Estimates near the origin for functional calculus on analytic semigroups

TL;DR

This paper derives precise lower bounds for the functional calculus of sectorial operator semigroups near zero, connecting these bounds to the algebraic structure of the generated Banach algebra.

Contribution

It introduces sharp estimates for the functional calculus near the origin, extending previous results and covering both quasinilpotent and non-quasinilpotent cases.

Findings

Sharp lower estimates established for functional calculus near zero

Results linked to the existence of an identity element in the generated Banach algebra

Extends and generalizes previous literature on operator semigroup calculus

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 a sector; here $F$ is given as the Fourier--Borel transform of an analytic functional. The results are linked to the existence of an identity element 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.