  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.
