Operator norm attainment and Birkhoff–James orthogonality

Abstract

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 X such that T attains norm only on ±D, where D is a connected closed subset of SX, 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), T⊥BA implies that there exists x∈D such that Tx⊥BAx. Here we explore the converse of the above result. We prove that in a real normed linear space X 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}. 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 X is strictly convex, then the set of operators in L(X) which satisfy the BŠ property is dense in L(X).