Weakly and almost weakly stable C_0-semigroups

TL;DR

This paper surveys the asymptotic behavior of C_0-semigroups on Banach spaces, focusing on weak and almost weak stability, providing conditions, examples, and discussing their fundamental differences.

Contribution

It offers a comprehensive overview of weak and almost weak stability of C_0-semigroups, including new conditions and illustrative examples.

Findings

Weak stability is characterized by absence of eigenvalues on the imaginary axis.

Almost weak stability is equivalent to weak stability for most time values.

The paper discusses fundamental differences and open questions in the field.

Abstract

In this paper we survey results concerning the asymptotic properties of C_0-semigroups on Banach spaces with respect to the weak operator topology. The property "no eigenvalues of the generator on the imaginary axis" is equivalent to weak stability for most time values; a phenomenon called "almost weak stability". Further, sufficient conditions actually implying weak stability are also given. By several examples we explain weak and almost weak stability and illustrate the fundamental difference between them. Many historical and bibliographical remarks position the material in the literature. We conclude the paper with some open questions and comments.