Sums of quadratic endomorphisms of an infinite-dimensional vector space

TL;DR

This paper proves that any endomorphism of an infinite-dimensional vector space can be expressed as the sum of four idempotent or four square-zero endomorphisms, establishing optimal decompositions in this context.

Contribution

It establishes the minimal number of idempotent and square-zero endomorphisms needed to decompose any endomorphism of an infinite-dimensional vector space, proving the result's optimality.

Findings

Every endomorphism splits as the sum of four idempotents.

Every endomorphism splits as the sum of four square-zero endomorphisms.

The decomposition results are optimal in the general case.

Abstract

We prove that every endomorphism of an infinite-dimensional vector space splits as the sum of four idempotents and as the sum of four square-zero endomorphisms, a result that is optimal in general.