Generalized frame operator distance problems

Abstract

Let S∈Md(C)+ be a positive semidefinite d×d complex matrix and let a=(ai)i∈Ik ∈R>0 k, indexed by Ik={1,...,k}, be a k-tuple of positive numbers. Let Td(a) denote the set of families G={gi}i∈Ik ∈(Cd)k such that ‖gi‖2=ai, for i∈Ik; thus, Td(a) is the product of spheres in Cd endowed with the product metric. For a strictly convex unitarily invariant norm N in Md(C), we consider the generalized frame operator distance function Θ(N,S,a) defined on Td(a), given by Θ(N,S,a)(G)=N(S−SG) where SG=∑_{i∈Ik} gigi^⁎ ∈ Md(C)+. In this paper we determine the geometrical and spectral structure of local minimizers G0∈Td(a) of Θ(N,S,a). In particular, we show that local minimizers are global minimizers, and that these families do not depend on the particular choice of N.