Invariant subspaces of operator Lie algebras and the theory of \(K\)-algebras
type: Item
instance of: scholarly article
title: Invariant subspaces of operator Lie algebras and the theory of \(K\)-algebras
zbMATH Open document ID: 1040.47052
DOI: 10.1023/A:1021778213915
publication date: 2002-01-01T00:00:00Z
review text: Let \(B(X)\) denote the space of bounded linear operators on a Banach space \(X\). A closed subspace \( Y \subset X\) is said to be invariant for a set \(M \subset B(X)\) if \(T(Y) \subset Y\) for all \(T \in M\). It is said to be hyperinvariant if it is invariant for \(M\) and its commutator. Considering \(B(X)\) as a Lie algebra under \([T,S] = TS-ST\), two of the results announced in this paper are: 1. If a closed Lie algebra of compact operators contains a nonzero ideal having no nonzero finite rank operator, then this algebra has a nontrivial invariant subspace. 2. If a Lie algebra of compact operators contains a nonzero ideal of quasinilpotent operators (also called Volterra ideal), then it has a nontrivial hyperinvariant subspace.
reviewed by: T.S.S.R.K. Rao
Mathematics Subject Classification ID: 47L10
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 17B60
Mathematics Subject Classification ID: 17B65
zbMATH DE Number: 1930684
zbMATH Keywords: Lie algebra of compact operators
zbMATH Keywords: Volterra ideal
zbMATH Keywords: hyperinvariant subspace
author: V. S. Shulman
author: Yurii V. Turovskii
MaRDI profile type: Publication
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1930684