Regularity of semigroups via the asymptotic behaviour at zero

Abstract

An interesting result by T. Kato and A. Pazy says that a contractive semigroup (T(t)) t≥0 on a uniformly convex space X is holomorphic iff \(\limsup_{t \downarrow0} \|T(t) - \operatorname{Id}\| < 2\). We study extensions of this result which are valid on arbitrary Banach spaces for semigroups which are not necessarily contractive. This allows us to prove a general extrapolation result for holomorphy of semigroups on interpolation spaces of exponent θ∈(0,1). As an application we characterize boundedness of the generator of a cosine family on a UMD-space by a zero-two law. Moreover, our methods can be applied to \(\mathcal{R}\)-sectoriality: We obtain a characterization of maximal regularity by the behaviour of the semigroup at zero and show extrapolation results.