Estimates near the origin for functional calculus on analytic semigroups

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.