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

Document Type: Special issue: Trends in Operators on Banach Spaces

Author

Ben Gurion University of the Negev

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.

Keywords