Multivariable Weighted Composition Operators: Lack of Point Spectrum, and Cyclic Vectors

  • Isabelle Chalendar
  • Elodie Pozzi
  • Jonathan R. Partington
Part of the Operator Theory: Advances and Applications book series (OT, volume 202)


We study weighted composition operators T α,ω on L 2([0, 1] d ) where d ≥ 1, defined by
$$ T_{\alpha ,\omega } f(x_1 , \ldots ,x_d ) = \omega (x_1 , \ldots ,x_d )f(\{ x_1 + \alpha _1 \} , \ldots ,\{ x_d + \alpha _d \} ), $$
where α=(α1,...α d )∈ℝ d and where. denotes the fractional part.

In the case where a is an irrational vector, we give a new and larger class of weights ω for which the point spectrum of T α,ω is empty. In the case of α∈ℚ d and ω(x 1, ..., x d) = x 1 ...x d, we give a complete characterization of the cyclic vectors of T α,ω.


Weighted composition operator Invariant subspace Point spectrum Cyclic vector 

Mathematics Subject Classification (2000)

Primary: 47A15 47A10 47A16 Secondary: 47B33 47A35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    G. Androulakis and A. Flattot. Hyperinvariant subspace for weighted composition operator on L p[0, 1]d). J. Operator Theory, to appear.Google Scholar
  2. [2]
    D.P. Blecher and A.M. Davie. Invariant subspaces for an operator on L 2(II) composed of a multiplication and a translation. J. Operator Theory, 23 (1990), no. 1, 115–123.zbMATHMathSciNetGoogle Scholar
  3. [3]
    I. Chalendar, A. Flattot, and N. Guillotin-Plantard. On the spectrum of multivariable weighted composition operators. Arch. Math. (Basel), 90 (2008), no. 4, 353–359.zbMATHMathSciNetGoogle Scholar
  4. [4]
    I. Chalendar and J.R. Partington. The cyclic vectors and invariant subspaces of rational Bishop operators. Research report, 2008.Google Scholar
  5. [5]
    I. Chalendar and J.R. Partington. Invariant subspaces for products of Bishop operators. Acta Sci. Math. (Szeged), 74 (2008), 717–725.MathSciNetGoogle Scholar
  6. [6]
    C.C. Cowen and B.D. MacCluer, Composition operators on spaces of analytic functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995.Google Scholar
  7. [7]
    A.M. Davie. Invariant subspaces for Bishop’s operators. Bull. London Math. Soc., 6 (1974), 343–348.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    A.V. Lipin, Spectral multiplicity of the solutions of polynomial operator equations. J. Soviet Math. 44 (1989), no. 6, 856–861zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    G.W. MacDonald. Invariant subspaces for Bishop-type operators. J. Funct. Anal., 91 (1990), no. 2, 287–311.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    G.W. MacDonald, Invariant subspaces for multivariate Bishop-type operators. J. Operator Theory 25 (1991), no. 2, 347–366.zbMATHMathSciNetGoogle Scholar
  11. [11]
    J.H. Shapiro, Composition operators and classical function theory. Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • Isabelle Chalendar
    • 1
  • Elodie Pozzi
    • 1
  • Jonathan R. Partington
    • 2
  1. 1.Isabelle Chalendar and Elodie Pozzi Université Lyon 1 INSA de Lyon, Ecole Centrale de Lyon CNRS, UMR 5208Institut Camille JordanVilleurbanne CedexFrance
  2. 2.School of MathematicsUniversity of LeedsLeedsUK

Personalised recommendations