Functions of dissipative operators under relatively bounded and relatively trace class perturbations

A.B. Aleksandrov and V.V. Peller

Vol. 340 (2026), No. 2, 199–228

DOI: 10.2140/pjm.2026.340.199

Abstract

We study the behaviour of functions of dissipative operators under relatively bounded and relatively trace class perturbations. We introduce and study the class of analytic relatively operator Lipschitz functions. An essential role is played by double operator integrals with respect to semispectral measures. We also study the class of analytic resolvent Lipschitz functions. Then we obtain a trace formula in the case of relatively trace class perturbations and show that the maximal class of functions for which the trace formula holds in the case of relatively trace class perturbations coincides with the class of analytic relatively operator Lipschitz functions. We also establish the inequality $\mathrm{\int \; <!--nolimits-->}<!--FUNCTION\; APPLICATION-->|\xi (t)|{(1+|t|)}^{-1}dt<\infty$ for the spectral shift function $\xi$ in the case of relatively trace class perturbations.

Keywords

dissipative operator, trace formula, relatively bounded perturbation, relatively trace class perturbation, semispectral measure, double operator integral, relatively operator Lipschitz function

Mathematical Subject Classification

Primary: 47A55

Secondary: 47A20, 47A60, 47B10, 47B15, 47B44

Milestones

Received: 6 May 2025

Revised: 25 September 2025

Accepted: 28 September 2025

Published: 26 November 2025