Support convergence in the single ring theorem

Abstract

We study the eigenvalues of non-normal square matrices of the form An = UnTnVn with Un, Vn independent Haar distributed on the unitary group and Tn real diagonal. We show that when the empirical measure of the eigenvalues of Tn converges, and Tn satisfies some technical conditions, all these eigenvalues lie in a single ring. © 2011 Springer-Verlag.