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

TL;DR

This paper introduces a weaker version of the Bhatia-eemrl property for operators, studies its density in various classical spaces, and compares it with the original property in the context of norm attainment and orthogonality.

Contribution

It defines the adjusted Bhatia-eemrl property, analyzes its density in classical Banach spaces, and contrasts it with the original property regarding norm attainment and orthogonality.

Findings

The adjusted Bhatia-eemrl property is norm-dense in $c_0$ and $L_1[0,1]$ spaces.

The set of functionals with the adjusted property on $C[0,1]$ is not norm-dense but is weak-* dense in $C(K)^*$.

The set of operators with the adjusted property is contained within the norm-attaining operators.

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-\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 $K$.