Norm derivatives and geometry of bilinear operators

TL;DR

This paper explores the geometry of Banach spaces through norm derivatives and Birkhoff-James orthogonality, providing characterizations of orthogonality relations and properties of bilinear operators.

Contribution

It offers new characterizations of orthogonality and symmetry points in Banach spaces using norm derivatives, and analyzes bilinear operators' properties.

Findings

Complete characterization of strong Birkhoff-James orthogonality in ℓ₁ⁿ and ℓ_∞ⁿ.

New orthogonality relation defined via norm derivatives.

Insights into smoothness and norm attainment of bilinear operators.

Abstract

We study the norm derivatives in the context of Birkhoff-James orthogonality in real Banach spaces. As an application of this, we obtain a complete characterization of the left-symmetric points and the right-symmetric points in a real Banach space in terms of the norm derivatives. We obtain a complete characterization of strong Birkhoff-James orthogonality in $\ell_1^n$ and $\ell_\infty^n$ spaces. We also obtain a complete characterization of the orthogonality relation defined by the norm derivatives in terms of some newly introduced variation of Birkhoff-James orthogonality. We further study Birkhoff-James orthogonality, approximate Birkhoff-James orthogonality, smoothness and norm attainment of bounded bilinear operators between Banach spaces.