Generalized frame operator distance problems

Let $Sinmathcal{M}_d(mathbb{C})^+$ be a positive semidefinite $dtimes d$ complex matrix and let $mathbf a=(a_i)_{iinmathbb{I}_k}in mathbb{R}_{>0}^k$, 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 $mathcal G={g_i}_{iinmathbb{I}_k}in (mathbb{C}^d)^k$ such that $|g_i|^2=a_i$, for $iinmathbb{I}_k$; 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 $$ Theta_{( N , , , S, , , mathbf a)}(mathcal G) =N(S-S_{mathcal G }) quad text{where} quad S_{mathcal G}=sum_{iinmathbb{I}_k} g_i,g_i^*inmathcal{M}_d(mathbb{C})^+,. $$ In this paper we determine the geometrical and spectral structure of local minimizers $mathcal G_0inmathbb T_{d}(mathbf a )$ 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 $N$.