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Winter 2019 Norm estimates for resolvents of linear operators in a Banach space and spectral variations
Michael Gil'
Adv. Oper. Theory 4(1): 113-139 (Winter 2019). DOI: 10.15352/aot.1801-1293


Let $P_t$ $(a\le t\le b)$ be a function whose values are projections in a Banach space. The paper is devoted to bounded linear operators $A$ admitting the representation $$A=\int_a^b \phi(t)dP_{t}+V,$$ where $\phi(t)$ is a scalar function and $V$ is a compact quasi-nilpotent operator such that $P_tVP_t=VP_t$ $(a\le t\le b)$. We obtain norm estimates for the resolvent of $A$ and a bound for the spectral variation of $A$. In addition, the representation for the resolvents of the considered operators is established via multiplicative operator integrals. That representation can be considered as a generalization of the representation for the resolvent of a normal operator in a Hilbert space. It is also shown that the considered operators are Kreiss-bounded. Applications to integral operators in $L^p$ are also discussed. In particular, bounds for the upper and lower spectral radius of integral operators are derived.


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Michael Gil'. "Norm estimates for resolvents of linear operators in a Banach space and spectral variations." Adv. Oper. Theory 4 (1) 113 - 139, Winter 2019.


Received: 11 January 2018; Accepted: 13 April 2018; Published: Winter 2019
First available in Project Euclid: 27 April 2018

zbMATH: 06946446
MathSciNet: MR3867337
Digital Object Identifier: 10.15352/aot.1801-1293

Primary: 47A10
Secondary: 47A11 , 47A30 , 47A55 , 47G10

Keywords: Banach space , ‎integral operator , invariant chain of projections , resolvent , spectral variation

Rights: Copyright © 2019 Tusi Mathematical Research Group


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Vol.4 • No. 1 • Winter 2019
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