Generalized frame operator distance problems (1812.10365v1)

Abstract: Let $S\in\mathcal{M}d(\mathbb{C})^+$ be a positive semidefinite $d\times d$ complex matrix and let $\mathbf a=(a_i){i\in\mathbb{I}k}\in \mathbb{R}{>0}^k$, indexed by $\mathbb{I}k={1,\ldots,k}$, be a $k$-tuple of positive numbers. Let $\mathbb T{d}(\mathbf a )$ denote the set of families $\mathcal G={g_i}{i\in\mathbb{I}_k}\in (\mathbb{C}^d)^k$ such that $|g_i|^2=a_i$, for $i\in\mathbb{I}_k$; thus, $\mathbb T{d}(\mathbf a )$ is the product of spheres in $\mathbb{C}^d$ endowed with the product metric. For a strictly convex unitarily invariant norm $N$ in $\mathcal{M}d(\mathbb{C})$, we consider the generalized frame operator distance function $\Theta{( N \, , \, S\, , \, \mathbf a)}$ defined on $\mathbb T_{d}(\mathbf a )$, given by $$ \Theta_{( N \, , \, S\, , \, \mathbf a)}(\mathcal G) =N(S-S_{\mathcal G }) \quad \text{where} \quad S_{\mathcal G}=\sum_{i\in\mathbb{I}k} g_i\,g_i^*\in\mathcal{M}_d(\mathbb{C})^+\,. $$ In this paper we determine the geometrical and spectral structure of local minimizers $\mathcal G_0\in\mathbb T{d}(\mathbf a )$ of $\Theta_{( N \, , \, S\, , \, \mathbf a)}$. In particular, we show that local minimizers are global minimizers, and that these families do not depend on the particular choice of $N$.