On the completely-Q property for linear transformations on Euclidean Jordan algebras

In this article, we introduce and study ultra and super SSM-(cone GUS)-properties for linear transformations on Euclidean Jordan algebras. We show that the Jordan quadratic strict semimonotonicity (JQSSM)-property is equivalent to the completely-Q-property for linear transformations on any Euclidean Jordan algebra. Also, we investigate some interconnections between the SSM-property, the Jordan linear SSM (JLSSM)-property, and the completely-Q property on a Euclidean Jordan algebra. Moreover, we give a characterization of the completely-Q property on the Jordan spin algebra and study the completely-Q property for some special transformations on a Euclidean Jordan algebra.