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Surjectivity of isometries between weighted spaces of holomorphic functions

  • Christopher Boyd
  • Pilar RuedaEmail author
Original Paper
  • 13 Downloads

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.

Keywords

Weighted spaces Surjectivity of isometries Bloch space Analytic functions 

Mathematics Subject Classification

Primary 46E15 46B04 Secondary 46E10 46B20 

Notes

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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Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity College DublinDublin 4Ireland
  2. 2.Departamento de Análisis Matemático, Facultad de MatemáticasUniversitat de ValènciaValenciaSpain

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