On spectra of composition operators

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

In this paper we consider composition operators Cφ on the Hilbert Hardy space over the unit disc, induced by analytic selfmaps φ. We use the fact that the operator C∗φCφ is asymptotically Toeplitz to obtain information on the essential spectrum and spectrum of Cϕ, which we are able to describe in select cases (including the case of some hypercyclic composition operators or that of composition operators with the property that the asymptotic symbol of C∗φCφ is constant a.e.). One of our tools is the Nikodym derivative of the pull-back measure induced by φ. An alternative formula for the essential norm of a composition operator (valid in select cases), in terms of the aforementioned Nikodym derivative, is established. Estimates of the spectra of adjoints of composition operators are obtained. Based on them, we describe the spectrum of composition operators induced by maps fixing a point, whose iterates exhibit a strong form of attractiveness to that point.

Original languageEnglish (US)
Pages (from-to)277-303
Number of pages27
JournalOperators and Matrices
Volume9
Issue number2
DOIs
StatePublished - Jun 1 2015

Fingerprint

Composition Operator
Hypercyclic Operators
Essential Norm
Derivative
Essential Spectrum
Pullback
Otto Toeplitz
Hardy Space
Iterate
Unit Disk
Hilbert
Valid
Alternatives
Operator
Estimate

Keywords

  • Composition operator
  • Spectrum

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory

Cite this

On spectra of composition operators. / Matache, Valentin.

In: Operators and Matrices, Vol. 9, No. 2, 01.06.2015, p. 277-303.

Research output: Contribution to journalArticle

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