Sous-espaces biinvariants pour certains shifts pondérés

El-Fallah, O.; Kellay, Karim

Annales de l'Institut Fourier, Volume 48 (1998) no. 5, pp. 1543-1558.

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

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

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

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

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

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

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