### Norm estimates for resolvents of linear operators in a Banach space and spectral variations

Michael Gil'

#### Abstract

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.

#### Article information

Source
Adv. Oper. Theory, Volume 4, Number 1 (2019), 113-139.

Dates
Accepted: 13 April 2018
First available in Project Euclid: 27 April 2018

https://projecteuclid.org/euclid.aot/1524816021

Digital Object Identifier
doi:10.15352/aot.1801-1293

Mathematical Reviews number (MathSciNet)
MR3867337

Zentralblatt MATH identifier
06946446

#### Citation

Gil', Michael. Norm estimates for resolvents of linear operators in a Banach space and spectral variations. Adv. Oper. Theory 4 (2019), no. 1, 113--139. doi:10.15352/aot.1801-1293. https://projecteuclid.org/euclid.aot/1524816021

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