The Engel-Jacobson theorem revisited
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 \(\sigma\) ].
scientific article; zbMATH DE number 4051805