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Differences of Stević–Sharma operators

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Abstract

A generalization of the products of composition, multiplication and differentiation operators is the Stević–Sharma operator \(T_{u_1,u_2,\varphi }\), defined by \(T_{u_1,u_2,\varphi }f=u_1\cdot f\circ \varphi +u_2\cdot f'\circ \varphi \), where \(u_1,u_2,\varphi \) are holomorphic functions on the unit disk \({\mathbb {D}}\) in the complex plane \({\mathbb {C}}\) and \(\varphi ({\mathbb {D}})\subset {\mathbb {D}}\). We are interested in the difference of Stević–Sharma operators which has never been considered so far. In this paper, we characterize its boundedness, compactness and order boundedness between Banach spaces of holomorphic functions. As an important special case, we obtain the above characterizations of the difference of weighted composition operators. Furthermore, we show the equivalence of order boundedness and Hilbert-Schmidtness for the difference of composition operators between Hardy or weighted Bergman spaces.

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Acknowledgements

This work was supported by NSF of China (Nos. 11771340, 11431011).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Shuming Wang, Maofa Wang & **n Guo

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  1. Shuming Wang
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  2. Maofa Wang
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  3. **n Guo
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Correspondence to **n Guo.

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Communicated by Jari Taskinen.

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Wang, S., Wang, M. & Guo, X. Differences of Stević–Sharma operators. Banach J. Math. Anal. 14, 1019–1054 (2020). https://doi.org/10.1007/s43037-019-00051-z

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  • DOI: https://doi.org/10.1007/s43037-019-00051-z

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