  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.
