The matrix function $e^{tA+B}$ is representable as the Laplace transform of a matrix measure

Abstract

Given a pair $A,B$ of matrices of size $n\times n$, we consider the matrix function $e^{At+B}$ of the variable $t\in \mathbb{C}$. If the matrix $A$ is Hermitian, the matrix function $e^{At+B}$ is representable as the bilateral Laplace transform of a matrix-valued measure $M(d\lambda )$ compactly supported on the real axis: $$e^{At+B}=\int e^{\lambda t}M(d\lambda ).$$ The values of the measure $M(d\lambda )$ are matrices of size $n\times n$, the support of this measure is contained in the convex hull of the spectrum of $A$. If the matrix $B$ is also Hermitian, then the values of the measure $M(d\lambda )$ are Hermitian matrices. The measure M(dλ) is not necessarily non-negative.

14 pages