# Kato smoothness and functions of perturbed self-adjoint operators

## TL;DR

This paper develops new estimates for the difference of functions of perturbed self-adjoint operators, broadening the class of functions and norms for which these estimates are valid, using scattering theory and Kato smoothness.

## Contribution

It introduces a novel framework for Schatten class valued smoothness and double operator integrals, extending previous results to a wider class of functions including unbounded ones.

## Key findings

- Established new operator norm estimates for $f(H_1)-f(H_0)$.
- Extended the class of functions $f$ for which estimates hold, including some unbounded functions.
- Developed a new notion of Schatten class valued smoothness and a framework for double operator integrals.

## Abstract

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.