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

Abstract

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 =(A 1,A 2,…,A n) and B =(B 1,B 2,…,B n) be n-tuples in B( H). Define the elementary operators ▵ AB and ▵ * AB :B(H)→B(H) by ▵ AB (X)=∑ i=1 nA iXB i−X and ▵ * AB (X)=∑ i=1 nA i *XB i *−X . This note considers the range-kernel orthogonality of the restrictions of ▵ AB and ▵ * AB to Schatten p-classes C p . It is proved that: (a) if 1< p<∞ , S∈ C p and ∑ i=1 n A * i A i , ∑ i=1 n A i A * i , ∑ i=1 n B * i B i and ∑ i=1 n B i B * i are all ⩽1, then min{∥▵ AB (X)+S∥ p,∥▵ * AB (X)+S∥ p}⩾∥S∥ p for all X∈ C p if and only if ▵ AB (S)=0=▵ * AB (S) ; (b) if p=2 and S∈ C 2 , then ∥▵ AB (X)+S∥ 2 2=∥▵ AB (X)∥ 2 2+∥S∥ 2 2 and ∥▵ * AB (X)+S∥ 2 2=∥▵ * AB (X)∥ 2 2+∥S∥ 2 2 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∈ C 1 , then the inequality of (a) holds (with p=1 and) for all X∈ C 1 if and only if ▵ AB (S)=0=▵ * AB (S) .