On weak orbits of operators
type: Item
instance of: scholarly article
title: On weak orbits of operators
zbMATH Open document ID: 1204.47001
author: Vladimír Müller
author: Cătălin Badea
published in: Topology and its Applications
publication date: 2009-04-23T00:00:00Z
review text: 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 \(\;n\geq 0\).
Mathematics Subject Classification ID: 47A05
Mathematics Subject Classification ID: 47B15
zbMATH DE Number: 5546622
zbMATH Keywords: completely nonunitary contraction
zbMATH Keywords: weak orbits
zbMATH Keywords: Fourier coefficients
MaRDI profile type: Publication
full work available at URL: j.topol.2008.12.020
OpenAlex ID: W2060513340
cites work: Q3994587
cites work: Upper and lower Fredholm spectra. II
cites work: Q5556050
cites work: Subadditive mean ergodic theorems
cites work: Seventy years of Rajchman measures
cites work: Local spectral radius formula for operators in Banach spaces
cites work: Power bounded operators and supercyclic vectors
cites work: Q5700970
cites work: Power bounded operators and supercyclic vectors II
cites work: Q5600360
cites work: On the orbits of an operator with spectral radius one
DOI: 10.1016/J.TOPOL.2008.12.020
description: scientific article; zbMATH DE number 5546622