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Images of Minimal-Vector Sequences Under Weighted Composition Operators on L2(\( \mathbb{D} \))

  • Paul S. Bourdon
  • Antoine Flattot
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 202)

Abstract

Let X be either the unit interval in ℝ or the unit disk \( \mathbb{D} \) in ℂ. Chalendar, Flattot, and Partington [2] study weighted composition operators Tw,ϒ on L2(X), where w ∈ L(X), ϒ : X → X is injective, and Tw,ϒf = wf for f ∈ L2(X). They introduce a (strict) partial order ≻ on X associated with Tw,ϒ and use it to obtain a sufficient condition for convergence of the sequence (Tn w,ϒyn) where (yn) is a backward minimal-vector sequence for Tw,ϒ. For the L2(\( \mathbb{D} \)) case, they give a detailed analysis of the situation where ϒ is linear-fractional. Through further study of the partial order ≻, we are able to generalize results from [2] that apply when ϒ is linear-fractional, replacing the linear-fractional hypotheses with univalence. In particular, our work yields generalizations of an invariant-subspace theorem in [2].

Keywords

Minimal-vector hyperinvariant subspace weighted composition operator 

Mathematics Subject Classification (2000)

Primary 47A15 47B33 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • Paul S. Bourdon
    • 1
  • Antoine Flattot
    • 2
  1. 1.Department of MathematicsWashington and Lee UniversityLexingtonUSA
  2. 2.Department of MathematicsUniversity of South CarolinaColumbiaUSA

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