Stahl's theorem explained

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

\operatorname{tr} denote the trace of a matrix. If A and B are n x n Hermitian matrices and B is positive semidefinite, define f (t) = \operatorname{tr}(\exp(A - tB)) , for all real t\geq 0 . Then f can be represented as the Laplace transform of a non-negative Borel measure \mu on [ 0 , inf ty) . In other words, for all real t\geq 0 ,f = \int _{[0,infty)} e ^{-ts} d\mu(s) , for some non-negative measure \mu depending upon A and B .^[6]

References

1. Stahl, Herbert R.. Proof of the BMV conjecture. Acta Mathematica. 211. 2. 2013. 255–290. 10.1007/s11511-013-0104-z . 1107.4875.
2. Bessis, D.. Moussa, P.. Villani, M.. Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics. Journal of Mathematical Physics. 16. 11. 2318–2325. 1975. 10.1063/1.522463. 1975JMP....16.2318B. free.
3. Lieb, Elliott. Seiringer, Robert. Equivalent forms of the Bessis-Moussa-Villani conjecture. Journal of Statistical Physics. 115. 1–2. 185–190. 2004. 10.1023/B:JOSS.0000019811.15510.27. 2004JSP...115..185L. math-ph/0210027.
4. Eremenko, A. È.. Herbert Stahl's proof of the BMV conjecture. Sbornik: Mathematics. 206. 1. 2015. 87–92. 10.1070/SM2015v206n01ABEH004447. 2015SbMat.206...87E. 1312.6003.
5. Heinävaara, Otte. Tracial joint spectral measures. 2023 . math.FA . 2310.03227.
6. Book: Clivaz. Fabien. Stahl's Theorem (aka BMV Conjecture): Insights and Intuition on its Proof. 254. 2016. 107–117. 0255-0156. 10.1007/978-3-319-29992-1_6. Operator Theory: Advances and Applications. 978-3-319-29990-7 . 1702.06403.