Hyperinvariant subspaces for bilateral weighted shifts (Q1063840)

Let \(H\) be a complex Hilbert space with the orthonormal basis \(\{e_ n\}_{n\in \mathbb{Z}}\) and let \(U\) be an invertible bilateral weighted shift defined by \(Ue_ n = p_ n e_{n+1}\), \(p_ n>0\), \(n\in\mathbb{Z}\). Then \(U\) has a proper hyperinvariant subspace, if the set \([r_-, R_-] \cup [r_+, R_+]\) is not a singleton set included in \(\{r\} \cup \{R\}\), where \[ \begin{alignedat}{2} r_+&=\varliminf_{k\to\infty} (p_0 \ldots p_{k-}) ^{1/k},\quad& R_+&=\varlimsup_{k\to\infty}(p_0 \ldots p_{k-}) ^{1/k},\\ r_-&=\varliminf_{k\to\infty}(p_{-k} \ldots p_-) ^{1/k},\quad& R_-&=\varlimsup_{k\to\infty}(p_{-k} \ldots p_-)^{1/k},\\ r&=| U^{-1}|^{-1}_{sp},\quad& R&=| U|_{sp}. \end{alignedat} \] On the other side the used technique is an analysis of real sequences and has as byproduct a new proof of a related, on this area well known, theorem of Chevreau-Pearcy-Shields.