Functions of perturbed commuting dissipative operators

TL;DR

This paper develops sharp Lipschitz estimates for differences of functions of pairs of commuting maximal dissipative operators, using advanced double operator integral techniques to handle the complex nature of such operators.

Contribution

It introduces new techniques for obtaining Lipschitz estimates for functions of commuting dissipative operators, extending previous methods to a more complex operator class.

Findings

Established sharp Lipschitz type estimates for operator differences.

Derived H"older type estimates for these differences.

Provided Schatten--von Neumann norm estimates.

Abstract

The main objective of the paper is to obtain sharp Lipschitz type estimates for the norm of operator differences $f(L_1,M_1)-f(L_2,M_2)$ for pairs $(L_1,M_1)$ and $(L_2,M_2)$ of commuting maximal dissipative operators. To obtain such estimates, we use double operator integrals with respect to semi-spectral measures associated with the pairs $(L_1,M_1)$ and $(L_2,M_2)$. Note that the situation is considerably more complicated than in the case of functions of two commuting contractions and to overcome difficulties we had to elaborate new techniques. We deduce from the main result H\"older type estimates for operator differences as well as their estimates in Schatten--von Neumann norms.