CHARACTERIZATION OF GLOBALLY-UNIQUELY-SOLVABLE PROPERTY OF A CONE-PRESERVING Z-TRANSFORMATION ON EUCLIDEAN JORDAN ALGEBRAS

Abstract

Let V be a Euclidean Jordan algebra with a symmetric cone K. We show that for a Z-transformation L with the additional property L(K) subset of K (which we will call 'cone-preserving'), GUS double left right arrow strictly copositive on K double left right arrow monotone + P. Specializing the result to the Stein transformation SA(X) := X - AXA(T) on the space of real symmetric matrices with the property S-A (S-+(n)) subset of S-+(n) we deduce that S-A GUS double left right arrow I +/- A positive definite.