Sums of weighted differentiation composition operators from weighted Bergman spaces to weighted Zygmund and Bloch-type spaces

Abstract

Let \({\mathcal {H}}({\mathbb {D}})\) be the space of analytic functions on the unit disc \({\mathbb {D}}\) and let \({\mathcal {S}}({\mathbb {D}})\) denote the set of all analytic self maps of the unit disc \({\mathbb {D}}\). Let \(\Psi =(\psi _j)_{j=0}^k\) be such that \(\psi _j\in {\mathcal {H}}({\mathbb {D}})\) and \(\varphi \in {\mathcal {S}}({\mathbb {D}})\). To treat the Stević–Sharma type operators and the products of composition operators, multiplication operators, differentiation operators in a unified manner, Wang et al. considered the following sum operator:

$$\begin{aligned} T_{\Psi ,\varphi }^kf= \sum \limits _{j=0}^k\psi _j\cdot f^{(j)}\circ \varphi = \sum \limits _{j=0}^k{\mathfrak {D}}_{\psi _j,\varphi }^jf, \quad f\in {\mathcal {H}}({\mathbb {D}}). \end{aligned}$$

We characterize the boundedness and compactness of the operators \(T_{\Psi ,\varphi }^k\) from the weighted Bergman spaces \(A_{v,p}\) to the weighted Zygmund-type spaces \({\mathcal {Z}}_w\) and the weighted Bloch-type spaces \({\mathcal {B}}_w\). Besides, giving examples of bounded, unbounded, compact and non-compact operators \(T_{\Psi ,\varphi }^k\), we give an example of two unbounded weighted differentiation composition operators \({\mathfrak {D}}_{\psi _0,\varphi }^0, \ {\mathfrak {D}}_{\psi _1,\varphi }^1:A_{v,p}\longrightarrow {\mathcal {Z}}_w( {\mathcal {B}}_w)\) such that their sum operator \({\mathfrak {D}}_{\psi _0,\varphi }^0+ {\mathfrak {D}}_{\psi _1,\varphi }^1= T_{\Psi ,\varphi }^1:A_{v,p}\longrightarrow {\mathcal {Z}}_w( {\mathcal {B}}_w)\) is bounded.

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Acknowledgements

The authors would like to thank the anonymous referee for his careful reading of the manuscript and providing valuable suggestions which help in improving the original manuscript. J. S. Manhas is supported by SQU Grant no. IG/SCI/MATH/20/08.

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Correspondence to Jasbir S. Manhas.

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Communicated by Eva A. Gallardo-Gutierrez.

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Manhas, J.S., Al Ghafri, M.S. Sums of weighted differentiation composition operators from weighted Bergman spaces to weighted Zygmund and Bloch-type spaces. Adv. Oper. Theory 6, 51 (2021). https://doi.org/10.1007/s43036-021-00147-0

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Keywords

  • Weighted composition operators
  • Weighted differentiation composition operators
  • Weighted Bloch spaces
  • Weighted Zygmund spaces
  • Weighted Bergman spaces
  • Bounded and compact operators

Mathematics Subject Classification

  • 47B33; 47B38
  • 46E15