Stahl's theorem

In matrix analysis Stahl's theorem is a theorem proved in 2011 by Herbert Stahl concerning Laplace transforms for special matrix functions.^[1] It originated in 1975 as the Bessis-Moussa-Villani (BMV) conjecture by Daniel Bessis, Pierre Moussa, and Marcel Villani.^[2] In 2004 Elliott H. Lieb and Robert Seiringer gave two important reformulations of the BMV conjecture.^[3] In 2015, Alexandre Eremenko gave a simplified proof of Stahl's theorem.^[4]

In 2023, Otte Heinävaara proved a structure theorem for Hermitian matrices introducing tracial joint spectral measures that implies Stahl's theorem as a corollary.^[5]

Statement of the theorem

Let ${\displaystyle \operatorname {tr} }$ denote the trace of a matrix. If ${\displaystyle A}$ and ${\displaystyle B}$ are ${\displaystyle n\times n}$ Hermitian matrices and ${\displaystyle B}$ is positive semidefinite, define ${\displaystyle \mathbf {f} (t)=\operatorname {tr} (\exp(A-tB))}$, for all real ${\displaystyle t\geq 0}$. Then ${\displaystyle \mathbf {f} }$ can be represented as the Laplace transform of a non-negative Borel measure ${\displaystyle \mu }$ on ${\displaystyle [0,\infty )}$. In other words, for all real ${\displaystyle t\geq 0}$,

${\displaystyle \mathbf {f} }$(t) = ${\displaystyle \int _{[0,\infty )}e^{-ts}\,d\mu (s)}$,

for some non-negative measure ${\displaystyle \mu }$ depending upon ${\displaystyle A}$ and ${\displaystyle B}$.^[6]