer-Verlag, 1984. DingJNew perturbation results on pseudo-inverses of linear operators in Banach spacesLinear Algb Appl2003362229235195546710.1016/S0024-3795(02)00493-7 DingJHuangQOn the stable perturbation and Nashed’s condition for generalized inversesNumer Func Anal Optim2020411417611768420044110.1080/01630563.2020.1813164 CazassaP GChristensenOPerturbation of operators and applications to frame theoryJ Fourier Anal Appl199735543557149193310.1007/BF02648883 ChenGXueYPerturbation analysis for the operator equation Tx = b in Banach spacesJ Math Anal Appl1997212107125146018810.1006/jmaa.1997.5482 J Ding (4872_CR5) 2020; 41 S L Campbell (4872_CR1) 1979 J Ding (4872_CR4) 2003; 362 4872_CR6 G Chen (4872_CR3) 1997; 212 P G Cazassa (4872_CR2) 1997; 3 4872_CR7 |
| References_xml | – reference: T Kato. Perturbation Theory for Linear Operators, Springer-Verlag, 1984. – reference: DingJNew perturbation results on pseudo-inverses of linear operators in Banach spacesLinear Algb Appl2003362229235195546710.1016/S0024-3795(02)00493-7 – reference: ChenGXueYPerturbation analysis for the operator equation Tx = b in Banach spacesJ Math Anal Appl1997212107125146018810.1006/jmaa.1997.5482 – reference: CampbellS LMeyerC DGeneralized Inverses of Linear Transformations1979LondonPitman – reference: CazassaP GChristensenOPerturbation of operators and applications to frame theoryJ Fourier Anal Appl199735543557149193310.1007/BF02648883 – reference: DingJHuangQOn the stable perturbation and Nashed’s condition for generalized inversesNumer Func Anal Optim2020411417611768420044110.1080/01630563.2020.1813164 – reference: M Z Nashed. Perturbations and approximations for generalized inverses and linear operator equations, Generalized Inverses and Applications, Academic Press, 1976, 325–396. – volume-title: Generalized Inverses of Linear Transformations year: 1979 ident: 4872_CR1 – volume: 3 start-page: 543 issue: 5 year: 1997 ident: 4872_CR2 publication-title: J Fourier Anal Appl doi: 10.1007/BF02648883 – volume: 212 start-page: 107 year: 1997 ident: 4872_CR3 publication-title: J Math Anal Appl doi: 10.1006/jmaa.1997.5482 – ident: 4872_CR6 – ident: 4872_CR7 doi: 10.1016/B978-0-12-514250-2.50013-5 – volume: 41 start-page: 1761 issue: 14 year: 2020 ident: 4872_CR5 publication-title: Numer Func Anal Optim doi: 10.1080/01630563.2020.1813164 – volume: 362 start-page: 229 year: 2003 ident: 4872_CR4 publication-title: Linear Algb Appl doi: 10.1016/S0024-3795(02)00493-7 |
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| Snippet | 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... 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... |
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| StartPage | 363 |
| SubjectTerms | Applications of Mathematics Banach spaces Linear operators Mathematics Mathematics and Statistics Operators (mathematics) Perturbation methods |
| Title | A generalization of Banach’s lemma and its applications to perturbations of bounded linear operators |
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