On the regularity of scalar type spectral $C_0$-semigroups

Abstract

We show that, for the $C_0$-semigroups of scalar type spectral operators, a well-known necessary condition for the generation of eventually norm-continuous $C_0$-semigroups, formulated exclusively in terms of the location of the spectrum of the semigroup's generator in the complex plane, is also sufficient and, in fact, characterizes the generators of immediately norm-continuous such semigroups. Combining characterizations of the immediate differentiability and the Gevrey ultradifferentiability of scalar type spectral $C_0$-semigroups with the generation theorem, found earlier by the author, we arrive at respective characterizations of the generation of such semigroups. We further establish characterizations of the generation of eventually differentiable and immediately compact scalar type spectral $C_0$-semigroups also in terms of the generator's spectrum and show that, for such semigroups, eventual compactness implies immediate. All the obtained results are instantly transferred to the $C_0$-semigroups of normal operators.