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

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. are 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 on the diagonal is the product of two involutory infinite matrices.