Some remarks on Birkhoff-James orthogonality of linear operators

TL;DR

This paper investigates Birkhoff-James orthogonality of linear operators in Hilbert and Banach spaces, providing new characterizations, norm retrieval methods, and approximation results using semi-inner-products and geometric ideas.

Contribution

It introduces novel characterizations of Euclidean spaces and methods to recover operator norms via orthogonality sets, advancing the understanding of operator geometry.

Findings

Characterization of Euclidean spaces using orthogonality

Method to retrieve operator norms from orthogonality sets

Best approximation results in operator spaces

Abstract

We study Birkhoff-James orthogonality of compact (bounded) linear operators between Hilbert spaces and Banach spaces. Applying the notion of semi-inner-products in normed linear spaces and some related geometric ideas, we generalize and improve some of the recent results in this context. In particular, we obtain a characterization of Euclidean spaces and also prove that it is possible to retrieve the norm of a compact (bounded) linear operator (functional) in terms of its Birkhoff-James orthogonality set. We also present some best approximation type results in the space of bounded linear operators.