Spectral Properties of $ C_{0}$-Semigroups of Positive Operators on C$^*$-Algebras

TL;DR

This paper investigates the spectral properties of positive $C_0$-semigroups on C$^*$-algebras, establishing that the growth bound equals the spectral bound of the generator and analyzing spectral values.

Contribution

It proves that for positive $C_0$-semigroups on C$^*$-algebras, the growth bound equals the spectral bound, and the spectral bound is a spectral value if the spectrum is non-empty.

Findings

Growth bound equals spectral bound for positive $C_0$-semigroups.

Spectral bound is a spectral value when the spectrum is non-empty.

Spectral properties are characterized for generators on C$^*$-algebras.

Abstract

Let $ (T(t))_{t\geq0} $ be a positive $C_{0}$-semigroup with generator $A$ on a C$^*$-algebra or on the predual of a W$^*$-algebra. Then the growth bound $\omega_{0}$ equals $s(A)$. If the spectrum of $A$ is not empty, then $s(A)$, the spectral bound of $A$, is a spectral value.