A non-analytic growth bound for laplace transforms and semigroups of operators

Let f : ℝ<inf>+</inf> → ℂ be an exponentially bounded, measurable function. We introduce a growth bound ζ(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 definition extends to vector and operator-valued cases. For a C<inf>0</inf>-semigroup T of operators, ζ(T) is closely related to the critical growth bound of T.