Birkhoff-James orthogonality and applications : A survey

Abstract

In the last few decades, the concept of Birkhoff-James orthogonality has been used in several applications. In this survey article, the results known on the necessary and sufficient conditions for Birkhoff-James orthogonality in certain Banach spaces are mentioned. Their applications in studying the geometry of normed spaces are given. The connections between this concept of orthogonality, and the Gateaux derivative and the subdifferential set of the norm function are provided. Several interesting distance formulas can be obtained using the characterizations of Birkhoff-James orthogonality, which are also mentioned. In the end, some new results are obtained.

Orthonormal pairs of operators

“…Various notions of orthogonality for vectors in a Banach space were introduced already by Birkhoff [3] and James [10] (recent surveys by Bottazzi, Conde and Sain and by Grover and Sushil are [5] and [9]), which were investigated even in the context of Hilbert C * -modules by Arambašić and Rajić [1]. One possible natural definition of orthogonality, investigated by Eskandari, Moslehian and Popovici in [7] and called Pythagoras orthogonality, is the following: two vectors x, y in a normed space are orthogonal, which is denoted as x ⊥ P y, if there exists a linear isometry f from the linear span of x and y into a Hilbert space such that the vectors f (x) and f (y) are orthogonal in the usual sense.…”