Joint functional calculus in algebra of polynomial tempered distributions

TL;DR

This paper develops a functional calculus for generators of semigroups within the algebra of polynomial tempered distributions, providing differential properties and applications to second derivative operators.

Contribution

It introduces a new functional calculus based on polynomial tempered distributions for countable generator systems, with proven differential properties and characterization of its image.

Findings

Established a differential property of the calculus.

Described the image using the commutant of polynomial shift semigroup.

Applied the calculus to functions of second derivative operators.

Abstract

In this paper we develop a functional calculus for a countable system of generators of contraction strongly continuous semigroups. As a symbol class of such calculus we use the algebra of polynomial tempered distributions. We prove a differential property of constructed calculus and describe its image with the help of the commutant of polynomial shift semigroup. As an application, we consider a function of countable set of second derivative operators.