Sesión Análisis funcional y complejoOn frequencies of parabolic Koenigs domains
Carlos Gómez-Cabello
Universidad del País Vasco, España - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
In recent times, $C_0$-semigroups of operators have attracted an increasing attention in the Analysis community. Indeed, some challenging problems concerning interesting properties of operators become more manageable when such an operator is assumed to belong to a $C_0$-semigroup. That is the case, for instance, of the computation of the point spectrum of composition operators. In this talk, we will consider $C_0$-semigroups $\{C_{\varphi_t}\}_{t\geq0}$ of composition operators on the Hardy spaces $H^p$. More precisely, those induced by parabolic semigroups of analytic functions on the unit disc $\mathbb{D}$, with Koenigs function $h$. We are interested in the study of the point spectrum of the operators $C_{\varphi_t}$. To do so, it suffices to look at the point spectrum of the infinitesimal generator of the semigroup of composition operators. This reduces to characterising those $\lambda\in\mathbb{C}$ such that $e^{\lambda h}\in H^p$. We will derive containment relations for the point spectrum and provide sufficient conditions for its complete characterization. We shall briefly discuss the techniques used in the proofs. They rely heavily on the geometric properties of $h(\mathbb{D})$ and on careful estimates of the harmonic measure of some boundary subsets of $h(\mathbb{D})$.
Trabajo en conjunto con: F. Javier González-Doña (Universidad Carlos III de Madrid, España).