Invariant Extremal Projections for Operator-Ordered Families
Philip Kennerberg
TL;DR
The paper identifies an invariant extremal projection principle for families of operators ordered by Löwner domination, encoded via covariance envelopes. It proves that, for any fixed linear representation, the worst-case quadratic deviation over the envelope is attained at a canonical extremal source in the envelope’s closure, allowing minimax problems to collapse to ordinary quadratic minimization constrained by a covariance-order. The authors develop an operator-theoretic approximation framework with finite-dimensional compressions to handle nonconvex, nonclosed envelopes and establish existence and uniqueness results for the minimizers, plus structural descriptions of envelopes in stationary settings. The approach extends Rayleigh-Ritz-like ideas beyond Hilbert spaces by working with second-order structures and dual pairings, yielding a robust mechanism to analyze stability sets without relying on spectral decompositions, compactness, or convexity. The results have implications for invariant projections in systems governed by operator domination, with explicit coordinate formulas and spectral descriptions in the stationary/LTI context.
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 Löwner 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.