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

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.