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

Gil’, Michael

Adv. Oper. Theory 4, No. 1, 113-139 (2019).

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

Keywords:

Banach space; resolvent; spectral variation; integral operator; invariant chain of projections

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