Similarity automorphism invariance of some P-properties of linear transformations on Euclidean Jordan algebras

Recently, Gowda and Sznajder [Gowda, M.S., Sznajder, R.: Automorphism invariance of P- and GUS-properties of linear transformations on Euclidean Jordan algebras. Math. Oper. Res. 31, 109–123 (2006)] have introduced and studied automorphism invariance of some P-properties for linear transformations. This paper deals with this automorphism invariance of some other complementarity properties, such as E0,P0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {E}_0,\,\hbox {P}_0$$\end{document}, S, Z-properties. Particularly, we answer Gowda and Sznajder in positive that order P-property is algebra automorphism invariant in simple Jordan algebras. By replacing transposition with the invertibility in the concept of automorphism invariance, we propose a notion of similarity automorphism invariance. Most complementarity properties of linear transformations are also shown to be similarity invariant under algebra automorphisms and cone automorphisms.