Range kernel orthogonality of derivations

AbstractLet H be a separable infinite dimensional complex Hilbert space, and let B(H) denote the algebra of operators on H into itself. The generalised derivation δA,B:B(H)→B(H) is defined by δA,B(X)=AX−XB; let ▵A,B:B(H)→B(H) be defined by ▵A,B(X)=AXB−X, and let dA,B denote δA,B or ▵A,B. Let S∈Cp (the Schatten p-class, 1<p<∞). Given that the pair (A,B) of operators satisfies the property kerdA,B|Cp⊆kerdA*,B*|Cp, we prove a necessary and sufficient condition for ||dA*,B*(X)+S||p⩾||S||p to hold for all X∈Cp