The non-analytic growth bound of a C0-semigroup and inhomogeneous Cauchy problems

The non-analytic growth bound ζ(T) of a C0-semigroup T measures the extent to which T can be approximated by a holomorphic function, and it is related to spectral properties of the generator A in regions of far from the real axis. We show that ζ(T) can be characterised by means of Fourier multiplier properties of the resolvent of A far from the real axis, and also by existence and uniqueness of mild solutions of inhomogeneous Cauchy problems of the form u′(t)=Au(t)+f(t) on where the Carleman spectra of f and u are far from the origin. The corresponding results for the exponential growth bound ω0(T) have been established earlier by other authors.