Birkhoff-James Orthogonality and Its Application in the Study of Geometry of Banach Space

TLDR: In this article, the authors considered the problem of defining the Birkhoff-James orthogonality of bounded linear operators in the setting of infinite-dimensional Banach spaces.

Abstract: The geometry of real finite-dimensional Banach spaces, popularly known as Minkowski Geometry, is referred to as a geometry “next” to the Euclidean Geometry in the famous fourth Hilbert problem. One of the major striking differences between Euclidean Geometry and the geometry of Banach spaces is that there is no unique notion of orthogonality in the later case. It is easy to observe that similar to the notion of Birkhoff-James orthogonality, the definitions of left symmetric and right symmetric points extend in an obvious way to bounded linear operators. The chapter deals solely with Birkhoff-James orthogonality Birkhoff; James, arguably the most “natural” and important notion of orthogonality defined in a normed linear space. Birkhoff-James orthogonality is intimately connected with various geometric properties of a Banach space, including strict convexity, uniform convexity and smoothness. Birkhoff-James orthogonality of bounded linear operators is considered in the setting of infinite-dimensional Banach spaces.