Functional calculus for $C_0$-semigroups using infinite-dimensional systems theory

TL;DR

This paper introduces a novel approach to functional calculus for $C_0$-semigroups' generators using infinite-dimensional systems theory, providing new insights and proofs in the field.

Contribution

It applies systems theory concepts to develop a functional calculus for semigroup generators, offering a new perspective and simplified proofs of existing results.

Findings

New functional calculus framework for $C_0$-semigroup generators

Simplified proofs of known functional calculus results

Connections established between systems theory and operator theory

Abstract

In this short note we use ideas from systems theory to define a functional calculus for infinitesimal generators of strongly continuous semigroups on a Hilbert space. Among others, we show how this leads to new proofs of (known) results in functional calculus.