Birkhoff-James orthogonality to a subspace of operators defined between Banach spaces

TL;DR

This paper investigates Birkhoff-James orthogonality of linear operators to subspaces in Banach spaces, providing characterizations under various conditions including reflexivity, finite dimensionality, and Hilbert space settings.

Contribution

It offers new characterizations of operator orthogonality to subspaces in Banach and Hilbert spaces, extending existing theory to broader contexts.

Findings

Complete characterization when domain is reflexive and subspace is finite dimensional

Orthogonality conditions in arbitrary Banach spaces with additional assumptions

Analysis of orthogonality in Hilbert spaces with respect to operator and numerical radius norms

Abstract

This paper deals with study of Birkhoff-James orthogonality of a linear operator to a subspace of operators defined between arbitrary Banach spaces. In case the domain space is reflexive and the subspace is finite dimensional we obtain a complete characterization. For arbitrary Banach spaces, we obtain the same under some additional conditions. For arbitrary Hilbert space $ \mathbb{H},$ we also study orthogonality to subspace of the space of linear operators $L(\mathbb{H}), $ both with respect to operator norm as well as numerical radius norm.