On weak orbits of operators
Let \(X\) be a complex Banach space, \(T\in B(X) , \) and \(\{a_n\} \) be a sequence of nonnegative reals with \(a_n \to 0. \) Then it is known that such operators have large orbits in the sense that there exists \(x \in X \) such that \( \| T^n x \| \geq a_n r(T^n) \) for all \(n\). The question whether the same holds for weak orbits was answered positively under some aditional conditions by the second author [Proc.\ Am.\ Math.\ Soc.\ 133, No.\,10, 2997--3004 (2005; Zbl 1090.47004)] and \textit{J.\,van Neerven} [Czech.\ Math.\ J.\ 45, No.\,3, 495--502 (1995; Zbl 0859.47003)]. In the article under review, the authors prove a generalization of these results in the Hilbert space setting; namely, the prove the following Theorem. Let \(H\) be a complex Hilbert space, \(T\in B(H)\) satisfy \(T^n\to 0\) in the weak operator topology, and assume \(1\in \sigma(T)\). If \(\{a_n\} \) is a sequence of positive reals with \(a_n \to 0\) and \(\epsilon >0\), then there exists \(x \in H \) with \(\| x \| < \sup_n a_n +\epsilon\) such that \[ \Re \langle T^nx , x \rangle > a_n \text{ for all }\;n\geq 0. \] As a corollary, they obtain that, for completely nonunitary contractions \(T\in B(H)\) with \(r(T)=1\), the assertion \(|\langle T^nx , x \rangle | > a_n\) holds for all \(\;n\geq 0\).
scientific article; zbMATH DE number 5546622