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

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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.

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

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Correspondence to José Bonet.

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The research of this paper was partially supported by the projects MTM2016-76647-P and GV Prometeo/2017/102.

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Bonet, J. A note about the spectrum of composition operators induced by a rotation. RACSAM 114, 63 (2020).

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  • Composition operator
  • Space of analytic functions
  • Rotation
  • Diophantine number

Mathematics Subject Classification

  • 47B33
  • 47A10
  • 46E10
  • 11K60