Strong Birkhoff–James Orthogonality of Compact Operators on Hilbert Spaces

Let $H$ be an infinite-dimensional Hilbert space, $AinBbb K(H)$, $Ane 0$ and $Bin B(H).$ We prove that $A$ and $B$ satisfy
the inequality $|A+CBD|ge |A|$ for all $C,Din B(H)$ if and only if the rank of the operator $B^*$ is strictly less than the dimension of the kernel of the operator $A^*A-|A|^2I.$