Semigroups of scalar type operators on Banach spaces

Abstract:

The main result is that if $\{ T(t):t \geqslant 0\}$ is a strongly continuous semigroup of scalar type operators on a weakly complete Banach space $X$ and if the resolutions of the identity for $T(t)$ are uniformly bounded in norm, then the infinitesimal generator is scalar type. Moreover, there exists a countably additive spectral measure $K( \cdot )$ such that $T(t) = \smallint \exp (\lambda t)dK(\lambda )$, for $t \geqslant 0$. This is a direct generalization of the well-known theorem of Sz.-Nagy about semigroups of normal operators on a Hilbert space. Similar spectral representations are given for representations of locally compact abelian groups and for semigroups of unbounded operators. Connections with the theory of hermitian and normal operators on Banach spaces are established. It is further shown that $R$ is the infinitesimal generator of a semigroup of hermitian operators on a Banach space if and only if iR is the generator of a group of isometries.