Polynomial differentiation composition operators from weighted Dirichlet spaces to weighted Bloch spaces

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Abstract

Let H($\mathbb{D}$) be the class of all analytic functions on $\mathbb{D}$, ϕ be an analytic self-map of $\mathbb{D}$, u₀,u₁,...,u_n ∈ H($\mathbb{D}$), and $\vec{u}=\left(u_{0}, u_{1}, \ldots, u_{n}\right)$. In this paper, we studied the operators

$P_{\vec{u}, \varphi}^{n}=\sum_{j=0}^{n} D_{u_{j}, \varphi}^{j},$

where $D_{u_{j}, \varphi}^{j}$ is the generalized weighted composition operator

$D_{u_{j}, \varphi}^{j} f=u_{j} \cdot f^{(j)} \circ \varphi, f \in H(\mathbb{D}) .$

To be precise, we characterized the boundedness and compactness of the operators $P_{\vec{u}, \varphi}^{n}$ acting from the weighted Dirichlet spaces $\mathscr{D}_{\alpha}^{p}$ to the weighted Bloch spaces for p ≥ 1, and we also calculated the essential norm of such operators for p > 1.

Keywords

weighted Dirichlet space / weighted Bloch space / polynomial differentiation composition operator / boundedness / essential norm / compactness

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