Golden–Thompson inequality

In physics and mathematics, the Golden–Thompson inequality, proved independently by Golden (1965) and Thompson (1965), says that for Hermitian matrices A and B,

\operatorname {tr}\,e^{{A+B}}\leq \operatorname {tr}\left(e^{A}e^{B}\right)

where tr is the trace, and e^A is the matrix exponential. This trace inequality is of particular significance in statistical mechanics, and was first derived in that context.

Bertram Kostant (1973) used the Kostant convexity theorem to generalize the Golden–Thompson inequality to all compact Lie groups.

References

Bhatia, Rajendra (1997), Matrix analysis, Graduate Texts in Mathematics, 169, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94846-1, MR 1477662
J.E. Cohen, S. Friedland, T. Kato, F. Kelly, Eigenvalue inequalities for products of matrix exponentials, Linear algebra and its applications, Vol. 45, pp. 55–95, 1982. doi:10.1016/0024-3795(82)90211-7
Golden, Sidney (1965), "Lower bounds for the Helmholtz function", Phys. Rev., Series II, 137: B1127–B1128, MR 0189691, doi:10.1103/PhysRev.137.B1127
Kostant, Bertram (1973), "On convexity, the Weyl group and the Iwasawa decomposition", Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, 6: 413–455, ISSN 0012-9593, MR 0364552
D. Petz, A survey of trace inequalities, in Functional Analysis and Operator Theory, 287–298, Banach Center Publications, 30 (Warszawa 1994).
Thompson, Colin J. (1965), "Inequality with applications in statistical mechanics", Journal of Mathematical Physics, 6: 1812–1813, ISSN 0022-2488, MR 0189688, doi:10.1063/1.1704727

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