On asymptotics for $C_0$-semigroups

TL;DR

This paper extends spectral and stability results from Hilbert space semigroups to Banach space semigroups of scalar type spectral operators, providing exponential estimates and characterizations of stability.

Contribution

It generalizes stability theorems and exponential estimates from Hilbert spaces to Banach spaces for scalar type spectral operator semigroups.

Findings

Exponential stability estimates with optimal constants.

Characterization of uniform exponential stability in Banach spaces.

Extension of spectral bound conditions to scalar type spectral operators.

Abstract

We stretch the spectral bound equal growth bound condition along with a generalized Lyapunov stability theorem, known to hold for $C_0$-semigroups of normal operators on complex Hilbert spaces, to $C_0$-semigroups of scalar type spectral operators on complex Banach spaces. For such semigroups, we obtain exponential estimates with the best stability constants. We also extend to a Banach space setting a celebrated characterization of uniform exponential stability for $C_0$-semigroups on complex Hilbert spaces and thereby acquire a characterization of uniform exponential stability for scalar type spectral and eventually norm-continuous $C_0$-semigroups.