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)


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].


Minimal-vector hyperinvariant subspace weighted composition operator 

Mathematics Subject Classification (2000)

Primary 47A15 47B33 


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  1. [1]
    S. Ansari and P. Enflo, Extremal vectors and invariant spaces, Trans. Amer. Math. Soc. 350 (1998), 539–558.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Isabelle Chalendar, Antoine Flattot, and Jonathan R. Partington, The method of minimal vectors applied to weighted composition operators, Oper. Theory Adv. Appl., 171 (2007), 89–105.CrossRefMathSciNetGoogle Scholar
  3. [3]
    Isabelle Chalendar and Jonathan R. Partington, Convergence properties of minimal vectors for normal operators and weighted shifts, Proc. Amer. Math. Soc. 133 (2004), 501–510.CrossRefMathSciNetGoogle Scholar
  4. [4]
    C.C. Cowen and B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995.Google Scholar
  5. [5]
    G. Koenigs, Recherches sur les intégrales de certaines équations fonctionnelles, Ann. Sci. Ecole Norm. Sup. (3) 1 (1884), Supplément, 3–41.Google Scholar
  6. [6]
    J.H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, New York, 1993.zbMATHGoogle Scholar
  7. [7]
    V.G. Troitsky, Minimal vectors in arbitrary Banach spaces, Proc. Amer. Math. Soc. 132 (2004), 1177–1180.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    G. Valiron, Sur l’itération des fonctions holomorphes dans un demi-plan, Bull. Sci. Math. (2) 55 (1931), 105–128.Google Scholar
  9. [9]
    Emilie B. Wiesner, Backward minimal points for bounded linear operators on finite-dimensional vector spaces, Linear Algebra Appl. 338 (2001), 251–259.zbMATHCrossRefMathSciNetGoogle Scholar

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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