Generators of $C_0$-semigroups of weighted composition operators

TL;DR

This paper characterizes when certain differential operators generate $C_0$-semigroups of weighted composition operators on Banach spaces of analytic functions, extending previous results and exploring the role of flows of analytic functions.

Contribution

It provides a comprehensive criterion for generators of $C_0$-semigroups of weighted composition operators on a broad class of Banach spaces, generalizing earlier findings.

Findings

Operators of the form $Af=Gf'+g f$ generate $C_0$-semigroups iff they generate $C_0$-semigroups.

On certain Banach spaces, no non-trivial holomorphic flow induces such semigroups.

The results include classical Hardy spaces and relate to cocycles of flows of analytic functions.

Abstract

We prove that in a large class of Banach spaces of analytic functions in the unit disc $\mathbb{D}$ an (unbounded) operator $Af=G\cdot f'+g\cdot f$ with $G,\, g$ analytic in $\mathbb{D}$ generates a $C_0$-semigroup of weighted composition operators if and only if it generates a $C_0$-semigroup. Particular instances of such spaces are the classical Hardy spaces. Our result generalizes previous results in this context and it is related to cocycles of flows of analytic functions on Banach spaces. Likewise, for a large class of non-separable Banach spaces $X $ of analytic functions in $\mathbb{D}$ contained in the Bloch space, we prove that no non-trivial holomorphic flow induces a $C_0$-semigroup of weighted composition operators on $X$. This generalizes previous results regarding $C_0$-semigroup of (unweighted) composition operators.