On a set of norm attaining operators and the strong Birkhoff-James orthogonality
Geunsu Choi, Mingu Jung, Sun Kwang Kim
TL;DR
The paper introduces the adjusted Bhatia-Šemrl property for norm attaining operators and situates it between the classical Bhatia-Šemrl property and norm attainment, using strong Birkhoff–James orthogonality. It then establishes dense-operator results across three canonical domain spaces: for $X=c_0$ and $X=L_1[0,1]$, the adjusted property is norm-dense (including in vector-valued targets when $Y$ has property quasi-$eta$), while for $X=C[0,1]$ it is not norm-dense but is weak-$*$-dense in the operator space via measure-theoretic arguments. The work highlights a robust alternative to the original property that yields stronger density results in classical spaces, and it raises questions about the extent of weak-$*$ density across general Banach domains. Overall, the adjusted BŠ property broadens the scope of norm-attaining operator theory and clarifies how strong orthogonality interacts with norm attainment in different geometric settings.
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 $K$.