Reverse triangle inequality in Hilbert $C^{\ast }$-modules

Abstract

We prove several versions of reverse triangle inequality in Hilbert $C^{\ast }$-modules. We show that if $e_1,...,e_m$ are vectors in a Hilbert module $\mathfrak{X}$ over a $C^{\ast }$-algebra $\mathfrak{A}$ with unit 1 such that $<e_i,e_j>=0(1\le i\ne j\le m)$ and $\Vert e_i\Vert =1(1\le i\le m)$, and also $r_k,{\rho }_k\in \mathbb{R}(1\le k\le m)$ and $x_1,...,x_n\in \mathfrak{X}$ satisfy $$0\le r_k^2\Vert x_j\Vert \le Re<r_k{e}_k,x_j>,0\le {\rho }_k^2\Vert x_j\Vert \le Im<{\rho }_k{e}_k,x_j>,$$ then [\sum_{k=1}^m(r_k^2+ρ_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+iρ_k)e_k .

12 Pages; to appear in J. Inequal. Pure Appl. Math (JIPAM)

References in corpus (2)

• Reverses of the Triangle Inequality in Inner Product Spaces
• Refinements of Reverse Triangle Inequalities in Inner Product Spaces

Cited by in corpus (1)

• Characterization of a generalized triangle inequality in normed spaces