  We prove that a (bounded linear) operator acting on an infinite-dimensional,
separable, complex Hilbert space can be written as a product of two
quasi-nilpotent operators if and only if it is not a semi-Fredholm operator.
This solves the problem posed by Fong and Sourour in 1984. We also consider
some closely related questions. In particular, we show that an operator can be
expressed as a product of two nilpotent operators if and only if its kernel and
co-kernel are both infinite-dimensional. This answers the question implicitly
posed by Wu in 1989.
