Kato smoothness and functions of perturbed self-adjoint operators

We consider the difference $f(H_1)-f(H_0)$ for self-adjoint operators $H_0$ and $H_1$ acting in a Hilbert space. We establish a new class of estimates for the operator norm and the Schatten class norms of this difference. Our estimates utilise ideas of scattering theory and involve conditions on $H_0$ and $H_1$ in terms of the Kato smoothness. They allow for a much wider class of functions $f$ (including some unbounded ones) than previously available results do. As an important technical tool, we propose a new notion of Schatten class valued smoothness and develop a new framework for double operator integrals.