Range-kernel orthogonality of the elementary operator X→∑i=1nAiXBi−X

Let H be a separable infinite dimensional complex Hilbert space and let B(H) denote the algebra of operators on H into itself. Let A=(A1,A2,…,An) and B=(B1,B2,…,Bn) be n-tuples in B(H). Define the elementary operators AB and *AB:B(H)→B(H) by AB(X)=∑i=1nAiXBi−X and *AB(X)=∑i=1nAi*XBi*−X. This note considers the range-kernel orthogonality of the restrictions of AB and *AB to Schatten p-classes . It is proved that: (a) if 1