Criterion of holomorphy with respect to a coupling constant of continuous functions of a perturbed self-adjoint operator

TL;DR

This paper establishes conditions on the spectral measure of a self-adjoint operator ensuring that the operator function remains holomorphic with respect to a coupling constant, with the strongest results for rank-one perturbations.

Contribution

It provides necessary and sufficient spectral measure conditions for holomorphy of operator functions under perturbations, especially sharp results for rank-one operators.

Findings

Spectral measure conditions for holomorphy of operator functions.

Characterization of holomorphy with respect to a coupling constant.

Optimal results achieved for rank-one perturbations.

Abstract

Sufficient and necessary conditions on the spectral measure of a self-adjoint operator $A$, acting in a Hilbert space, are obtained, under which for any continuous scalar function the operator function $\phi(A+\gamma B)$ is holomorphic with rspect to the coupling constant $\gamma$ in a neighborhood of $\gamma=0$, where $B$ is a self-adjoint operator. The sharpest results are obtained in the case where $B$ is a rank-one operator.