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Surjectivity of isometries between weighted spaces of holomorphic functions

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Abstract

We examine the surjectivity of isometries between weighted spaces of holomorphic functions. We show that for certain classical weights on the open unit disc all isometries of the weighted space of holomorphic functions, \({ {\mathcal {H}}}_{v_o}( \varDelta )\), are surjective. Criteria for surjectivity of isometries of \({ \mathcal H}_v(U)\) in terms of a separation condition on points in the image of \({ {\mathcal {H}}}_{v_o}(U)\) are also given for U a bounded open set in \({\mathbb {C}}\). Considering the weight \(v(z)= 1-|z|^2\) and the isomorphism \(f\mapsto f'\) we are able to show that all isometries of the little Bloch space are surjective.

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Acknowledgements

The authors wish to thank Richard Smith for his reference to the Invariance of Domains, Joseph Cima for his correspondence regarding [10] and Manuel Maestre and Jose Bonet for the alternative proof of Theorem 4 and their advice regarding the paper. The second author is supported by the Ministerio de Economía y Competitividad and FEDER, project MTM2016-77054-C2-1-P.

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  1. School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland

    Christopher Boyd

  2. Departamento de Análisis Matemático, Facultad de Matemáticas, Universitat de València, Burjassot, 46100, Valencia, Spain

    Pilar Rueda

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  1. Christopher Boyd
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  2. Pilar Rueda
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Correspondence to Pilar Rueda.

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Boyd, C., Rueda, P. Surjectivity of isometries between weighted spaces of holomorphic functions. RACSAM 113, 2461–2477 (2019). https://doi.org/10.1007/s13398-018-00617-w

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