Eventual Positivity of Operator Semigroups

10.12.2015, 12:30 Uhr  –  Campus Golm, Haus 5, Raum 1.12
Forschungsseminar Diskrete Spektraltheorie

Jochen Glück (Universität Ulm)

Let T = (T(t)) be a C_0-semigroup on some function space or, more generally, on a Banach lattice E. The semigroup T is called positive if T(t)f \ge 0 for each 0 \le f \in E and for each t \geg 0. Those semigroups are of enormous importance in applications and accordingly, their properties have been intensively studied. In particular, a lot is known about the characterization of those semigroups in terms of their generator, about their spectral theory and about their asymptotic behavior. Only recently, a somewhat weaker property than positivity, namely eventual positivity of semigroups, began to draw some attention. Here, a C_0-semigroup T is called eventually positive if for each 0 \le f \in E there is a time t_0 \ge 0 such that T(t)f \ge 0 for all t \ge t_0. In finite dimensions, one of the motivations to consider eventually positive semigroups was to generalize the Perron-Frobenius spectral theorem to a wider class of operators/semigroups. During the last decade, it turned out the eventually positive semigroups also occur within applications in an infinite-dimensional setting. Examples of such occurrences are the semigroup generated by the bi-Laplace operator on certain domains and the Dirichlet-to-Neumann semigroup on a circle.Those applications motivate the development of a general theory of eventually positive semigroup. In this talk we sketch a first approach to such a theory, focusing especially on the characterization of eventually positive $C_0$-semigroups in terms of spectral properties of their generator. We then demonstrate by a couple of examples how our theory can be applied to prove eventual positivity of several disparate examples of semigroups.