Generators of semigroups on Banach spaces inducing holomorphic semiflows

TL;DR

This paper investigates the relationship between generators of semigroups of composition operators on Banach spaces of analytic functions and holomorphic semiflows, focusing on the conditions under which the generator can be represented as a differential operator with a holomorphic symbol.

Contribution

It establishes conditions under which generators of composition semigroups induce holomorphic semiflows, extending the understanding of their structure on Banach spaces.

Findings

Characterization of generators inducing holomorphic semiflows

Conditions for the differential operator representation

Extension of known results on composition operators

Abstract

Let $A$ be the generator of a $C_0$-semigroup $T$ on a Banach space of analytic functions on the open unit disc. If $T$ consists of composition operators, then there exists a holomorphic function $G:{\mathbb D}\to{\mathbb C}$ such that $Af=Gf'$ with maximal domain. The aim of the paper is the study of the reciprocal implication.