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
editLet denote the trace of a matrix. If and are Hermitian matrices and is positive semidefinite, define , for all real . Then can be represented as the Laplace transform of a non-negative Borel measure on . In other words, for all real ,
- (t) = ,
for some non-negative measure depending upon and .[6]
References
edit- ^ 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.255-290&rft.date=2013&rft_id=info:arxiv/1107.4875&rft_id=info:doi/10.1007/s11511-013-0104-z&rft.au=Stahl, Herbert R.&rfr_id=info:sid/en.wikipedia.org:Stahl's theorem" class="Z3988">
- ^ 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:10.1063/1.522463.2318-2325&rft.date=1975&rft_id=info:doi/10.1063/1.522463&rft_id=info:bibcode/1975JMP....16.2318B&rft.au=Bessis, D.&rft.au=Moussa, P.&rft.au=Villani, M.&rft_id=https://doi.org/10.1063%2F1.522463&rfr_id=info:sid/en.wikipedia.org:Stahl's theorem" class="Z3988">
- ^ 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.0000019811.15510.27.1–2&rft.pages=185-190&rft.date=2004&rft_id=info:arxiv/math-ph/0210027&rft_id=info:doi/10.1023/B:JOSS.0000019811.15510.27&rft_id=info:bibcode/2004JSP...115..185L&rft.au=Lieb, Elliott&rft.au=Seiringer, Robert&rfr_id=info:sid/en.wikipedia.org:Stahl's theorem" class="Z3988">
- ^ 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/SM2015v206n01ABEH004447.87-92&rft.date=2015&rft_id=info:arxiv/1312.6003&rft_id=info:doi/10.1070/SM2015v206n01ABEH004447&rft_id=info:bibcode/2015SbMat.206...87E&rft.au=Eremenko, A. È.&rfr_id=info:sid/en.wikipedia.org:Stahl's theorem" class="Z3988">
- ^ Heinävaara, Otte (2023). "Tracial joint spectral measures". arXiv:2310.03227 [math.FA].
- ^ 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 0255-0156.107-117&rft.date=2016&rft_id=info:arxiv/1702.06403&rft.issn=0255-0156&rft_id=info:doi/10.1007/978-3-319-29992-1_6&rft.isbn=978-3-319-29990-7&rft.aulast=Clivaz&rft.aufirst=Fabien&rfr_id=info:sid/en.wikipedia.org:Stahl's theorem" class="Z3988">