Abstract
In this work we consider the powers T of a linear bounded operator T and strongly continuous operator semigroups (T(t)) on a Banach space X. We look for conditions assuring "stability", i.e., convergence to zero, with respect to a natural topology. For this purpose we proceed as follows. In Chapter 1 we give an overview on (nontrivial) functional analytic methods, as for example the Jacobs–Glicksberg–de Leeuw decomposition theorem, spectral mapping theorems and an inverse Laplace transform. In Chapter 2 we discuss the "discrete time" case and first describe polynomial and power boundedness of an operator T. In Section 2 the stability with respect to the strong operator topology is treated. Weak and almost weak stability is studied in Sections 3, 4 and 5 including abstract characterisations and concrete examples. We show in particular that a "typical" contraction as well as a "typical" unitary or isometric operator on an infinite-dimensional separable Hilbert space is almost weakly but not weakly stable. We proceed analogously in Chapter 3 for a Co-semigroup(T(t)). We first characterise boundedness and polynomial boundedness in terms of the cogenerator or the resolvent of the generator. We then shortly discuss exponential stability in Section 2. For strongly stable semigroups the classical theorems of Foias–Sz.-Nagy and Arendt–Batty–Lyubich–Vu are cited and supplemented. In Sections 4–6 we study weakly and almost weakly stable semigroups. Together with various (and different) characterisations we give new concrete and abstract examples (in form of category theorems).
Author and committee
dc:creator, dc:contributor.*- Author
-
- Eisner, Tatjana
Identifiers
dc:identifier.*- Identifier
- hdl:10900/49075