A complete characterization of Birkhoff-James orthogonality in infinite dimensional normed space

TL;DR

This paper provides a comprehensive characterization of Birkhoff-James orthogonality for bounded linear operators and functionals in infinite dimensional real normed spaces, with applications to smoothness conditions.

Contribution

It offers a complete characterization of Birkhoff-James orthogonality for operators and functionals, extending understanding in infinite dimensional normed spaces.

Findings

Characterization of Birkhoff-James orthogonality for operators

Characterization of orthogonality of bounded linear functionals with strictly convex duals

Necessary and sufficient conditions for smoothness of operators

Abstract

In this paper, we study Birkhoff-James orthogonality of bounded linear operators and give a complete characterization of Birkhoff-James orthogonality of bounded linear operators on infinite dimensional real normed linear spaces. As an application of the results obtained, we prove a simple but useful characterization of Birkhoff-James orthogonality of bounded linear functionals defined on a real normed linear space, provided the dual space is strictly convex. We also provide separate necessary and sufficient conditions for smoothness of bounded linear operators on infinite dimensional normed linear spaces.