Infinite-Dimensional Triangularization

TL;DR

This paper extends the concept of triangularization from matrices to linear transformations on arbitrary vector spaces, establishing equivalent conditions, properties, and generalizations including simultaneous triangularization and topological closure.

Contribution

It introduces a comprehensive theory of triangularization for infinite-dimensional spaces, generalizing classical finite-dimensional results and providing new insights into the structure of linear transformations.

Findings

Triangularizable transformations are characterized by polynomial annihilation of finite subspaces.

Finite collections of commuting triangularizable transformations are simultaneously triangularizable.

The set of triangularizable transformations has a well-defined closure in the standard topology.

Abstract

The goal of this paper is to generalize the theory of triangularizing matrices to linear transformations of an arbitrary vector space, without placing any restrictions on the dimension of the space or on the base field. We define a transformation T of a vector space V to be "triangularizable" if V has a well-ordered basis such that T sends each vector in that basis to the subspace spanned by basis vectors no greater than it. We then show that the following conditions (among others) are equivalent: (1) T is triangularizable, (2) every finite-dimensional subspace of V is annihilated by f(T) for some polynomial f that factors into linear terms, (3) there is a maximal well-ordered set of subspaces of V that are invariant under T, (4) T can be put into a crude version of the Jordan canonical form. We also show that any finite collection of commuting triangularizable transformations is simultaneously triangularizable, we describe the closure of the set of triangularizable transformations in the standard topology on the algebra of all transformations of V , and we extend to transformations that satisfy a polynomial the classical fact that the double-centralizer of a matrix is the algebra generated by that matrix.