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

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.