Mean ergodic composition operators on generalized Fock spaces

Abstract

Every bounded composition operator C ψ dened by an analytic symbol ψ on the complex plane when acting on generalized Fock spaces F p ϕ , 1 ≤ p ≤ ∞, is power bounded. Mean ergodic and uniformly mean ergodic composition operators on the spaces are characterized. The set of periodic points of these operators is also determined. Theorem 1.1. (Theorem 2.1 of [18], Theorem 2.1 of [19]) Let 1 ≤ p ≤ ∞ and ψ be a nonconstant analytic map on the complex plane C. Then the operator C ψ : F p ϕ → F p ϕ is 1 *Corresponding author Key words and phrases. Composition operators, generalized Fock spaces, power bounded operator, mean ergodic operator, uniformly mean ergodic operator.

Spectrums and uniform mean ergodicity of weighted composition operators on Fock spaces

“…The monographs [14,28] provides basic information on ergodic theory. Inspired by all these works, the authors and J. Bonet in [26] studied the mean ergodicity of composition operators acting on generalized Fock spaces, and concluded that all bounded composition operators on Fock spaces F p are power bounded whenever 1 ≤ p ≤ ∞. In this section, we show that this conclusion is no longer true in general for the weighted composition operators W (u,ψ) .…”

“…(ii) Let 1 ≤ p ≤ ∞,and ψ(z) = az with |a| = 1. If both u(0) and a are roots of unity, then W (u,ψ) is uniformly mean ergodic on F p .By a result in[26], the composition operator C ψ is not uniformly mean ergodic on F ∞ whenever |a| = 1. Now the weight function u makes it possible to enrich uniformity by taking the value |u(0)| smaller.Proof.…”

Spectrums and Uniform Mean Ergodicity of Weighted Composition Operators on Fock Spaces

“…The monographs [14,28] provide basic information on ergodic theory. Inspired by all these works, the authors and J. Bonet [26] studied the mean ergodicity of composition operators acting on generalized Fock spaces and concluded that all bounded composition operators on Fock spaces F p are power bounded whenever 1 ≤ p ≤ ∞. In this section, we show that this conclusion is no longer true in general for the weighted composition operators W (u,ψ) .…”

Gaussian integral means in the Fock space

“…For further information related to the theory of Fock spaces we refer to mentioned monograph [14]. See also the recent works [8,10] on composition operators on the Fock space.…”