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]

References

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