Sesión Análisis Funcional y ComplejoPositive shadowing of composition operators on the Hardy space of the disk.
Ben-Hur Eidt
USP - Universidade de São Paulo, Brasil - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
The (positive) shadowing property is a concept from dynamical systems that originated in the works [3] and [4]. In the context of bounded linear operators on Banach spaces, this property has received attention in recent years, culminating in a recent characterization of hyperbolicity in terms of shadowing and expansivity ( [1] , [2] ). Motivated by these results, it is natural to investigate this property for more concrete operators.
In this presentation, we will consider a holomorphic self-map of the open unit disk \(\mathbb{D}\), denoted by \(\phi\). We will describe the positive shadowing property for composition operators \(C_{\phi}f=f\circ \phi\) on the Hardy space \(H^2(\mathbb{D})\) using some general criteria and canonical forms. In particular, we obtain a description of all linear fractional self-maps, \(\phi\), of \(\mathbb{D}\) such that \(C_{\phi}\) has the positive shadowing property. If time permits, we will also discuss shadowing on \(H^p(\mathbb{D})\) for \(p \neq 2\).
Trabajo en conjunto con: Artur Blois (Unicamp), Paulo Lupatini (Unicamp) y Osmar R. Severiano (Unicamp).
Referencias
[1] N. C. Bernardes Jr. and A. Messaoudi. “Shadowing and structural stability for operators”. In: Ergodic Theory Dynam. Systems 41.4 (2021), pp. 961–980.
[2] N. C. Bernardes Jr. et al. “Expansivity and shadowing in linear dynamics”. In: J. Math. Anal. Appl. 461.1 (2018), pp. 796–816.
[3] R. Bowen. “ω-limit sets for axiom A diffeomorphisms”. In: J. Differential Equations 18.2 (1975), pp. 333–339.
[4] J. G. Sina˘ı. “Gibbs measures in ergodic theory”. In: Uspehi Mat. Nauk 27.4(166) (1972), pp. 21–64.