Functons of perturbed pairs of dissipative operators

TL;DR

This paper develops a functional calculus for pairs of dissipative operators and establishes Lipschitz estimates for functions of these operators within Schatten classes, extending the understanding of operator perturbations.

Contribution

It introduces a way to define functions of pairs of dissipative operators and proves Lipschitz-type bounds for their perturbations in Schatten classes.

Findings

Operators' differences in Schatten classes imply bounded differences of their functions.

Lipschitz estimates hold for functions in the Besov space $B_{ ext{infty,1}}^1$.

The results extend perturbation theory to non-commuting dissipative operator pairs.

Abstract

Let $f$ be a function in the inhomogeneous analytic Besov space $B_{\infty,1}^1$. For a pair $(L,M)$ of not necessarily commuting maximal dissipative operators, we define the function $f(L,M)$ of $L$ and $M$ as a densely defined linear operator. We prove for $p\in[1,2]$ that if $(L_1,M_1)$ and $(L_2,M_2)$ are pairs of not necessarily commuting maximal dissipative operators such that both differences $L_1-L_2$ and $M_1-M_2$ belong to the Schatten--von Neumann class $\boldsymbol{S}_p$ than for an arbitrary function $f$ in the inhomogeneous analytic Besov space $B_{\infty,1}^1$, the operator difference $f(L_1,M_1)-f(L_2,M_2)$ belongs to $\boldsymbol{S}_p$ and the following Lipschitz type estimate holds: $$ \|f(L_1,M_1)-f(L_2,M_2)\|_{\boldsymbol{S}_p} \le\operatorname{const}\|f\|_{B_{\infty,1}^1}\max\big\{\|L_1-L_2\|_{\boldsymbol{S}_p},\|M_1-M_2\|_{\boldsymbol{S}_p}\big\}. $$