A Non-Analytic Growth Bound for Laplace Transforms and Semigroups of Operators

Let f :R - C be an exponentially bounded, measurable function. We introduce a growth bound (zeta)(f) which measures the extent to which f can be approximated by holomorphic functions. This growth bound is related to the location of the domain of holomorphy of the Laplace transform of f far from the real axis. The denition extends to vector and operator-valued cases. For a C(0) -semigroup T of operators, (zeta)(T) is closely related to the critical growth bound of T .