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Interior functions and bicyclic vectors. (Fonctions intérieures et vecteurs bicycliques.) (French) Zbl 1021.47005
Summary: We consider weights $$\omega$$ on $$\mathbb{Z}$$ such that $$\omega(n)\to 0$$ as $$n\to +\infty$$, $$\omega(n) \to+\infty$$ as $$n\to-\infty$$, and satisfying some regularity conditions. Setting $l^2_\omega= \left\{u= (u_n)_{n \in\mathbb{Z}}: \|u\|_\omega= \left(\sum_{n\in\mathbb{Z}} |u_n|^2 \omega (n)^2 \right)^{1/2}<+ \infty\right\},$ and denoting by $$S_\omega: (u_n)_{n\in \mathbb{Z}} \to(u_{n-1})_{n \in\mathbb{Z}}$$ the usual shift on $$l^2_\omega$$, we show that if $\sum_{n\geq 1}{n\over \ln\omega (-n)}\bigl(2 \omega(n)^{-2}- \omega(n-1)^{-2}- \omega(n+1)^{-2}) <+\infty,$ then there exists a singular inner function $$U$$ such that $$\widehat U=(\widehat U(n))_{n\geq 0}$$ is not bicyclic in $$l^2_\omega$$, that is, the closure of $$\text{span} \{S^n_\omega\widehat U:n\in \mathbb{Z}\}$$ is a proper subspace of $$l^2_\omega$$.

##### MSC:
 47A16 Cyclic vectors, hypercyclic and chaotic operators 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47A15 Invariant subspaces of linear operators
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