Representation of the function Tr(exp(A - λB)) as a Laplace transform with positive weight and some matrix inequalities

Abstract

The conjecture that Tr(exp(A- lambda B)) can be written as a Laplace transform with positive measure rho is considered for finite Hermitian matrices A and B by means of Bernstein's theorem. An explicit formula is given for the moments of rho in terms of divided differences of exp(A) and elements of B. For a large class of matrices A and B the moments of rho take their maximum and minimum values when A and B commute and so upper and lower bounds for the moments of rho are established; further analysis suggests that this is generally true if B is positive definite and A and B are bounded. Some inequalities for the divided differences of the exponential are derived. Also, if A and B are both positive definite, upper and lower bounds are derived for Tr(AnBn) and Tr(AB)n in terms of the eigenvalues of A and B. Applications to problems of statistical mechanics and possibly Euclidean field theory are mentioned.