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. x, u(x), u²(x), … 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 ±1 on the diagonal is the product of two involutory infinite matrices.

设V是域上具有可数维度的向量空间，设u是它的局部有限自同态，即 x , u ( x ) , u ² ( x ) , ... 对V中的所有x线性相关。我们给出了u可分解为两个零平方自同态之和的几个充分必要条件。此外，如果u是可逆的，我们给出u可分解为两次对合的乘积的充分必要条件，以及u的可分解性成索引 2 的两个单能自同态的乘积。我们的结果实质上扩展了有限维设置中已知的结果。特别地，我们得到每个在域中有元素的严格上三角无限矩阵是两个平方零无限矩阵（虽然可能是非三角形的）之和，并且每个上三角无限矩阵（在域中有元素) 对角线上只有 ±1 是两个对合无限矩阵的乘积。