Wandering subspace theorems
type: Item
instance of: scholarly article
title: Wandering subspace theorems
zbMATH Open document ID: 1079.47015
author: Anders Olofsson
published in: Integral Equations and Operator Theory
publication date: 2005-04-29T00:00:00Z
review text: Let \(H\) be a Hilbert space and \(L(H)\) be the space of all bounded linear operators on \(H\). For an operator \(T \in L(H)\) and a set \(M \subset H\), let \([M]_{T}\) denote the smallest closed \(T\)-invariant subspace of \(H\) containing the set \(M\). In this paper, the author deals with the relation \((\ast)\) \(H=[M]_{T}\), where \(M=H \circleddash T(H)\). A closed subspace \(M\) of \(H\) is said to be a wandering subspace for \(T\) if \(M \perp T^{k}(M)\) for \(k \geq 1\). A result satisfying the relation \((\ast)\) is called a wandering subspace theorem. An operator \(T \in L(H)\) is called expansive if \(\| Tx \| \geq \| x \|\) for \(x \in H\). In Section \(2\) and Section \(3\), the author studies expansive operators \(T \in L(H)\) such that \[ \| T^{k}x \|^{2} \leq c_{k}(\| Tx \|^{2}-\| x \|^{2})+c \| x \|^{2},\quad x \in H,\;k \geq 2 \] and \(\sum_{k \geq 2} {1 \over c_{k}}= \infty\) for some positive constants \(c_{k}\) \((k \geq 2)\) and \(c\). In Section \(4\), he investigates operators \(T \in L(H)\) such that \[ \| Tx + y \|^{2} \leq 2 (\| x \|^{2} + \| Ty \|^{2}),\quad x,y \in H. \]
reviewed by: Ömer Gök
Mathematics Subject Classification ID: 47A16
Mathematics Subject Classification ID: 47A15
zbMATH DE Number: 2162376
zbMATH Keywords: wandering subspace
zbMATH Keywords: cyclic vector
zbMATH Keywords: expansive operators
zbMATH Keywords: contractive operators
zbMATH Keywords: Wold decomposition
MaRDI profile type: Publication
full work available at URL: s00020-003-1322-0
OpenAlex ID: W1987207644
DOI: 10.1007/S00020-003-1322-0
description: scientific article; zbMATH DE number 2162376