Regularity of semigroups via the asymptotic behaviour at zero (Q372351)

A short proof of Kato-Beurling's theorem for holomorphic semigroups is given. This result is applied to obtain extrapolation properties for consistent semigroups on \(L_p\) spaces and interpolation spaces of exponent \(\theta\in(0,1)\). For a cosine function on an UMD-space, the author proves a zero-two law: Let \(\{U(t)\}_{t\in \mathbb R}\) be a \(C_0\)-group and \(C(t)=0.5(U(t)+U(-t))\) be an associated cosine function. If \(\limsup_{t\to 0}\|C(t)-I\|<2\), then \(\{C(t)\}_{t\in\mathbb R}\) is uniformly continuous and the left hand side is equal to zero. Kato-Beurling's theorem, extrapolation results and maximum regularity characteristics for \(\mathcal R\)-analytic semigroups are presented as well.