Invariant Extremal Projections for Operator-Ordered Families

TL;DR

This paper introduces an extremal projection principle for operator-ordered families, enabling simplified quadratic optimization within covariance envelopes without relying on convexity or compactness.

Contribution

It establishes an envelope extremal principle for operator-ordered families, linking maximal quadratic functionals to extremal configurations via an operator-theoretic approximation scheme.

Findings

Maximal quadratic functional values match extremal configurations.

Reduction of minimax problems to well-posed quadratic minimization.

Structural properties of covariance envelopes are characterized.

Abstract

We study an extremal projection principle for families of operators ordered by domination, induced by fixed bounded linear mappings acting on a source with an additive baseline. Stability is defined through domination of second--order structure, leading to a covariance envelope of admissible sources ordered by the L\"owner relation. Our main result establishes an envelope extremal principle: the maximal value of the quadratic functional over the entire envelope coincides with that of a single extremal configuration, which may lie only in the closure of the admissible class. This identification is obtained without convexity, compactness, or any global Hilbert space structure governing all components of the system, and relies instead on an operator--theoretic approximation scheme. As a consequence, minimax optimization over stability sets reduces to an ordinary quadratic minimization problem with well--posed existence and uniqueness properties for the associated minimizing operators. Structural properties of covariance envelopes are also derived, including density, closure, and spectral characterizations in stationary settings.