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

AbstractLet 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 Cp. It is proved that: (a) if 1<p<∞ , S∈Cp and ∑i=1nA*iAi, ∑i=1nAiA*i, ∑i=1nB*iBi and ∑i=1nBiB*i are all ⩽1, then min{∥▵AB(X)+S∥p,∥▵*AB(X)+S∥p}⩾∥S∥p for all X∈Cp if and only if ▵AB(S)=0=▵*AB(S);(b) if p=2 and S∈C2, then ∥▵AB(X)+S∥22=∥▵AB(X)∥22+∥S∥22 and ∥▵*AB(X)+S∥22=∥▵*AB(X)∥22+∥S∥22 if and only if ▵AB(S)=0=▵*AB(S); and(c) if A and B are the n-tuples of (a) such that ▵BB(S)=0=▵*BB(S) for some injective S∈C1, then the inequality of (a) holds (with p=1 and) for all X∈C1 if and only if ▵AB(S)=0=▵*AB(S)