Mean Ergodic Weighted Composition Operator 𝝀𝑪𝝋 on Bloch Space

Subject Areas : Statistics

Fakhreddin  falahat ¹ (Department of Mathematics, Islamic Azad University, Shiraz Branch, Shiraz, Iran)
Zahra  Kamali ² (faculty membrer of Islamic Azad University, Shiraz branch)

Received: 2020-11-29 Accepted : 2021-09-09 Published : 2022-03-21

Keywords: عملگر ترکیبی وزن‌دار, عملگر ارگودیک میانگین, نقطه دنجوی-ولف, فضای بلوچ,

Abstract :

Investigating the mean ergodicity of composition operators on various Banach Spaces has always been of interest to mathematicians and many authors studied this topics intensively, in many different spaces, such as, the space of all holomorphic functions on unit disk, Hardy space and Bloch space. In this paper, for a self map of the unit disk, φ and λ∈ℂ, we consider weighted composition operator, (λ𝐶φ)𝑓=λ𝑓𝑜φ , for every 𝑓 in Bloch space and Little Bloch space and inquiry the conditions under which the weighted composition operator 𝜆𝐶𝜑, is mean ergodic or uniformly mean ergodic on the Bloch and Little Bloch Space. In fact, we will show, if |λ|>1,𝜆𝐶𝜑, cannot be power bounded, mean ergodic or uniformly mean ergodic, in contrast, if |λ|<1, 𝜆𝐶𝜑, is always power bounded, mean ergodic or uniformly mean ergodic. In the case, |λ|=1, we will see that it depends directly to the Denjoy-Wolff point of 𝜑.

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