Equality Cases in Matrix Exponential Inequalities

The Golden–Thompson inequality states that for any Hermitian matrices A and B, $\operatorname{tr} e^A e^B \geqq \operatorname{tr} e^{A + B} $ and the Bernstein inequality states that for any matrix A, $\operatorname{tr} e^{A^ * + A} \geqq \operatorname{tr} e^{A^ * } e^A $. In this paper, $\{\operatorname{tr} ( e^{X/2^k } e^{Y/2^k } )^{2k} \}$ is shown to be a monotonic sequence when X and Y are Hermitian matrices or when $X = Y^ * $. Then we prove that (i) equality holds in the Golden–Thompson inequality if and only if A and B commute, and (ii) equality holds in the Bernstein inequality if and only if A is normal.