Modular Birkhoff-James orthogonality in \(B(\mathbb{X,Y})\) and \(K(\mathbb{X,Y})\) (Q2192882)

The authors continue their work on variations of Birkhoff-James orthogonality in spaces of operators (bounded linear operators, \({\mathcal B}(X,Y)\) and spaces of compact operators, \({\mathcal K}(X,Y)\)) in real Banach spaces. Considering, say, \({\mathcal B}(X)\) as a right module in \({\mathcal B}(X,Y)\), \(T,S \in {\mathcal B}(X,Y)\) are said to be right modular orthogonal if \(T\) is Birkhoff-James orhtogonal to \(SA\) for all \(A \in {\mathcal B}(X)\). Fixing an operator \(A\), it turns out that if you expect symmetry (left) to hold for all operators, then \(A\) is of rank one, and the range of \(A\) consists of left symmetric points (in the Birkhoff-James orthogonality) in~\(Y\).