The theory of Besov functional calculus: developments and applications to semigroups

TL;DR

This paper advances the theory of Besov functional calculus for semigroup generators, establishing its optimality, clarifying its structure, and deriving new spectral mapping theorems with broad implications for semigroup analysis.

Contribution

It extends and deepens the Besov calculus for semigroup generators, providing optimal constructions, structural insights, and generalizations of key semigroup results.

Findings

Optimal construction of Besov functional calculus

Structural characterization of the algebra al B

New spectral mapping theorems for operator semigroups

Abstract

We extend and deepen the theory of functional calculus for semigroup generators, based on the algebra $\mathcal B$ of analytic Besov functions, which we initiated in a previous paper. In particular, we show that our construction of the calculus is optimal in several natural senses. Moreover, we clarify the structure of $\mathcal B$ and identify several important subspaces in practical terms. This leads to new spectral mapping theorems for operator semigroups and to wide generalisations of a number of basic results from semigroup theory.