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

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Raj Kuldip; Devi Manisha; Esi Ayhan; Aiyub Muhammed

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Paper Information

Journal: INTERNATIONAL JOURNAL OF NONLINEAR ANALYSIS AND APPLICATIONS Year:2025 | Volume:16 | Issue:1 Page(s): 1-10

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Title

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

Author(s)

Raj Kuldip | Devi Manisha | Esi Ayhan | Aiyub Muhammed | Issue Writer Certificate

Keywords

Dirichlet-type space

Bloch-type spaces

generalized weighted composition operator

boundedness

Compactness

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.

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