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

Ulrich Groh

TL;DR

This work addresses the relationship between the spectral data of the generator $A$ of a positive $C_0$-semigroup and the long-time behavior of the semigroup in the noncommutative setting of $C^*$-algebras and their duals. It develops a unified integral resolvent representation $\mathcal{R}(\lambda,A)x = \lim_{t\to\infty} \int_{0}^{t} e^{-\lambda s} T(s)x \, ds$ for $\Re(\lambda) > s(A)$ and proves that the growth bound $\omega_0$ equals the spectral bound $s(A)$, with $s(A)$ belonging to the spectrum whenever $\sigma(A)$ is nonempty. The paper also establishes that the spectral radius of positive operators on a C$^*$-algebra lies in the spectrum, and, in the unital case, yields eigenvectors for $s(A)$ in the adjoint, thereby linking spectral properties to asymptotic behavior. These results extend to preduals of W$^*$-algebras and to positive $C_0$-groups, providing a sharp, operator-algebraic description of long-time stability and convergence for positive dynamics in noncommutative settings.

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 $ω_{0}$ equals $s(A)$. If the spectrum of $A$ is not empty, then $s(A)$, the spectral bound of $A$, is a spectral value.

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

TL;DR

This work addresses the relationship between the spectral data of the generator of a positive -semigroup and the long-time behavior of the semigroup in the noncommutative setting of -algebras and their duals. It develops a unified integral resolvent representation for and proves that the growth bound equals the spectral bound , with belonging to the spectrum whenever is nonempty. The paper also establishes that the spectral radius of positive operators on a C-algebra lies in the spectrum, and, in the unital case, yields eigenvectors for in the adjoint, thereby linking spectral properties to asymptotic behavior. These results extend to preduals of W-algebras and to positive -groups, providing a sharp, operator-algebraic description of long-time stability and convergence for positive dynamics in noncommutative settings.

Abstract

Let be a positive -semigroup with generator on a C-algebra or on the predual of a W-algebra. Then the growth bound equals . If the spectrum of is not empty, then , the spectral bound of , is a spectral value.
Paper Structure (3 sections, 9 theorems, 43 equations)

This paper contains 3 sections, 9 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Positive Operators
  3. Positive Semigroups

Key Result

Lemma 2.1

Let $T$ be a positive operator on a C$^{\star}$-algebra $\mathfrak{A}$, let $\lambda \in \rho(T)$ such that $\abs{\lambda} > r(T)$. Then for all $0 \leqslant \varphi \in \mathfrak{A}^{*}$ and $0 \leqslant x \in \mathfrak{A}$, we have

Theorems & Definitions (17)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Theorem 3.3
  • ...and 7 more