## Norm estimates for resolvents of linear operators in a Banach space and spectral variations.(English)Zbl 06946446

Summary: Let $$P_t$$ ($$a\leq t\leq b$$) 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 $$P_tVP_t=VP_t$$ ($$a\leq t\leq b$$). 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.

### MSC:

 47A10 Spectrum, resolvent 47A55 Perturbation theory of linear operators 47G10 Integral operators 47A11 Local spectral properties of linear operators 47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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