Sums of three quadratic endomorphisms of an infinite-dimensional vector space

Abstract

Let V be an infinite-dimensional vector space over a field. In a previous article [5], we have shown that every endomorphism of V splits into the sum of four square-zero ones but also into the sum of four idempotent ones. Here, we study decompositions into sums of three endomorphisms with prescribed split annihilating polynomials with degree 2. Except for endomorphisms that are the sum of a scalar multiple of the identity and of a finite-rank endomorphism, we achieve a simple characterization of such sums. In particular, we give a simple characterization of the endomorphisms that split into the sum of three square-zero ones, and we prove that every endomorphism of V is a linear combination of three idempotents.