Some New Results for Linear Transformations on Euclidean Jordan Algebras

Generalizing the w-P property of a matrix, Tao [Some w-P properties for linear transformations on Euclidean Jordan algebras, to appear in Pacific J Optimization] recently introduced and studied the w-P and the w-uniqueness properties for linear transformations defined on Euclidean Jordan algebras. In this paper, we study further to these properties. In particular, we specialize them to the space S-n of all n x n real symmetric matrices and the space H-n of all n x n complex Hermitian matrices for Lyapunov and Stein transformations. We also present a sufficient condition for the w-uniqueness property on S-n. In addition, we give a characterization of the w-P and the column sufficiency properties for a matrix-induced transformation on Euclidean Jordan algebras.