On a set of norm attaining operators and the strong Birkhoff-James orthogonality

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-\v{S}emrl property for operators which is weaker than the Bhatia-\v{S}emrl property. The set of operators with the adjusted Bhatia-\v{S}emrl property is contained in the set of norm attaining ones as it was in the case of the Bhatia-\v{S}emrl property. It is known that the set of operators with the Bhatia-\v{S}emrl property is norm-dense if the domain space $X$ of the operators has the Radon-Nikod\'ym 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-\v{S}emrl property, we show that the set of operators with the adjusted Bhatia-\v{S}emrl 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-\v{S}emrl property on $C[0,1]$ is not norm-dense but such a set is weak-$*$-dense in $C(K)^*$ for any compact Hausdorff $K$.