Sous-espaces biinvariants pour certains shifts pondérés

El-Fallah, O. ; Kellay, Karim

Annales de l'Institut Fourier, Tome 48 (1998), p. 1543-1558 / Harvested from Numdam

Résumé

Nous étudions les sous-espaces biinvariants du shift usuel sur les espaces à poids

Lω2=f∈L2(𝕋):∥f∥ω=∑n∈ℤ|f(n)|ω2(n)1/2<+∞,

où $\omega (n)=(1+n)^p, n\ge 0$ et ${\omega (n)\over (1+\vert n\vert )^p}{\mathrel {\mathop {\hspace{0.0pt}n\rightarrow -\infty }\limits ^{\rightarrow }}} +\infty $, pour un certain entier $p\ge 1$. Nous montrons que la trace analytique de tout sous-espace biinvariant est de type spectral, lorsque $\sum _{n\ge 2}{1\over n\log \omega (-n)}$ diverge, mais que ceci n’est plus valable lorsque $\sum _{n\ge 2}{1\over n\log \omega (-n)}$ converge.

Abstract

We study the biinvariant subspaces for the usual shift on the weighted spaces

Lω2={f∈L2(𝕋):∥f∥ω=∑n∈ℤ|f(n)|ω2(n)1/2<+∞},

where $\omega (n)=(1+n)^p, n\ge 0$ and ${\omega (n)\over (1+\vert n\vert )^p}{\mathrel {\mathop {\hspace{0.0pt}n\rightarrow -\infty }\limits ^{\rightarrow }}} +\infty $ for some integer $p\ge 1$. We show that the analytic part of all biinvariant subspaces is spectral if $\sum _{n\ge 2}{1\over n\log \omega (-n)}$ diverges, but that this does not hold when $\sum _{n\ge 2}{1\over n\log \omega (-n)}$ converges.

Détail

Publié le : 1998-01-01

MR 99k:47012 | Zbl 0919.47020 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.1666

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