The Engel-Jacobson theorem revisited
type: Item
instance of: scholarly article
title: The Engel-Jacobson theorem revisited
zbMATH Open document ID: 0645.15010
DOI: 10.1016/0021-8693(87)90227-4
author: H. Radjavi
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: The Jacobson extension of Engel's theorem says that: A set \({\mathcal N}\) of nilpotent linear operators on a finite-dimensional vector space for which to each (ordered) pair A, B in \({\mathcal N}\) there correspond a scalar c such that AB-cBA\(\in {\mathcal N}\), is triangularizable. In this article the author replaces the term cB, in the above statement, with any noncommutative polynomial in A and B (which may change from pair to pair) and derives the same result. The author also gives a shorter proof for one of his earlier results on general triangularizability: An arbitrary set of operators is triangularizable if and only if it has permutable trace [i.e., \(tr(A_ 1A_ 2...A_ n)=tr(A_{\sigma (1)}A_{\sigma (2)}...A_{\sigma (n)})\) for every permutation \(\sigma\) ].
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 4051805
zbMATH Keywords: Lie algebra
zbMATH Keywords: Jacobson and Engel's theorem
zbMATH Keywords: nilpotent linear operators
zbMATH Keywords: noncommutative polynomial
zbMATH Keywords: general triangularizability
reviewed by: Ali-Akbar Jafarian
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90227-4
OpenAlex ID: W2035903596
cites work: Q4138841
cites work: On certain concepts in the theory of algebraic matric groups
cites work: A Trace Condition Equivalent to Simultaneous Triangularizability
cites work: Banach-Lie algebras of compact operators
description: scientific article; zbMATH DE number 4051805