Generalized frame operator distance problems

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 ${\mathbb{I}}_k=\{1,\dots ,k\}$, 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 $\Vert g_i{\Vert }^2=a_i$, 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 $${\Theta }_{(N,S,\mathbf{a})}(\mathcal{G})=N(S-S_{\mathcal{G}})\text{where}{S}_{\mathcal{G}}=\sum _{i\in {\mathbb{I}}_k}{g}_i{g}_i^{\ast }\in {\mathcal{M}}_d(\mathbb{C}{)}^{+}.$$ 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 $N$.

24 pages. There exists text overlap with other arxiv manuscripts in the preliminary sections