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Birkhoff-James orthogonality: A Minimax Theorem approach

Hranislav Stanković

TL;DR

This paper addresses the problem of establishing an elementary proof of the Bhatia-Šemrl Theorem for bounded linear operators on a Hilbert space by employing the Asplund-Ptak minimax theorem. The authors show that Birkhoff-James orthogonality $A \perp B$ is equivalent to the existence of a sequence of unit vectors $x_n$ with $\|Ax_n\| \to \|A\|$ and $\langle Ax_n, Bx_n\rangle \to 0$, deriving this from the minimax identity $\sup_{\|x\|=1} \inf_{\lambda} \|Ax+\lambda Bx\| = \inf_{\lambda} \|A+\lambda B\|$. They extend the argument to the finite-dimensional case and discuss connections to $r$-orthogonality and related characterizations. The contribution provides a streamlined, accessible proof framework that unifies previous approaches and highlights the power of minimax methods in operator geometry.

Abstract

In this paper, we present an elementary proof of the Bhatia-Šemrl Theorem, utilizing the Minimax Theorem for bounded linear operators by Asplund and Ptak [1]. Some related results are also discussed.

Birkhoff-James orthogonality: A Minimax Theorem approach

TL;DR

This paper addresses the problem of establishing an elementary proof of the Bhatia-Šemrl Theorem for bounded linear operators on a Hilbert space by employing the Asplund-Ptak minimax theorem. The authors show that Birkhoff-James orthogonality is equivalent to the existence of a sequence of unit vectors with and , deriving this from the minimax identity . They extend the argument to the finite-dimensional case and discuss connections to -orthogonality and related characterizations. The contribution provides a streamlined, accessible proof framework that unifies previous approaches and highlights the power of minimax methods in operator geometry.

Abstract

In this paper, we present an elementary proof of the Bhatia-Šemrl Theorem, utilizing the Minimax Theorem for bounded linear operators by Asplund and Ptak [1]. Some related results are also discussed.

Paper Structure

This paper contains 2 sections, 3 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Proof of Bhatia-Šemrl Theorem

Key Result

Theorem 1.1

AsplundPtak71 Let $\mathcal{H},\mathcal{K}$ be at least two-dimensional inner-product spaces over a field $\mathbb{K}\in\{\mathbb{C}, \mathbb{R}\}$, and let $A,B\in\mathfrak{B}(\mathcal{H},\mathcal{K})$. Then,

Theorems & Definitions (7)

  • Theorem 1.1: Minimax Theorem
  • Remark 1.1
  • Theorem 2.1
  • proof
  • Remark 2.1
  • Theorem 2.2
  • proof