Eigenvectors and cyclic vectors of bilateral weighted shifts. II: Simply invariant subspaces
An injective bilateral weighted shift T on \(\ell^ 2\) is unitarily equivalent to multiplication by the variable on a space \(L^ 2(\beta)\) of formal Laurent series. By means of this aspect the following question raised by B. Chevreau is investigated: If \(\sigma_ p(T^*)\) is a circle, can one conclude that f is not in the closed linear span of \(\{T^ nf:n=1,2,...\}\) for any \(f\neq 0?\) One result is, that in case \(L^ 2(\beta)\) consists only of quasi- analytic functions on the boundary \(\partial D\) of the unit disc, then \(\partial D\subset \sigma_ p(T^*)\), and all cyclic invariant subspaces \({\mathcal M}\) have the property \(\cap \{T^ n{\mathcal M}:n=0,1,2,...\}=\{0\}\), and hence Chevreau's conclusion holds. Some corollaries and examples are given and it is demonstrated, that the condition of quasi-analyticity is essential. [For part I see Revista Un. Math. Argentina 26, 24-41 (1972; Zbl 0249.47024)].
quasi-analytic functions on the boundary of the unit disc
scientific article; zbMATH DE number 3842550