Lifshitz-Krein trace formula for Hirsch functiuonal calculus on Banach spaces

TL;DR

This paper extends the Lifshitz-Krein trace formula to Banach spaces, providing a spectral shift function for nonpositive operators and exploring perturbation effects on operator monotonic functions.

Contribution

It introduces a simple spectral shift function definition and proves trace formulas for perturbations of operator monotonic functions on Banach spaces.

Findings

Spectral shift function defined for pairs of nonpositive operators on Banach spaces.

Trace formulas of Lifshitz-Kre
21n type established for nuclear perturbations.

Lipschitz properties of operator monotonic functions analyzed.

Abstract

We give a simple definition of a spectral shift function for pairs of nonpositive operators on Banach spaces and prove trace formulas of Lifshitz-Kre\u{\i}n type for a perturbation of an operator monotonic (negative complete Bernstein) function of negative and nonpositive operators on Banach spaces induced by nuclear perturbation of an operator argument. The Lipschitzness of such functions is also investigated. The results may be regarded as a contribution to a perturbation theory for Hirsch functional calculus.