The Birkhoff-James orthogonality and norm attainment for multilinear maps

TL;DR

This paper investigates the density of multilinear maps with the Bhatia-9emrl property in complex Banach spaces, extending previous operator results and highlighting cases where density fails.

Contribution

It extends the study of the Bhatia-9emrl property to multilinear maps on complex Banach spaces and identifies conditions where density does or does not hold.

Findings

Density holds for multilinear maps on some complex Banach spaces.

Density does not hold when the domain space is c_0 for arbitrary range.

Only the zero operator has the Bhatia-9emrl property from c_0 to certain Banach spaces.

Abstract

Very recently, motivated by the result of Bhatia and \v{S}emrl which characterizes the Birkhoff-James orthogonality of operators on a finite dimensional Hilbert space in terms of norm attaining points, the Bhatia-\v{S}emrl property was introduced. The main purpose of this article is to study the denseness of the set of multilinear maps with the Bhatia-\v{S}emrl property which is contained in the set of norm attaining ones. Contrary to the most of previous results which were shown for operators on real Banach spaces, we prove the denseness for multilinear maps on some complex Banach spaces. We also show that the denseness of operators does not hold when the domain space is $c_0$ for arbitrary range. Moreover, we find plenty of Banach spaces $Y$ such that only the zero operator has the Bhatia-\v{S}emrl property in the space of operators from $c_0$ to $Y$.