Browsing by Author "Mirka, Anssi"

In this Thesis I present the general theory of semigroups of linear operators. From the philosophical point of view I begin by connecting deterministic evolution in time to dynamic laws that are stated in terms of a differential equation. This leads us to associate semigroups with the models for autonomic deterministic motion. From the historical point of view I reflect upon the history of the exponential function and its generalizations. I emphasize their role as solutions to certain linear differential equations that characterize both exponential functions and semigroups. This connection then invites us to consider semigroups as generalizations of the exponential function. I believe this angle of approach provides us with motivation as well as useful ideas. From the mathematical point of view I construct the basic elements of the theory. First I consider briefly uniformly and strongly continuous semigroups. After that I move on to the more general σ(X, F)-continuous case. Here F is a so called norming subspace of the dual X^*. I prove the existence of both the infinitesimal generator S of the semigroup and the resolvent (λ - S)^(-1) as well as some of their basic properties. Then I turn to the other direction and show how to create a semigroup starting from its generator. That is the content of the famous Hille—Yosida Theorem. From the practical point of view I give some useful characterizations of the generator in terms of dissipativity and accretivity. These techniques also lead us to an effortless proof of Stone's Theorem on unitary groups. Finally, from an illustrational point of view I give two applications. The first is about multiplicative semigroups on L^p spaces, where the setting is simple enough to allow intuition to accompany us. The second takes on a problem of generating a particular stochastic weak*-continuous semigroup. It serves to illustrate some of our results.