Some Results on Operator Semigroups and Evolution Problems

In this thesis we address certain questions arising in the functional analytic study of dynamical systems and differential equations. First, we discuss the operator theoretic counterparts of the central ergodic theoretical notions of strong and weak mixing. These concepts correspond to particular types of asymptotic behaviour of operator semigroups in the weak operator topology. In particular, we carry over classical theorems of Halmos and Rohlin for measure preserving transformations to the Hilbert space operator setting. Further, we illustrate operator semigroup methods and results on a class of telegraph systems with various boundary conditions. We study both linear and nonlinear boundary value problems. The stability of linear telegraph systems is discussed by applying theorems from the previous chapters. For the existence of solutions, we are particularly interested in time-dependent boundary conditions, since this case has little been investigated so far.

Autorentext

Dr. András Serény has obtained his PhD degree in Mathematics and its Applications at the Central European University, Budapest, in 2008. With coauthors, he has published several papers, mostly about the asymptotic behaviour of operator semigroups and the non-linear telegraph equation. In recent years, he has been working in the software industry.