  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.
