Perturbations of Operator Functions in a Hilbert Space

Abstract

Let $A$ and $\tilde A$ be linear bounded operators in a separable Hilbert space, and $f$ be a function analytic on the closed convex hull of the spectra of $A$ and $\tilde A$. Let $SN_2$ and $SN_1$ be the ideals of Hilbert-Schmidt and nuclear operators, respectively. In the paper, a sharp estimate for the norm of $f(A)-f(\tilde A)$ is established, provided $A$ and $\tilde A$ have the so called Hilbert-Schmidt property. In particular, $A$ has the Hilbert-Schmidt property, if one of the following conditions holds: $A-A^*\in SN_2$, or $AA^*-I\in SN_1$. Here $A^*$ is adjoint to $A$, and $I$ is the unit operator. Our results are new even in the finite dimensional case.