From norm derivatives to orthogonalities in Hilbert $C^*$-modules

TL;DR

This paper introduces a method to compute the norm derivative in Hilbert $C^*$-modules, uses it to characterize orthogonality concepts, and provides new proofs and solutions related to operator equations.

Contribution

It develops a formula for the norm derivative in Hilbert $C^*$-modules, applies it to characterize orthogonality, and offers simplified proofs and solutions for classical and generalized equations.

Findings

Computed the norm derivative for elements in Hilbert $C^*$-modules.

Characterized various orthogonality concepts using the norm derivative.

Provided a simpler proof of Birkhoff--James orthogonality and solved a generalized Daugavet equation.

Abstract

Let $\big(\mathscr{X}, \langle\cdot, \cdot\rangle\big)$ be a Hilbert $C^*$-module over a $C^*$-algebra $\mathscr{A}$ and let $\mathcal{S}(\mathscr{A})$ be the set of states on $\mathscr{A}$. In this paper, we first compute the norm derivative for elements $x$ and $y$ of $\mathscr{X}$ as follows \begin{align*} \rho_{_{+}}(x, y) = \max\Big\{\mbox{Re}\,\varphi(\langle x, y\rangle): \, \varphi \in \mathcal{S}(\mathscr{A}), \varphi(\langle x, x\rangle) = \|x\|^2\Big\}. \end{align*} We then apply it to characterize different concepts of orthogonality in $\mathscr{X}$. In particular, we present a simpler proof of the classical characterization of Birkhoff--James orthogonality in Hilbert $C^*$-modules. Moreover, some generalized Daugavet equation in the $C^*$-algebra $\mathbb{B}(\mathcal{H})$ of all bounded linear operators acting on a Hilbert space $\mathcal{H}$ is solved.