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

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.