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

Abstract

<abstract><p>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</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ 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}. $\end{document} </tex-math></disp-formula></p> <p>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.</p></abstract>