The structure of almost-invariant half-spaces for some operators
type: Item
instance of: scholarly article
title: The structure of almost-invariant half-spaces for some operators
zbMATH Open document ID: 1308.47007
author: Ben Wallis
author: Gleb Sirotkin
published in: Journal of Functional Analysis
publication date: 2014-08-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1509.01485
review text: A closed subspace \(Y\) of an infinite-dimensional complex Banach space \(X\) is called a half-space if \(\dim Y=\infty\) and \(\text{codim} Y=\infty\). This half space is called almost-invariant for a linear bounded operator \(T\) acting on \(X\) if \(TY\subset Y+F\) for some finite-dimensional subspace \(F\subset X\). The minimal possible value of \(\dim F\) for \(F\) in the previous property is called the defect of \(Y\) with respect to \(T\). The authors study the connections between almost-invariant half-spaces and some properties of \(T\). We mention the following two results of the paper: every weakly compact (every quasinilpotent) operator admits an almost-invariant half-space of defect \(\leq1\).
reviewed by: Vladimir S. Pilidi
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 6335937
zbMATH Keywords: Banach space
zbMATH Keywords: half-space
zbMATH Keywords: almost-invariant half-space
zbMATH Keywords: spectrum
zbMATH Keywords: invariant subspace
MaRDI profile type: Publication
OpenAlex ID: W2008428234
arXiv ID: 1509.01485
cites work: Q4781239
cites work: Q3154380
cites work: Q5317928
cites work: Almost invariant half-spaces of operators on Banach spaces
cites work: A hereditarily indecomposable \(\mathcal L_{\infty}\)-space that solves the scalar-plus-compact problem
cites work: On the invariant subspace problem for Banach spaces
cites work: A general approach to Read's type constructions of operators without non-trivial invariant closed subspaces
cites work: An example of a minimal action of the free semi-group \(\mathbb{F}^{+}_{2}\) on the Hilbert space
cites work: Handbook of the geometry of Banach spaces. Volume 1
cites work: Базисные последовательности, биортогональные системы и нормирующие множества в пространствах Банаха и Фреше
cites work: Q5680636
cites work: Q4136031
cites work: On almost-invariant subspaces and approximate commutation
cites work: Almost invariant half-spaces of algebras of operators
cites work: Every operator has almost-invariant subspaces
cites work: A Solution to the Invariant Subspace Problem
cites work: Infinite matrices with ``few'' non-zero entries and without non-trivial invariant subspaces
DOI: 10.1016/J.JFA.2014.07.007
description: scientific article; zbMATH DE number 6335937