Schur ideals and homomorphisms of the semidefinite cone

Abstract

The semidefinite cone Kn consisting of all n by n real symmetric positive semidefinite matrices is considered. A set I in Kn is said to be a Schur ideal if it is closed under addition, multiplication by nonnegative scalars, and Schur multiplication by any element of Kn. A Schur homomorphism of Kn is a mapping of Kn to itself that preserves addition, (nonnegative) scalar multiplication and Schur products. This paper is concerned with Schur ideals and homomorphisms of Kn. It shows that in the topology induced by the trace inner product, Schur ideals in Kn need not be closed, all finitely generated Schur ideals are closed, and in K2, a Schur ideal is closed if and only if it is a principal ideal. It also characterizes Schur homomorphisms of Kn and, in particular, shows that any Schur automorphism of Kn is of the form Φ(X) = P XPT for some permutation matrix P.