Reverse triangle inequality in Hilbert $C^*$-modules

TL;DR

This paper establishes various forms of reverse triangle inequalities within Hilbert $C^*$-modules, extending classical results to a more general algebraic setting and characterizing conditions for equality.

Contribution

It introduces new reverse triangle inequalities in Hilbert $C^*$-modules with orthogonal vectors and provides conditions for equality, expanding the theoretical framework of inequalities in operator algebra contexts.

Findings

Derived multiple reverse triangle inequalities for Hilbert $C^*$-modules.

Identified conditions under which equality holds in these inequalities.

Extended classical inequalities to the setting of $C^*$-algebra modules.

Abstract

We prove several versions of reverse triangle inequality in Hilbert $C^*$-modules. We show that if $e_1, ..., e_m$ are vectors in a Hilbert module ${\mathfrak X}$ over a $C^*$-algebra ${\mathfrak A}$ with unit 1 such that $<e_i,e_j>=0 (1\leq i\neq j \leq m)$ and $\|e_i\|=1 (1\leq i\leq m)$, and also $r_k,\rho_k\in\Bbb{R} (1\leq k\leq m)$ and $x_1, ..., x_n\in {\mathfrak X}$ satisfy $$0\leq r_k^2 \|x_j\|\leq {Re}< r_ke_k,x_j> ,\quad0\leq \rho_k^2 \|x_j\| \leq {Im}< \rho_ke_k,x_j> ,$$ then [\sum_{k=1}^m(r_k^2+\rho_k^2)]^{{1/2}}\sum_{j=1}^n \|x_j\|\leq\|\sum_{j=1}^nx_j\|, and the equality holds if and only if \sum_{j=1}^n x_j=\sum_{j=1}^n\|x_j\|\sum_{k=1}^m(r_k+i\rho_k)e_k .