Non power bounded generators of strongly continuous semigroups

TL;DR

This paper investigates conditions under which operators that are not power bounded can still generate strongly continuous semigroups, extending classical results and applying to function and sequence spaces.

Contribution

It introduces weaker conditions than power boundedness for operator generation of semigroups, using elazko's conditions, and demonstrates applications to classical spaces.

Findings

Weaker conditions than power boundedness can ensure semigroup generation.

Application of these conditions to classical function and sequence spaces.

Extension of classical semigroup generation results.

Abstract

It is folklore that a power bounded operator on a sequentially complete locally convex space generates a uniformly continuous $C_0$-semigroup which is given by the corresponding power series representation. Recently, Doma\'nski asked if in this result the assumption of being power bounded can be relaxed. We employ conditions introduced by \.{Z}elazko to give a weaker but still sufficient condition for generation and apply our results to operators on classical function and sequence spaces.