Operator norm attainment and Birkhoff–James orthogonality

Recently in [Sain D., Paul K., Operator norm attainment and inner product spaces, Linear Algebra Appl. 439 (2013) 2448–2452] it was proved that if T   is a linear operator on a finite dimensional real normed linear space XX such that T attains norm only on ±D, where D   is a connected closed subset of SXSX, then T   satisfies the Bhatia–Šemrl (BŠ) property [Bhatia R., Šemrl P., Orthogonality of matrices and distance problem, Linear Algebra Appl. 287 (1999) 77–85], i.e., for A∈L(X)A∈L(X), T⊥BAT⊥BA implies that there exists x∈Dx∈D such that Tx⊥BAxTx⊥BAx. Here we explore the converse of the above result. We prove that in a real normed linear space XX of dimension 2, a linear operator T satisfies the BŠ property if and only if the set of unit vectors on which T   attains norm is connected in the corresponding projective space RP1≡SX/{x∼−x}RP1≡SX/{x∼−x}. Motivated by the result in 2 dimensions, we conjecture that this characterization of BŠ property is true in general n  -dimensional real normed linear spaces. We further prove that if the space XX is strictly convex, then the set of operators in L(X)L(X) which satisfy the BŠ property is dense in L(X)L(X).