Orthogonality of the range and the kernel of some elementary operators

We prove the orthogonality of the range and the kernel of an important class of elementary operators with respect to the unitarily invariant norms associated with norm ideals of operators. This class consists of those mappings $E:B(H)\to B(H)$, $E(X)=AXB+CXD$, where $B(H)$ is the algebra of all bounded Hilbert space operators, and $A$, $B$, $C$, $D$ are normal operators, such that $AC=CA$, $BD=DB$ and $\ker A\cap \ker C=\ker B\cap \ker D=\{0\}$. Also we establish that this class is, in a certain sense, the widest class for which such an orthogonality result is valid. Some other related results are also given.