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 ${\mathcal{W}}_{\varphi ,\xi }^n$ and is  defined by
$${\mathcal{W}}_{\varphi ,\xi }^{n}f(\upsilon )=\varphi (\upsilon )f^{(n)}(\xi (\upsilon )),f\in H(\mathbb{D});\upsilon \in \mathbb{D},$$
where $\varphi \in H(\mathbb{D})$ and $\xi$ is a holomorphic self-map of $\mathbb{D}$. This operator is basically the combination of multiplication operator $M_{\varphi }$, 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