# Stahl's theorem

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] Statement of the theorem Let {displaystyle operatorname {tr} } denote the trace of a matrix. If A and B are n × n Hermitian matrices and B is positive semidefinite, define {displaystyle mathbf {f} } (t) = {displaystyle operatorname {tr(exp(A-tB))} } , for all real t ≥ 0. Then {displaystyle mathbf {f} } can be represented as the Laplace transform of a non-negative Borel measure μ on {displaystyle {[0,infty )}} . In other words, for all real t ≥ 0, {displaystyle mathbf {f} } (t) = {displaystyle int _{[0,infty )}e^{-ts},dmu (s)} , for some non-negative measure μ depending upon A and B.[5] References ^ 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. ^ 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. ^ 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. ^ 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. ^ 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. Categories: Conjectures that have been provedTheorems in analysisTheorems in measure theory

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