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Interior functions and bicyclic vectors. (Fonctions intĂ©rieures 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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