Exposed faces and duality for symmetric and unitarily invariant norms

Abstract

Let ψ be a unitarily invariant norm on the space of (real or complex) n×m matrices, and g the associated symmetric gauge function: thus ψ(A)g(s(A)), where s(A) is the decreasing sequence of singular values of A. Denote by Bψ and Bg the closed unit balls of ψ and g. In a previous paper we showed a close relationship between the faces of Bψ and those of Bg. In particular, to each face of Bψ we associated a standard face of Bg, and we used this association to completely describe the matrices that are members of an individual face of ψ. In the present paper, we consider the duality operator that transforms each exposed face of Bψ into an exposed face of the unit ball of ψ's dual and the duality operator that does the same with exposed faces of Bg. We show that these two operators are very nicely related. Among other results, we prove the following. Given an exposed face E of Bψ, the standard face associated with the dual of E is precisely the dual of the standard face of Bg associated with E. E is an exposed face of Bψ if and only if its associated standard face is an exposed face of Bg. As a by-product we completely determine the subdifferential of ψ in terms of the subdifferential of g, and we completely characterize the matrices that are dual to a given matrix A with respect to ψ. © 1994.