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 [math]\displaystyle{ \operatorname{tr} }[/math] denote the trace of a matrix. If [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] are [math]\displaystyle{ n\times n }[/math] Hermitian matrices and [math]\displaystyle{ B }[/math] is positive semidefinite, define [math]\displaystyle{ \mathbf{f}(t) = \operatorname{tr}(\exp(A-tB)) }[/math], for all real [math]\displaystyle{ t\geq 0 }[/math]. Then [math]\displaystyle{ \mathbf{f} }[/math] can be represented as the Laplace transform of a non-negative Borel measure [math]\displaystyle{ \mu }[/math] on [math]\displaystyle{ [0,\infty) }[/math]. In other words, for all real [math]\displaystyle{ t\geq 0 }[/math],
- [math]\displaystyle{ \mathbf{f} }[/math](t) = [math]\displaystyle{ \int_{[0,\infty)} e^{-ts}\, d\mu(s) }[/math],
for some non-negative measure [math]\displaystyle{ \mu }[/math] depending upon [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math].[6]
References
- ↑ Stahl, Herbert R. (2013). "Proof of the BMV conjecture". Acta Mathematica 211 (2): 255–290. doi:10.1007/s11511-013-0104-z.
- ↑ 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. doi:10.1063/1.522463. Bibcode: 1975JMP....16.2318B.
- ↑ Lieb, Elliott; Seiringer, Robert (2004). "Equivalent forms of the Bessis-Moussa-Villani conjecture". Journal of Statistical Physics 115 (1–2): 185–190. doi:10.1023/B:JOSS.0000019811.15510.27. Bibcode: 2004JSP...115..185L.
- ↑ Eremenko, A. È. (2015). "Herbert Stahl's proof of the BMV conjecture". Sbornik: Mathematics 206 (1): 87–92. doi:10.1070/SM2015v206n01ABEH004447. Bibcode: 2015SbMat.206...87E.
- ↑ Heinävaara, Otte. Tracial joint spectral measures.
- ↑ Clivaz, Fabien (2016). Stahl's Theorem (aka BMV Conjecture): Insights and Intuition on its Proof. Operator Theory: Advances and Applications. 254. pp. 107–117. doi:10.1007/978-3-319-29992-1_6. ISBN 978-3-319-29990-7.
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