Birkhoff-James orthogonality in complex Banach spaces and Bhatia-\v{S}emrl Theorem revisited

TL;DR

This paper investigates Birkhoff-James orthogonality in complex Banach spaces, highlighting geometric differences between smooth and non-smooth points, and provides a new proof of the Bhatia-cemrl Theorem on operator orthogonality.

Contribution

It offers a novel geometric perspective on orthogonality and presents a new proof of a key theorem in operator theory within complex Banach spaces.

Findings

Distinguishes geometric properties of smooth and non-smooth points

Provides a new proof of the Bhatia-cemrl Theorem

Enhances understanding of orthogonality in complex Banach spaces

Abstract

We explore Birkhoff-James orthogonality of two elements in a complex Banach space by using the directional approach. Our investigation illustrates the geometric distinctions between a smooth point and a non-smooth point in a complex Banach space. As a concrete outcome of our study, we obtain a new proof of the Bhatia-\v{S}emrl Theorem on orthogonality of linear operators.