Faces of the unit ball of a unitarily invariant norm

AbstractLet ψ be a unitarily invariant norm on the space of all n×m complex matrices, and let g:Rn→R be the symmetric gauge function associated with ψ. That is to say, we have ψ(A)g(s(A)) for any A, where s(A) is the nonincreasing sequence of singular values of A. In this article we consider the relationship between the facial structures of the closed unit balls of g and ψ, which we denote by Bg and Bψ. Our main result gives a complete characterization of the faces of Bψ in terms of the faces of Bg. For, with each face E of Bψ we associate a standard face FE of Bg (i.e.,FE is a face of Bg whose barycenter is a nonnegative, nonincreasing n-vector). Conversely, each standard face F of Bg originates a set of n×m matrices, say the set MF, which is proven to be a face of Bψ. Our main result also asserts that any face E of Bψ is of the form QMFR, where Q and R are unitary matrices of appropriate orders and FFE. We then explore the main result and draw some of its consequences. For example, we show that the singular values of the barycenter of a face E of Bψ have the same multiplicities as the barycenter of FE. Formulas are given which relate the dimensions of E and FE. The relative interior and the relative boundary of E are characterized in terms of the relative interior and the relative boundary of FE. We completely describe the face of Bψ generated by a matrix A. Our results also cover the case of real matrices