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We prove that a (bounded linear) operator acting on an infinite-dimensional,separable, complex Hilbert space can be written as a product of twoquasi-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 considersome closely related questions. In particular, we show that an operator can beexpressed as a product of two nilpotent operators if and only if its kernel andco-kernel are both infinite-dimensional. This answers the question implicitlyposed by Wu in 1989.

An operator is a product of two quasi-nilpotent operators if and only if it is not semi-Fredholm