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The double commutant property for composition operators

  • Miguel LacruzEmail author
  • Fernando León-Saavedra
  • Srdjan Petrovic
  • Luis Rodríguez-Piazza
Article
  • 18 Downloads

Abstract

We investigate the double commutant property for a composition operator \(C_\varphi \), induced on the Hardy space \(H^2({\mathbb {D}})\) by a linear fractional self-map \(\varphi \) of the unit disk \({\mathbb {D}}.\) Our main result is that this property always holds, except when \(\varphi \) is a hyperbolic automorphism or a parabolic automorphism. Further, we show that, in both of the exceptional cases, \(\{C_\varphi \}^{\prime \prime }\) is the closure of the algebra generated by \(C_\varphi \) and \(C_\varphi ^{-1},\) either in the weak operator topology, if \(\varphi \) is a hyperbolic automorphism, or surprisingly, in the uniform operator topology, if \(\varphi \) is a parabolic automorphism. Finally, for each type of a linear fractional mapping, we settle the question when any of the algebras involved are equal.

Keywords

Composition operator Hardy space Linear fractional map Double commutant property 

Notes

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© Universitat de Barcelona 2019

Authors and Affiliations

  1. 1.Departamento de Análisis Matemático and Instituto de MatemáticasFacultad de Matemáticas, Universidad de Sevilla, Campus Reina MercedesSevilleSpain
  2. 2.Departamento de MatemáticasUniversidad de Cádiz, Avenida de la UniversidadJerez de la FronteraSpain
  3. 3.Department of MathematicsWestern Michigan UniversityKalamazooUSA

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