Archiv der Mathematik

, Volume 77, Issue 3, pp 253–264

# Fonctions intérieures et vecteurs bicycliques

• K. Kellay
Article

## 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