Range kernel orthogonality of derivations

Abstract

Let 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 δ A,B:B(H)→B(H) is defined by δ A,B(X)=AX−XB ; let ▵ A,B:B(H)→B(H) be defined by ▵ A,B(X)=AXB−X , and let d A,B denote δ A,B or ▵ A,B . Let S∈ C p (the Schatten p-class, 1<p<∞). Given that the pair (A,B) of operators satisfies the property kerd A,B| C p⊆ kerd A *,B * | C p , we prove a necessary and sufficient condition for ||d A *,B * (X)+S|| p⩾||S|| p to hold for all X∈ C p .