Abstract
We show that a linear transformation on a vector space is a sum of two commuting square-zero transformations if and only if it is a nilpotent transformation with index of nilpotency at most 3 and the codimension of im T ∩ ker T in ker T is greater than or equal to the dimension of the space im T 2 . We also characterize products of two commuting unipotent transformations with index 2.
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