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Linear Combination of Composition Operators on Bergman and Korenblum Spaces

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Abstract

Motivated by the recent work of Galindo et al. (J. Funct. Anal. 265, 629–643, 2013), in this paper, we give an elegant compactness criterion for any finite linear combination of composition operators on the weighted Bergman space in terms of power type characterization. More precisely, let T be any finite linear combination of composition operators, then

$$ T \ \text{is compact on} \ {A}^{p}_{\alpha}(\mathbf{D}) \ \text{if and only if} \ \lim\limits_{n\rightarrow\infty}\frac{ \| T z^{n}\|_{A^{-\gamma}} }{ \| z^{n}\|_{A^{-\gamma}} }=0, $$

which reveals that the compactness of T on the weighted Bergman space \(A^{p}_{\alpha }(\mathbf {D})\) and Korenblum space A−γ(D) are equivalent.

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Acknowledgments

The authors are sincerely grateful to the anonymous referees for their careful reading of the initial version of the manuscript and helpful suggestions. It is a pleasure to express our gratitude to professors Ruhan Zhao and Christopher Hammond for their valuable discussions on the example in Introduction.

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

    Xin Guo & Maofa Wang

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  1. Xin Guo
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  2. Maofa Wang
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Correspondence to Xin Guo.

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It was partially supported by National Science Foundation of China (11771340).

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Guo, X., Wang, M. Linear Combination of Composition Operators on Bergman and Korenblum Spaces. Potential Anal 57, 305–326 (2022). https://doi.org/10.1007/s11118-021-09917-0

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