Compactness of the differences of weighted composition operators from weighted Bergman spaces to weighted-type spaces on the unit ball (Q449588)

Let \(B\) denote the open unit ball in the complex vector space \(\mathbb C^{N}\) and \(H(B)\) denotes the space of all holomorphic functions on \(B\). Let \(v\) be a positive continuous function on \(B\). The weighted-type space \(H_{v}^{\infty}(B)\) consists of all \(f\in H(B)\) such that \(\| f\|_{H_{v}^{\infty}}=\sup_{z\in B}v(z)|f(z)|<\infty\).NEWLINENEWLINE With the norm \(\| \cdot \|_{H_{v}^{\infty}}\), \(H_{v}^{\infty}\) is a Banach space. For \(\varphi:B\to B\) a holomorphic self-map of \(B\) and \(u\in H(B)\), the weighted composition operator \(W_{\varphi,u}\) on \(H(B)\) is defined by \(W_{\varphi,u}f(z)=u(z)f(\varphi(z))\), \(z\in B\).NEWLINENEWLINEIn the present paper, the authors characterize the compactness of differences of weighted composition operators acting from the weighted Bergman space to the weighted-type space on the unit ball. For the case \(1<p<\infty\), they also find an asymptotically equivalent expression to the essential norm of these operators. The presentation of this paper is clear, and the results are new and interesting. They may be helpful for researchers working in related areas.