The Araki-Lieb-Thirring inequality and the Golden-Thompson inequality in Euclidean Jordan algebras

ABSTRACT

Motivated by the Araki-Lieb-Thirring inequality tr(A^1/2BA^1/2)^rp ≤ tr(A^r/2B^rA^r/2)^p for p ≥ 0, r ≥ 1 (tr(A^1/2BA^1/2)^rp ≥ tr(A^r/2B^rA^r/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 $\mathrm{t}\mathrm{r}\left(P_{b^{1/2}}\right(a){)}^{rp}\le \mathrm{t}\mathrm{r}(P_{b^{r/2}}\left(a^r\right){)}^p$ for p ≥ 0, r ≥ 1 ($\mathrm{t}\mathrm{r}\left(P_{b^{1/2}}\right(a){)}^{rp}\ge \mathrm{t}\mathrm{r}(P_{b^{r/2}}\left(a^r\right){)}^p$ 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 P_x and x° y denote, respectively, the quadratic representation and Jordan product of x and y in Euclidean Jordan algebras.

Keywords:

• Euclidean Jordan algebra
• majorization
• Araki-Lieb-Thirring inequality
• Golden-Thompson inequality

2010 Mathematics Subject Classifications:

• 15A42
• 17C20
• 17C55

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).

Additional information

Funding

The work was supported in part by National Natural Science Foundation of China (11971302 and 12071022).