Sous-espaces biinvariants pour certains shifts pondérés
Annales de l'Institut Fourier, Volume 48 (1998) no. 5, p. 1543-1558

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

${L}_{\omega }^{2}=\left\{f\in {L}^{2}\left(𝕋\right):\parallel f{\parallel }_{\omega }={\left(\sum _{n\in ℤ}|f\left(n\right)|{\omega }^{2}\left(n\right)\right)}^{1/2}<+\infty \right\},$

where $\omega \left(n\right)=\left(1+n{\right)}^{p},n\ge 0$ and $\frac{\omega \left(n\right)}{\left(1+|n|{\right)}^{p}}\stackrel{\to }{\phantom{\rule{0.0pt}{0ex}}n\to -\infty }+\infty$ for some integer $p\ge 1$. We show that the analytic part of all biinvariant subspaces is spectral if ${\sum }_{n\ge 2}\frac{1}{nlog\omega \left(-n\right)}$ diverges, but that this does not hold when ${\sum }_{n\ge 2}\frac{1}{nlog\omega \left(-n\right)}$ converges.

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

${L}_{\omega }^{2}=\left\{f\in {L}^{2}\left(𝕋\right):\parallel f{\parallel }_{\omega }={\left(\sum _{n\in ℤ}|f\left(n\right)|{\omega }^{2}\left(n\right)\right)}^{1/2}<+\infty \right\},$

$\omega \left(n\right)=\left(1+n{\right)}^{p},n\ge 0$ et $\frac{\omega \left(n\right)}{\left(1+|n|{\right)}^{p}}\stackrel{\to }{\phantom{\rule{0.0pt}{0ex}}n\to -\infty }+\infty$, pour un certain entier $p\ge 1$. Nous montrons que la trace analytique de tout sous-espace biinvariant est de type spectral, lorsque ${\sum }_{n\ge 2}\frac{1}{nlog\omega \left(-n\right)}$ diverge, mais que ceci n’est plus valable lorsque ${\sum }_{n\ge 2}\frac{1}{nlog\omega \left(-n\right)}$ converge.

@article{AIF_1998__48_5_1543_0,
author = {El-Fallah, O. and Kellay, Karim},
title = {Sous-espaces biinvariants pour certains shifts pond\'er\'es},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {48},
number = {5},
year = {1998},
pages = {1543-1558},
doi = {10.5802/aif.1666},
zbl = {0919.47020},
mrnumber = {99k:47012},
language = {fr},
url = {http://www.numdam.org/item/AIF_1998__48_5_1543_0}
}

El-Fallah, O.; Kellay, Karim. Sous-espaces biinvariants pour certains shifts pondérés. Annales de l'Institut Fourier, Volume 48 (1998) no. 5, pp. 1543-1558. doi : 10.5802/aif.1666. http://www.numdam.org/item/AIF_1998__48_5_1543_0/

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