Bernstein functions of several semigroup generators on Banach spaces under bounded perturbations. II

TL;DR

This paper extends the analysis of Bernstein functions of semigroup generators on Banach spaces to the multidimensional case, proving Frechet differentiability and generalizing the Livschits-Krein trace formula.

Contribution

It establishes Frechet differentiability of operator Bernstein functions and generalizes the Livschits-Krein trace formula to multiple dimensions.

Findings

Proved Frechet differentiability of multidimensional Bernstein functions.

Generalized Livschits-Krein trace formula for several semigroup generators.

Extended previous one-dimensional results to a multidimensional framework.

Abstract

The paper deals with multidimensional Bochner-Phillips functional calculus. In the previous paper by the author bounded perturbations of Bernstein functions of several commuting semigroup generators on Banach spaces where considered, conditions for Lipschitzness and estimates for the norm of commutators of such functions where proved. Also in the one-dimensional case the Frechet differentiability of Bernstein functions of semigroup generators on Banach spaces where proved and a generalization of Livschits-Kre\u{i}n trace formula derived. The aim of the present paper is to prove the Frechet differentiability of operator Bernstein functions and the Livschits-Kre\u{i}n trace formula in the multidimensional setting.