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Differences of differential operators between weighted-type spaces

Mohammed Said Al Ghafri|Jasbir Singh Manhas
대한수학회 (The Korean Mathematical Society)|2021.07|대한수학회논문집 (Communications of the KMS)|제36권 제3호|pp. 465-483|DOI 10.4134/CKMS.c200095

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키워드

multiplication operatorsDifference operatorsdifferential operatorsweighted-type spacesBloch-type spacesbounded and compact operators
주제분류수학

초록

Let $\mathcal{H}(\mathbb{D})$ be the space of analytic functions on the unit disc $\mathbb{D}$. Let $\psi=(\psi_j)_{j=0}^n$ and $\Phi=(\Phi_j)_{j=0}^n$ be such that $\psi_j, \Phi_j \in \mathcal{H}(\mathbb{D})$. The linear differential operator is defined by $ T_{\psi}(f)=\sum_{j=0}^n \psi_j f^{(j)},$ $f\in \mathcal{H}(\mathbb{D})$. We characterize the boundedness and compactness of the difference operator $ (T_{\psi}-T_{\Phi})(f)=\sum_{j=0}^n \left(\psi_j-\Phi_j\right) f^{(j)}$ between weighted-type spaces of analytic functions. As applications, we obtained boundedness and compactness of the difference of multiplication operators between weighted-type and Bloch-type spaces. Also, we give examples of unbounded (non compact) differential operators such that their difference is bounded (compact).

Abstract

Let $\mathcal{H}(\mathbb{D})$ be the space of analytic functions on the unit disc $\mathbb{D}$. Let $\psi=(\psi_j)_{j=0}^n$ and $\Phi=(\Phi_j)_{j=0}^n$ be such that $\psi_j, \Phi_j \in \mathcal{H}(\mathbb{D})$. The linear differential operator is defined by $ T_{\psi}(f)=\sum_{j=0}^n \psi_j f^{(j)},$ $f\in \mathcal{H}(\mathbb{D})$. We characterize the boundedness and compactness of the difference operator $ (T_{\psi}-T_{\Phi})(f)=\sum_{j=0}^n \left(\psi_j-\Phi_j\right) f^{(j)}$ between weighted-type spaces of analytic functions. As applications, we obtained boundedness and compactness of the difference of multiplication operators between weighted-type and Bloch-type spaces. Also, we give examples of unbounded (non compact) differential operators such that their difference is bounded (compact).

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