Convexity of sums of eigenvalues of a segment of unitaries

Abstract

For an n×n unitary matrix u = ez with z skew-Hermitian, the angles of u are the arguments of its spectrum, i.e. the spectrum of −iz. For 1 ≤ m ≤ n, we show that sm(t), the sum of the first m angles of the path t ↦→ etxey of unitary matrices, is a convex function of t (provided the path stays in a vicinity of the identity matrix). This vicinity is described in terms of the operator norm of matrices, and it is optimal. We show that when all the maps t ↦→ sm(t) are linear, then x commutes with y. Several applications to unitarily invariant norms in the unitary group are given. Then we extend these applications to Ad-invariant Finsler norms in the special unitary group of matrices. This last result is obtained by proving that any Adinvariant Finsler norm in a compact semi-simple Lie group K is the supremum of a family of what we call orbit norms, induced by the Killing form of K.