Reverse inequality to Golden–Thompson type inequalities: Comparison of [formula omitted] and [formula omitted

Abstract

Let A and B be n-by- n Hermitian matrices. Then Tr e A e B ⩽ S ( α ) Tr e A + B where α is the condition number of e A and S ( t ) is the Specht ratio of the reverse arithmetic-geometric mean inequality. It is a sharp reverse result to the Golden–Thompson inequality. This can be extended to each eigenvalue. Equivalently there exists a unitary V such that e A / 2 e B e A / 2 ⩽ S ( α ) V e A + B V ∗ . We also show that there exists a unitary W such that W e A + B W ∗ ⩽ S ( α ) e A / 2 e B e A / 2 .