The kernel of the composition of two operators (Q1206931)

Let \({\mathcal H}\) be a Hilbert space, \(B({\mathcal H})\) the algebra of all bounded linear operators on \({\mathcal H}\), and \(A,B\in B({\mathcal H})\). Consider the statement: \[ \text{dimker} (AB)\geq \text{dimker}(A).\tag{S} \] In this article the author gives conditions on \(B\) such that (S) holds for every \(A\in B({\mathcal H})\), and conditions on \(A\) such that (S) holds for every \(B\in B({\mathcal H})\).