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Archiv der Mathematik

, Volume 77, Issue 3, pp 253–264 | Cite as

Fonctions intérieures et vecteurs bicycliques

  • K. Kellay
Article
  • 38 Downloads

Abstract.

We consider weights \( \omega \) on \( \Bbb Z \) such that \( \omega(n)\to 0 \) as \( n\to +\infty \), \( \omega(n)\to +\infty \) as \( {n\to -\infty} \), and satisfying some regularity conditions. Set¶¶\(l^2_\omega = \{u = (u_n)_{n \in \Bbb Z} : \|u\|_\omega = ( \sum_{n\in\Bbb Z} |u_n|^2 \omega(n)^{2} t)^{1\over 2}<+{\infty}\}\) ¶¶and denote by \( S_\omega : (u_n)_{n\in\Bbb Z}\to(u_{n-1})_{n\in\Bbb Z} \) the usual shift on \( l^{2}_{\omega} \). We show that if¶¶\( \sum_{n\geqq1} {n\over {\rm{ln}}{\omega(-n)}} (2\omega(n)^{-2} - \omega(n-1)^{-2} - \omega(n+1)^{-2}) \) < \( +\infty \)¶¶then there exists a singular inner function U such that \( \widehat{U} = (\widehat{U}(n))_{n\geqq 0} \) is not bicyclic in \( l^{2}_{\omega} \), that is, the closure of Span\( \left\{{S^n_\omega}\widehat U : {n\in\Bbb Z}\right\} \) is a proper subspace of \( l^2_\omega \).

Keywords

Regularity Condition Proper Subspace Usual Shift 

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Copyright information

© Birkhäuser Verlag, Basel 2001

Authors and Affiliations

  • K. Kellay
    • 1
    • 2
  1. 1.Département de mathématiques et de statistique, Université Laval, Québec (Québec), Canada G1K 7P4Canada
  2. 2.Current address: CMI, LATP, Université de Provence, 39, rue F. Joliot–Curie, 13453 Marseille cedex 13, France, kellay@cmi.univ-mrs.frFrance

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