# 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. It originated in 1975 as the Bessis-Moussa-Villani (BMV) conjecture by Daniel Bessis, Pierre Moussa, and Marcel Villani. In 2004 Elliott H. Lieb and Robert Seiringer gave two important reformulations of the BMV conjecture. In 2015 Alexandre Eremenko gave a simplified proof of Stahl's theorem. 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. 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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