Mean ergodic composition operators in spaces of homogeneous polynomials

Abstract

We study some dynamical properties of composition operators defined on the space P( m X) of m-homogeneous polynomials on a Banach space X when P( m X) is endowed with two different topologies: the one of uniform convergence on compact sets and the one defined by the usual norm. The situation is quite different for both topologies: while in the case of uniform convergence on compact sets every power bounded composition operator is uniformly mean ergodic, for the topology of the norm there is no relation between the latter properties. Several examples are given.2010 Mathematics Subject Classification. Primary 46G25, 47B33, 47A35, Secondary 46E50.

Mean ergodic composition operators on spaces of holomorphic functions on a Banach space

“…From[29, Proposition 9.16] we know that every bounded subset of H(B X ) is relatively compact, therefore H(B X ) is semi-Montel and, in particular, semi-reflexive. Then, as a consequence of [6, p. 917] (see also[21, Proposition 3.1]) we have that every power bounded operator is uniformly mean ergodic.…”

“…However, it seems that there is no previous literature about the dynamics of such operators. The present work can be considered a sequel of [21] by the same authors, where we study some dynamical properties (especially mean ergodicity) of composition operators in spaces of homogeneous polynomials. As in [21], the motivation and inspiration of our investigation comes from several previous works, as [7], where the authors characterise those composition operators C ϕ : H(U) → H(U) which are power bounded, where H(U) is the space of holomorphic functions on a connected domain of holomorphy U of C d .…”

“…The present work can be considered a sequel of [21] by the same authors, where we study some dynamical properties (especially mean ergodicity) of composition operators in spaces of homogeneous polynomials. As in [21], the motivation and inspiration of our investigation comes from several previous works, as [7], where the authors characterise those composition operators C ϕ : H(U) → H(U) which are power bounded, where H(U) is the space of holomorphic functions on a connected domain of holomorphy U of C d . It was proved in [7] that C ϕ is power bounded if and only if it is (uniformly) mean ergodic, and this happens if and only if the symbol ϕ has stable orbits.…”

Mean ergodic composition operators on spaces of smooth functions and distributions

“…In recent years there have been several articles studying mean ergodicity and related properties of (weighted) composition operators on various spaces of functions, such as spaces of holomorphic functions in finite dimensions [8], [5], [4], [12], [3], [16], [23], spaces of holomorphic functions on infinite Banach spaces [18], spaces of homogeneous polynomials on infinite dimensional Banach spaces [17], spaces of real analytic functions [9], the Schwartz space of rapidly decreasing functions on R [10], spaces of meromorphic functions [11], and within the general framework of function spaces defined by local properties [19]. This note is organized as follows.…”

Composition Operators on Classes of Holomorphic Functions on Banach Spaces

“…Moreover, we give a detailed introduction of the results we are going to study in the last section of this chapter. Chapter 2 is based on the already published paper [37] and treats mean ergodic composition operators when acting on spaces of m-homogeneous polynomials. The situation is quite different for the two topologies considered in this space: while in the case of uniform convergence on compact sets every power bounded composition operator is uniformly mean ergodic, for the topology of the norm there is no relation between the latter properties.…”