A note on the Ostrowski–Schneider type inertia theorem in Euclidean Jordan algebras

In a recent paper [7], Gowda et al. extended Ostrowski–Schneider type inertia results to certain linear transformations on Euclidean Jordan algebras. In particular, they showed that In(a)=In(x)In(a)=In(x) whenever a∘x>0a∘x>0 by the min–max theorem of Hirzebruch, where the inertia of an element x in a Euclidean Jordan algebra is defined byIn(x):=(π(x),ν(x),δ(x)),In(x):=(π(x),ν(x),δ(x)),with π(x)π(x), ν(x)ν(x), and δ(x)δ(x) denoting, respectively, the number of positive, negative, and zero eigenvalues, counting multiplicities. In this paper, we present a Peirce decomposition version of Wimmer’s result [13] and show that it is equivalent to the above result. In addition, we extend Higham and Cheng’s result ([8], Lemma 4.2) to the setting of Euclidean Jordan algebras.