Orthogonality and smoothness induced by the norm derivatives

TL;DR

This paper explores how derivatives of the norm induce orthogonality and smoothness in normed spaces, providing analytic characterizations and examining their relationships through examples including compact operators.

Contribution

It offers new analytic characterizations of orthogonality and smoothness induced by norm derivatives in normed spaces, linking these notions via support functionals and examples.

Findings

Characterization of orthogonality via support functionals

Connection between local smoothness and orthogonality sets

Illustrations with spaces of compact operators

Abstract

We study the concepts of orthogonality and smoothness in normed linear spaces, induced by the derivatives of the norm function. We obtain analytic characterizations of the said orthogonality relations in terms of support functionals in the dual space. We also characterize the related notions of local smoothness and establish its connection with the corresponding orthogonality set, which is analogous to the well-known relation between the Birkhoff-James orthogonality and the classical notion of smoothness. The similarities and the differences between the various notions of smoothness are illustrated by considering some particular examples, including $ \mathbb{K}(\mathbb{H}), $ the Banach space of all compact operators on a Hilbert space $ \mathbb{H}. $