Document Type : Research Paper

Authors

• Manisha Devi ¹
• Kuldip Raj ¹
• Ayhan Esi ²
• Muhammed Aiyub ³

¹ School of Mathematics Shri Mata Vaishno Devi University Katra-182320, J&K, India

² Engineering Faculty Malatya Turgut Ozal University Malatya,44040, Turkey

³ Department of Mathematics, Bahrain University, P. O. Box-32038, Bahrain

Abstract

Let $\mathbb{D}= \{\upsilon\in\mathbb{C}:|\upsilon|<1\}$ 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.

Keywords

• Dirichlet-type space
• Bloch-type spaces
• generalized weighted composition operator
• boundedness
• compactness