##
**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 |

### Keywords:

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

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