On the Convexification of Spectral Sets Induced by Non-Invariant Sets

TL;DR

This paper explores the convexification of spectral sets in finite-dimensional FTvN systems, providing new geometric characterizations for non-invariant sets and extending results to invariant cases.

Contribution

It introduces simple geometric characterizations of convex hulls of spectral sets induced by non-invariant sets, filling a gap in the existing literature.

Findings

Characterizations of convex hulls for non-invariant spectral sets

New convexification results for invariant spectral sets

Extension of convexification techniques to broader classes of spectral sets

Abstract

Given a finite-dimensional FTvN system $(\mathbb{V},\mathbb{W},\lambda)$, we study the convexification of the spectral set $\lambda^{-1}(\mathcal{C})$ induced by a set $\mathcal{C} \subseteq \mathbb{W}$. While the case of invariant $\mathcal{C}$ has been relatively well-studied, the results for non-invariant $\mathcal{C}$ are largely lacking in the literature. We fill this void by developing simple and geometric characterizations of the convex hull and closed convex hull of $\lambda^{-1}(\mathcal{C})$ when $\mathcal{C}$ has no invariance property. We further specialize our results to the case of invariant $\mathcal{C}$, and obtain new convexifications of $\lambda^{-1}(\mathcal{C})$ in this case.