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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)

Abstract

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.

Keywords

Weighted composition operator Invariant subspace Point spectrum Cyclic vector 

Mathematics Subject Classification (2000)

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

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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

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