On the Li-Stević Integral Type Operators from Weighted Bergman Spaces into β-Zygmund Spaces

Abstract

For an analytic self-map φ of the unit disk \({\mathbb{D}}\) and an analytic function g on \({\mathbb{D}}\), we define the following integral type operators: $$T_{\varphi}^{g}f(z) := \int_{0}^{z} f(\varphi(\zeta))g(\zeta) d\zeta\quad {\rm and}\quad C_{\varphi}^{g}f(z) := \int_{0}^{z}f^{\prime}(\varphi(\zeta))g(\zeta) d\zeta$$ . We give a characterization for the boundedness and compactness of these operators from the weighted Bergman space \({L_{a}^p(dA_{\alpha})}\) into the β-Zygmund space \({\mathcal{Z}_{\beta}}\). We will also estimate the essential norm of these type of operators. As an application of results, we characterize the above operator-theoretic properties of Volterra type integral operators and composition operators.