# On some properties of multidimensional Bochner-Phillips functional calculus

## TL;DR

This paper develops a multidimensional functional calculus for semigroup generators using Bernstein functions, establishing spectral mapping theorems, holomorphy conditions, and moment inequalities for these operators.

## Contribution

It introduces a new multidimensional calculus framework for semigroup generators based on Bernstein functions, with spectral and holomorphy results.

## Key findings

- Spectral mapping theorems for joint spectra are established.
- Conditions for holomorphy of semigroups are provided.
- Moment inequalities for operators in the calculus are proved.

## Abstract

The multidimensional functional calculus of semigroup generators, based on the class of Bernstein functions in several variables is developed, the spectral mapping theorems for joint spectra have been stated, the condition for holomorphy of semigroups, generated by operators which arises in the calculus is given, and the moment inequality for such operators is proved.