Recent trends in harmonic and complex analysis

Asymptotic behaviour of powers of composition operators

5 avr. 2017, 11:45

45m

Salle de séminaire (Université d'Orléans, Mathématiques)

Orateur

Prof. Isabelle Chalendar (UPEM)

Description

We study the asymptotic behaviour of the powers $T^n$ of a continuous composition operator $T$ on an arbitrary Banach space $X$ of holomorphic functions on the open unit disc of the complex plane. We show that for composition operators, one has the following dichotomy: either the powers converge uniformly or they do not converge even strongly. We also show that uniform convergence of the powers of an operator $T\in L(X)$ is very much related to the behaviour of the poles of the resolvent of $T$ on the unit circle and that all poles of the resolvent of the composition operator $T$ on $X$ are algebraically simple. Our results are applied to study the asymptotic behaviour of semigroups of composition operators associated with holomorphic semiflows.