Sums and products of two quadratic endomorphisms of a countable-dimensional vector space

TL;DR

This paper characterizes when certain infinite-dimensional linear operators can be decomposed into sums or products of special types of endomorphisms, extending finite-dimensional results to countably infinite-dimensional spaces.

Contribution

It provides necessary and sufficient conditions for decomposing locally finite endomorphisms into sums of square-zero endomorphisms and for invertible endomorphisms into products of involutions or unipotent endomorphisms.

Findings

Every strictly upper-triangular infinite matrix is the sum of two square-zero matrices.

Every upper-triangular infinite matrix with ±1 on the diagonal is a product of two involutions.

Results extend finite-dimensional decompositions to countably infinite-dimensional vector spaces.

Abstract

Let $V$ be a vector space with countable dimension over a field, and let $u$ be an endomorphism of it which is locally finite, i.e. $(u^k(x))_{k \geq 0}$ is linearly dependent for all $x$ in $V$. We give several necessary and sufficient conditions for the decomposability of $u$ into the sum of two square-zero endomorphisms. Moreover, if $u$ is invertible, we give necessary and sufficient conditions for the decomposability of $u$ into the product of two involutions, as well as for the decomposability of $u$ into the product of two unipotent endomorphisms of index $2$. Our results essentially extend the ones that are known in the finite-dimensional setting. In particular, we obtain that every strictly upper-triangular infinite matrix with entries in a field is the sum of two square-zero infinite matrices (potentially non-triangular, though), and that every upper-triangular infinite matrix (with entries in a field) with only $\pm 1$ on the diagonal is the product of two involutory infinite matrices.