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A note about the spectrum of composition operators induced by a rotation

  • José BonetEmail author
Original Paper
  • 22 Downloads

Abstract

A characterization of those points of the unit circle which belong to the spectrum of a composition operator \(C_{\varphi }\), defined by a rotation \(\varphi (z)=rz\) with \(|r|=1\), on the space \(H_0(\mathbb {D})\) of all analytic functions which vanish at 0, is given. Examples show that the spectrum of \(C_{\varphi }\) need not be closed. In these examples the spectrum is dense but point 1 may or may not belong to it, and this is related to Diophantine approximation.

Keywords

Composition operator Space of analytic functions Rotation Diophantine number 

Mathematics Subject Classification

47B33 47A10 46E10 11K60 

Notes

Acknowledgements

The author is grateful to J.L. Varona for providing him with useful references about number theory.

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Copyright information

© The Royal Academy of Sciences, Madrid 2020

Authors and Affiliations

  1. 1.Instituto Universitario de Matemática Pura y Aplicada IUMPAUniversitat Politècnica de ValènciaValenciaSpain

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