Lower estimates near the origin for functional calculus on operator semigroups

TL;DR

This paper establishes precise lower bounds for the functional calculus of operator semigroup generators near zero, connecting these bounds to algebraic properties like idempotents and identity elements.

Contribution

It provides new sharp lower estimates for the functional calculus of operator semigroup generators, extending existing results to both quasinilpotent and non-quasinilpotent cases.

Findings

Sharp lower bounds near zero for $F(-uA)$ established

Results linked to the existence of identity or idempotents in the algebra

Extends previous results to broader classes of semigroups

Abstract

This paper provides sharp lower estimates near the origin for the functional calculus $F(-uA)$ of a generator $A$ of an operator semigroup defined on the (strictly) positive real line; here $F$ is given as the Laplace transform of a measure or distribution. The results are linked to the existence of an identity element or an exhaustive sequence of idempotents in the Banach algebra generated by the semigroup. Both the quasinilpotent and non-quasinilpotent cases are considered, and sharp results are proved extending many in the literature.