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Weighted composition followed by differentiation between Bloch-type spaces

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Abstract

We investigate the behavior of \(DuC_\varphi :\mathcal {B}_\alpha \rightarrow \mathcal {B}_\beta \), that is, the product of a weighted composition operator \(uC_\varphi \) and the differentiation operator \(D\), between Bloch-type spaces with standard weights. For all \(0<\alpha ,\beta <\infty \), we characterize the boundedness and estimate the essential norm of \(DuC_\varphi \) in terms of the analytic function \(u:\mathbb {D} \rightarrow \mathbb {C}\) and the symbol function \(\varphi :\mathbb {D} \rightarrow \mathbb {D}\). As a corollary, we characterize the compactness of \(DuC_\varphi \).

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Acknowledgments

We are grateful to Professor Mikael Lindström for his helpful comments and suggestions on how to improve the paper. We would also like to thank the referees for comments concerning the background.

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Authors and Affiliations

  1. Department of Mathematical Sciences, P.O. Box 3000, 90014 , University of Oulu, Finland

    Olli Hyvärinen & Ilmari Nieminen

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  1. Olli Hyvärinen
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  2. Ilmari Nieminen
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Correspondence to Olli Hyvärinen.

Additional information

The research of O. Hyvärinen and I. Nieminen has been supported by grants from the Emil Aaltonen Foundation and the Väisälä Foundation, respectively.

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Hyvärinen, O., Nieminen, I. Weighted composition followed by differentiation between Bloch-type spaces. Rev Mat Complut 27, 641–656 (2014). https://doi.org/10.1007/s13163-013-0138-y

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  • DOI: https://doi.org/10.1007/s13163-013-0138-y

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