Semigrupos de operadores de composición

Abstract

The main problem we consider in this work is the study of semigroups of composition operators on spaces of analytic functions on the unit disc. Mainly, we focus on the problem in Hardy spaces, Hp(D), and Bergman spaces, Ap (D). We can divide this work in two different parts. Firstly, we study the concept of strongly continuous semigroup of operators on a Banach space, as well as the existence of its infinitesimal generator. Secondly, we pay attention to semigroups of composition operators on spaces of analytic functions on the unit disc and its construction from the concept of semigroup of analytic functions. If {ϕt}t≥0 is a semigroup of analytic self maps of the unit disc D with the composition as operation between them, and X is a Banach space of analytic functions on D, then Q(t)f = f ◦ ϕt, f ∈ X, t ≥ 0, defines a semigroup of composition operators whenever Q(t) ∈ B(X) for t ≥ 0. We will proof under what conditions they are strongly continuous and we will focus on cases such as Hardy spaces and Bergman spaces. Finally, we will calculate its infinitesimal generator. Results of the first part are classic and they can be found in many traditional books of Functional Analysis, such as [Rudin, W. Functional analysis. McGraw-Hill, Inc. Second Edition, 1991], while the rest of this report collects current results, so we will study recently published works of different authors.