Differences weighted composition operators acting between kind of weighted Bergman-type spaces and the Bers-type space -I-

Aydah Mohammed Ayed Al-Ahmadi; Aydah Mohammed Ayed Al-Ahmadi

doi:10.3934/math.2023831

AIMS Mathematics

2023, Volume 8, Issue 7: 16240-16251. doi: 10.3934/math.2023831

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Research article

Differences weighted composition operators acting between kind of weighted Bergman-type spaces and the Bers-type space -I-

Aydah Mohammed Ayed Al-Ahmadi ^,

Department of Mathematics, College of Sciences, Jouf University, Saudi Arabia

Received: 23 February 2023 Revised: 23 April 2023 Accepted: 26 April 2023 Published: 06 May 2023
MSC : 47B38

Let $ \mathcal{O}(\mathbb{D}) $ denote the class of all analytic or holomorphic functions on the open unit disk $ \mathbb{D} $ of $ \mathbb{C} $. Let $ \varphi $ and $ \psi $ are an analytic self-maps of $ \mathbb{D} $ and $ u, v\in \mathcal{O}(\mathbb{D}). $ The difference of two weighted composition operators is defined by

$ T_{\varphi, \psi}f(z): = \bigg(W_{\varphi, \;u}f- W_{\psi, \;v}f\bigg)(z) = u (z)(f\circ \varphi)(z) -v(z)(f\circ \psi)(z), \ f \in \mathcal{O}(\mathbb{D})\;\hbox{and}\;z\in \mathbb{D}. $

The boundedness and compactness of the differences of two weighted composition operators from $ {\cal H}_{\alpha}^\infty(\mathbb{D}) $ spaces into $ \mathcal{N}_{K}(\mathbb{D}) $ spaces (resp. from $ \mathcal{N}_{K}(\mathbb{D}) $ into $ {\cal H}_{\alpha}^\infty (\mathbb{D}) $) are investigate in this paper.
- weighted composition operator,
- compact difference bounded operator,
- weighted analytic space
Citation: Aydah Mohammed Ayed Al-Ahmadi. Differences weighted composition operators acting between kind of weighted Bergman-type spaces and the Bers-type space -I-[J]. AIMS Mathematics, 2023, 8(7): 16240-16251. doi: 10.3934/math.2023831

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Abstract

Let $ \mathcal{O}(\mathbb{D}) $ denote the class of all analytic or holomorphic functions on the open unit disk $ \mathbb{D} $ of $ \mathbb{C} $. Let $ \varphi $ and $ \psi $ are an analytic self-maps of $ \mathbb{D} $ and $ u, v\in \mathcal{O}(\mathbb{D}). $ The difference of two weighted composition operators is defined by

$ T_{\varphi, \psi}f(z): = \bigg(W_{\varphi, \;u}f- W_{\psi, \;v}f\bigg)(z) = u (z)(f\circ \varphi)(z) -v(z)(f\circ \psi)(z), \ f \in \mathcal{O}(\mathbb{D})\;\hbox{and}\;z\in \mathbb{D}. $

The boundedness and compactness of the differences of two weighted composition operators from $ {\cal H}_{\alpha}^\infty(\mathbb{D}) $ spaces into $ \mathcal{N}_{K}(\mathbb{D}) $ spaces (resp. from $ \mathcal{N}_{K}(\mathbb{D}) $ into $ {\cal H}_{\alpha}^\infty (\mathbb{D}) $) are investigate in this paper.

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