Compact and order bounded sum of weighted differentiation composition operators

TL;DR

This paper characterizes bounded, compact, and order bounded sums of weighted differentiation composition operators from Bergman type spaces to weighted Banach spaces of analytic functions, providing a comprehensive analysis of their properties.

Contribution

It introduces new characterizations for the boundedness, compactness, and order boundedness of sums of weighted differentiation composition operators.

Findings

Characterization of boundedness conditions

Criteria for compactness of the operators

Conditions for order boundedness

Abstract

In this paper, we characterize bounded, compact and order bounded sum of weighted differentiation composition operators from Bergman type spaces to weighted Banach spaces of analytic functions, where the sum of weighted differentiation composition operators is defined as $$ S^{n}_{\vec{u},\tau}(f)= \displaystyle\sum_{j=0}^{n}D_{u_{j} ,\tau}^{j}(f), \; \; f \in \mathcal{H}(\mathbb D).$$ Here $\mathcal{H}(\mathbb D)$ is the space of all holomorphic functions on $\mathbb D$, $\vec{u}=\{u_{j}\}_{j=0}^{n}$, $u_{j} \in \mathcal{H}(\mathbb{D})$, $\tau$ a holomorphic self-map of $\mathbb D$, $f^{(j)}$ the $j$th derivative of $f$ and weighted differentiation composition operator $D_{u_{j},\tau}^{j}$ is defined as $D_{u_{j},\tau}^{j}(f)=u_{j}C_{\tau}D^{j}(f)=u_{j}f^{(j)}\circ\tau, \; \; f \in \mathcal{H}(\mathbb D).$