Weighted composition semigroups on Banach spaces of holomorphic functions

TL;DR

This paper characterizes the generators of strongly continuous semigroups of weighted composition operators on Banach spaces of holomorphic functions, linking their structure to holomorphic functions g and G.

Contribution

It provides a complete description of the infinitesimal generators of such semigroups and establishes the reciprocal connection, advancing understanding of operator semigroup theory in complex analysis.

Findings

Generators have form $ extstyle ext{gf + Gf'}$ with holomorphic g, G.

Semigroup generation is characterized by the form of the generator.

Reciprocal implication confirms the structure of semigroups as weighted composition operators.

Abstract

We study, to certain Banach spaces $X$, families of weighted composition operators. Notably, we show that if this family form a strongly continuous semigroup, then its infinitesimal generator ($\Gamma, D(\Gamma)$) is given by $\Gamma f = gf+Gf^\prime$ with $D(\Gamma) = \{ f\in X \ | \ gf+Gf^\prime\in X \}$ where $g, G$ are holomorphic functions. Moreover, our second maim result is to study the reciprocal implication. That is if $(\Gamma , D(\Gamma))$, define like above, generate a strongly continuous semigroup, then this one is a family of weighted composition operators.