Regularity of Semigroups via the Asymptotic Behaviour at Zero

TL;DR

This paper extends Kato and Pazy's result on the regularity of semigroups to arbitrary Banach spaces, characterizing holomorphy and regularity through asymptotic behavior at zero, with applications to cosine families and R-sectoriality.

Contribution

It generalizes the zero-limit condition for semigroup holomorphy beyond contractive cases to all Banach spaces and applies this to various operator regularity concepts.

Findings

Characterization of semigroup holomorphy via zero asymptotic behavior.

Extrapolation results for holomorphy on interpolation spaces.

Zero-two law for boundedness of cosine family generators.

Abstract

An interesting result by T. Kato and A. Pazy says that a contractive semigroup (T(t)) on a uniformly convex space X is holomorphic iff limsup_{t \downarrow 0} ||T(t)-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 {\theta} in (0,1). As 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 R-sectoriality: We obtain a characterization of maximal regularity by the behaviour of the semigroup at zero and show extrapolation results.