A Besov algebra calculus for generators of operator semigroups and related norm-estimates

TL;DR

This paper develops a new bounded functional calculus for operator semigroup generators, extending classical methods and enabling improved norm-estimates on Hilbert and Banach spaces.

Contribution

It introduces a Besov algebra calculus that unifies and extends existing calculi, providing a direct approach to norm-estimates and their improvements.

Findings

Constructed a new bounded functional calculus for semigroup generators.

Unified and extended classical Hille-Phillips calculus.

Enabled improved norm-estimates for operators on Hilbert and Banach spaces.

Abstract

We construct a new bounded functional calculus for the generators of bounded semigroups on Hilbert spaces and generators of bounded holomorphic semigroups on Banach spaces. The calculus is a natural (and strict) extension of the classical Hille-Phillips functional calculus, and it is compatible with the other well-known functional calculi. It satisfies the standard properties of functional calculi, provides a unified and direct approach to a number of norm-estimates in the literature, and allows improvements of some of them.