ABSTRACT
Motivated by the Araki-Lieb-Thirring inequality tr(A1/2BA1/2)rp ≤ tr(Ar/2BrAr/2)p for p ≥ 0, r ≥ 1 (tr(A1/2BA1/2)rp ≥ tr(Ar/2BrAr/2)p for p ≥ 0, 0 ≤ r ≤ 1) for Hermitian positive semidefinite matrices and the Golden-Thompson inequality tr(exp (A + B)) ≤ tr(exp (A)exp (B)) for Hermitian matrices, in this paper, we extend these inequalities to the setting of Euclidean Jordan algebras in the form for p ≥ 0, r ≥ 1 (
for p ≥ 0, 0 ≤ r ≤ 1) for a, b ≥ 0 and tr(exp (a + b)) ≤ tr(exp (a)°exp (b)) for all a and b, where Px and x° y denote, respectively, the quadratic representation and Jordan product of x and y in Euclidean Jordan algebras.
Acknowledgments
The authors would like to thank the anonymous referee for his/her useful comments and suggestions, which helped to improve the presentation of this paper.
Disclosure statement
No potential conflict of interest was reported by the author(s).