Summary: Let \(B(H)\) denote the algebra of operators on the separable Hilbert space \(H\). Let \(C_ 2\) denote the (Hilbert) space of Hilbert-Schmidt operators on \(H\), with norm \(\|\cdot\|_ 2\) defined by \(\| S\|_ 2^ 2=(S,S)=\hbox{tr}(SS^*)\). Given \(A,B\in B(H)\), define the derivation \(C(A,B): B(H)\to B(H)\) by \(C(A,B)X=AX-XB\). We show that \[ \| C(A,B)X+S\|_ 2^ 2=\| C(A,B)X\|_ 2^ 2+\| S\|_ 2^ 2 \] holds for all \(X\in B(H)\) and for every \(S\in C_ 2\) such that \(C(A,B)S=0\) if and only if \(\overline{\text{ran}}S\) reduces \(A\), \(\hbox {ker}^ \perp S\) reduces \(B\), and \(A\mid\overline{\text{ran}}S\) and \(B\mid\hbox {ker}^ \perp S\) are unitarily equivalent normal operators. We also show that if \(A,B\in B(H)\) are contractions and \(R(A,B): B(H)\to B(H)\) is defined by \(R(A,B)X=AXB-X\), then \(S\in C_ 2\) and \(R(A,B)S=0\) imply \[ \| R(A,B)X+S\|_ 2^ 2=\| R(A,B)X\|^ 2_ 2+\| S\|_ 2^ 2 \] for all \(X\in B(H)\).