Generalized weighted composition operators acting between Dirichlet-type spaces and Bloch-type spaces

Let \mathbb{D}= \{\upsilon\in\mathbb{C}:|\upsilon|<۱\} be the open unit disk in the complex plane \mathbb{C} and let H(\mathbb{D}) be the space of all holomorphic functions on  \mathbb{D}. For a non-negative integer n and a function f \in H(\mathbb{D}), the n^{th}- order differentiation operator is defined as D^n f = f^{(n)}. The weighted composition operator together with n^{th}- order differentiation operator give rise to a new operator generally termed as generalized weighted composition operator denoted by \mathcal{W}^{n}_{\phi,\xi} and is  defined by\begin{equation*}\mathcal{W}^{n}_{\phi,\xi}f(\upsilon)  =\phi(\upsilon)f^{(n)}(\xi(\upsilon)),\quad f\in H(\mathbb{D}); \upsilon\in%\mathbb{D},\end{equation*}where \phi\in H(\mathbb{D}) and \xi is a holomorphic self-map of \mathbb{D}. This operator is basically the combination of multiplication operator M_{\phi}, composition operator  C_{\xi} and n^{th}- order differentiation operator D^{n}. We study the boundedness and compactness of this operator between Dirichlet-type spaces and Bloch-type spaces.

کلیدواژه ها:

Dirichlet-type space ، Bloch-type spaces ، generalized weighted composition operator ، boundedness ، Compactness