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

Renbo Zhao

Abstract

Given a finite-dimensional FTvN system $(\mathbb{V},\mathbb{W},λ)$, we study the convexification of the spectral set $λ^{-1}(\mathcal{C})$ 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 $λ^{-1}(\mathcal{C})$ when $\mathcal{C}$ has no invariance property. We further specialize our results to the case of invariant $\mathcal{C}$, and obtain new convexifications of $λ^{-1}(\mathcal{C})$ in this case.

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

Abstract

Given a finite-dimensional FTvN system , we study the convexification of the spectral set induced by a set . While the case of invariant has been relatively well-studied, the results for non-invariant 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 when has no invariance property. We further specialize our results to the case of invariant , and obtain new convexifications of in this case.
Paper Structure (9 sections, 18 theorems, 40 equations)

This paper contains 9 sections, 18 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Notations
  4. Preliminaries
  5. Main Results
  6. First Result
  7. Second Result
  8. Specializing the Results in Section \ref{['sec:second']} to Invariant Sets
  9. Proof of Lemma \ref{['lem:spectral_set']}

Key Result

Lemma 2.1

The following statements are equivalent:

Theorems & Definitions (49)

  • Definition 2.1: Orbit
  • Remark 2.1
  • Lemma 2.1: Gowda_19
  • Lemma 2.2: Gowda_19
  • Definition 2.2: Majorization
  • Lemma 2.3: Gowda_23
  • Lemma 2.4: Gowda_19
  • Definition 2.3: Reduced System
  • Remark 2.2
  • Definition 2.4: Invariant Set
  • ...and 39 more