Power convergence of Abel averages

TL;DR

This paper establishes necessary and sufficient conditions for the power convergence of Abel averages of semigroups in Banach spaces, extending classical results to unbounded operators and resolvent-based averages.

Contribution

It provides a comprehensive characterization of convergence conditions for Abel averages of both bounded and unbounded semigroups in complex Banach spaces.

Findings

Conditions for power convergence of Abel averages are fully characterized.

Results extend classical bounded operator convergence to unbounded operators.

Convergence criteria include cases involving resolvent-defined averages.

Abstract

Necessary and sufficient conditions are presented for the Abel averages of discrete and strongly continuous semigroups, $T^k$ and $T_t$, to be power convergent in the operator norm in a complex Banach space. These results cover also the case where $T$ is unbounded and the corresponding Abel average is defined by means of the resolvent of $T$. They complement the classical results by Michael Lin establishing sufficient conditions for the corresponding convergence for a bounded $T$.