Mean ergodicity of weighted composition operators on spaces of holomorphic functions

Power bounded weighted composition operators on function spaces defined by local properties

“…Moreover, for (unweighted) composition operators we show that the notions of topologizability, power boundedness and (uniform) mean ergodicity coincide on kernels of these partial differential operators. In particular, the results of this section generalize results obtained in [4] and [6], where the special case of the Cauchy-Riemann operator was considered. In the final section 5 we characterize, under mild additional assumptions on the weight and the symbol, those weighted composition operators which are generators of strongly continuous operator semigroups on the space of continuous functions C(X) equipped with the compact-open topology, where X is a locally compact, σ-compact, non-compact Hausdorff space.…”

“…The special case of the Cauchy-Riemann operator will give the space of holomorphic functions of a single variable equipped with the compact-open topology. In this context topologizability and power boundedness of weighted composition operators have been studied in [4]. As already mentioned in example 2.3 iv), for a non-constant polynomial with complex coefficients in d ≥ 2 variables P ∈ C[X 1 , .…”

“…Recently, mean ergodicity, power boundedness, and topologizablity of (weighted) composition operators and multiplication operators on various function spaces have been investigated by several authors, see e.g. [3], [4], [6], [7], [8], [9], [13] [25], [26], [27]. The purpose of this article is to give a general framework for studying power boundedness etc.…”

Dynamics and spectra of composition operators on the Schwartz space

“…For example, Fonf, Lin and Wojstaycyck proved in [13] that if X is a Banach space which has Schauder basis and it is not reflexive then there exists an operator which is power bounded but not mean ergodic. In the last years, power boundedness and mean ergodicity have been studied by several authors from a dynamical point of view, mainly in spaces of analytic functions [5,6,7,10]. Roughly speaking, power boundedness and mean ergodicity are related to small orbits, and hyperciclicity and superciclicity to big orbits.…”

Mean ergodic composition operators in spaces of homogeneous polynomials

“…If the domain is the unit disc, the authors in [3] characterise when C ϕ is mean ergodic or uniformly mean ergodic on the disc algebra or on the space of bounded holomorphic functions in terms of the asymptotic behaviour of the symbol. In [4] it is investigated the power boundedness and (uniform) mean ergodicity of weighted composition operators on the space of holomorphic functions on the unit disc in terms of the symbol and the multiplier. Finally, in [10] the author studies power boundedness and mean ergodicity for (weighted) composition operators on function spaces defined by local properties in a very general framework which extends previous works.…”