Functions of perturbed dissipative operators

TL;DR

This paper extends the theory of operator Lipschitz and H"older functions to maximal dissipative operators, providing sharp conditions, derivatives, and estimates for perturbations, including Schatten class and commutator cases.

Contribution

It introduces new sharp conditions and formulas for operator functions and derivatives specifically for maximal dissipative operators, expanding previous results from self-adjoint and unitary cases.

Findings

Sharp conditions for operator Lipschitz functions in the upper half-plane.

Operator H"older continuity of order a0a0a0a0 for analytic functions.

Explicit formulas for operator derivatives and estimates for Schatten class perturbations.

Abstract

We generalize our results of \cite{AP2} and \cite{AP3} to the case of maximal dissipative operators. We obtain sharp conditions on a function analytic in the upper half-plane to be operator Lipschitz. We also show that a H\"older function of order $\a$, $0<\a<1$, that is analytic in the upper half-plane must be operator H\"older of order $\a$. Then we generalize these results to higher order operator differences. We obtain sharp conditions for the existence of operator derivatives and express operator derivatives in terms of multiple operator integrals with respect to semi-spectral measures. Finally, we obtain sharp estimates in the case of perturbations of Schatten-von Neumann class $\bS_p$ and obtain analogs of all the results for commutators and quasicommutators. Note that the proofs in the case of dissipative operators are considerably more complicated than the proofs of the corresponding results for self-adjoint operators, unitary operators, and contractions that were obtained earlier in \cite{AP2}, \cite{AP3}, and \cite{Pe6}.