Cyclicity of bicyclic operators
Given an injective operator on a separable Banach space \(X\), let \(R^\infty(T)\) be the intersection of the ranges of all operators \(T^n\) (\(n\geq 1\)). The operator \(T\) is called \textit{bicyclic} if there is \(x_0\in R^\infty(T)\) such that the linear span of \(\{T^nx_0:\) \(n\in{\mathbb Z}\}\) is dense in \(X\). If there is a vector \(x_0\in X\) such that the linear span of the orbit \(\{T^nx_0:\) \(n\geq0\}\) is dense in \(X\), the operator \(T\) is said to be \textit{cyclic}. In the paper under review, the authors prove, under suitable conditions on the norms \(\| T^nx_0\| \) (\(n\in{\mathbb Z}\)), that a bicyclic operator is, in fact, cyclic. As an application, the authors characterize a special class of weights \(\omega\) such that the backward shift operator is cyclic on \(\ell^2_\omega(\mathbb Z)\). In addition, they also establish that if \(S\) is a bounded invertible shift on \(\ell^2_\omega(\mathbb Z)\), under some condition on the asymptotic behavior of \(\omega\), either \(S\) or \(S^*\) is cyclic, thus answering, in the affirmative, a question proposed by \textit{D.\,A.\thinspace Herrero} [Rev.\ Unión Mat.\ Argent.\ 26, 24--41 (1972; Zbl 0249.47024)] in the non-quasianalytic case. The paper concludes with some results on completeness of translates in certain Banach spaces of real functions.
scientific article; zbMATH DE number 5149407
scientific article