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

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.