Infinite-Dimensional Triangularizable Algebras

TL;DR

This paper generalizes classical theorems by characterizing triangularizable subsets of linear transformations on arbitrary vector spaces and describing their algebraic structure, extending results from finite-dimensional matrix theory to infinite dimensions.

Contribution

It introduces a new definition of triangularizability for subsets of End(V) and characterizes strictly triangularizable sets as topologically nilpotent, extending finite-dimensional results.

Findings

Strictly triangularizable sets are exactly topologically nilpotent.

Characterization of triangularizable subalgebras of End(V).

Generalization of Levitzki's and McCoy's theorems to infinite-dimensional spaces.

Abstract

Let End(V) denote the ring of all linear transformations of an arbitrary k-vector space V over a field k. We define a subset X of End(V) to be "triangularizable" if V has a well-ordered basis such that X sends each vector in that basis to the subspace spanned by basis vectors no greater than it. We then show that an arbitrary subset of End(V) is "strictly" triangularizable (defined in the obvious way) if and only if it is topologically nilpotent. This generalizes the theorem of Levitzki that every nilpotent semigroup of matrices is triangularizable. We also give a description of the triangularizable subalgebras of End(V), which generalizes of a theorem of McCoy classifying triangularizable algebras of matrices over algebraically closed fields.