Weakly and almost weakly stable C_0-semigroups

Details Weakly and almost weakly stable C_0-semigroups Weakly and almost weakly stable C_0-semigroups

2008-05-07

arxiv.org/abs/0805.1039v1

Int J. Dynam. Syst. Diff. Eq. 1 (2007), 44-57

Mathematics

Functional Analysis

22 pages; survey article

Scientific paper

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.

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