On a Set of Norm Attaining Operators and the Strong Birkhoff–James Orthogonality

Abstract

Continuing the study of recent results on the Birkhoff–James orthogonality and the norm attainment of operators, we introduce a property namely the adjusted Bhatia–Šemrl property for operators which is weaker than the Bhatia–Šemrl property. The set of operators with the adjusted Bhatia–Šemrl property is contained in the set of norm attaining ones as it was in the case of the Bhatia–Šemrl property. It is known that the set of operators with the Bhatia–Šemrl property is norm-dense if the domain space X of the operators has the Radon–Nikodým property like finite dimensional spaces, but it is not norm-dense for some classical spaces such as $$c_0$$ , $$L_1[0,1]$$ and C[0, 1]. In contrast with the Bhatia–Šemrl property, we show that the set of operators with the adjusted Bhatia–Šemrl property is norm-dense when the domain space is $$c_0$$ or $$L_1[0,1]$$ . Moreover, we show that the set of functionals having the adjusted Bhatia–Šemrl property on C[0, 1] is not norm-dense but such a set is weak- $$*$$ -dense in $$C(K)^*$$ for any compact Hausdorff K.