The minimax principle and related topics in the Jordan setting

TL;DR

This paper establishes a minimax principle for weakly compact JB*-triples, leading to new inequalities and theorems about singular values, spectral resolutions, and perturbations, extending classical results to this algebraic setting.

Contribution

It introduces a minimax principle for JB*-triples and derives related spectral and perturbation results, generalizing known theorems to this non-associative algebraic framework.

Findings

A minimax principle characterizes singular values in JB*-triples.

Weyl inequality and Cauchy-Poincaré interlacing theorem are established.

Perturbation stability of spectral resolutions and convex combinations is demonstrated.

Abstract

We prove a minimax principle for weakly compact JB$^*$-triples characterizing geometrically the singular values of an element. Among the consequences of this principle we present a Weyl inequality on the perturbation of the singular values and a Cauchy-Poincar\'e (interlacing) theorem. We also obtain a version of the Ky Fan maximum principle in the setting of weakly compact JB$^*$-triples. We study perturbations of the spectral resolutions showing that small perturbations of an element produces small perturbations of the corresponding spectral resolutions. As a consequence we obtain that weakly compact JB$^*$-triples satisfy the property that perturbations of a convex combination of elements in the closed unit ball coincide with a convex combination of perturbations of the elements also in the closed unit ball. All these results hold true when particularized to weakly compact JB$^*$-algebras.