Abstract
The BMV conjecture states that for \(n\times n\) Hermitian matrices \(A\) and \(B\) the function \(f_{A,B}(t)={{\mathrm{trace\,}}}e^{tA+B}\) is exponentially convex. Recently the BMV conjecture was proved by Herbert Stahl. The proof of Herbert Stahl is based on ingenious considerations related to Riemann surfaces of algebraic functions. In the present paper we give a purely “matrix” proof of the BMV conjecture for \(2\times 2\) matrices. This proof is based on the Lie product formula for the exponential of the sum of two matrices. The proof also uses the commutation relations for the Pauli matrices and does not use anything else.
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Notes
We omit the “trivial” factors \(M_{\,0}^{+}=I\), \(M_{\,0}^{-}=I\).
References
(1965) (in Russian). English translation: Akhiezer, N.I.: The Clasical Moment Problem. Oliver and Boyd, Edinburgh (1965)
Bernstein, S.N.: Sur les functions absolument monotones. Acta Math. 52 (1928), 1–66. (In French)
AH CCCP, 1952
Bessis, D., Moussa, P., Villani, M.: Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics. J. Math. Phys. 16(11), 2318–2325 (1975)
Eremenko, A.: Herbert Stahl’s proof of the BMV conjecture. arXiv:1312.6003
206(1), 97–102 (2015) (in Russian). English translation: Eremenko, A.: Herbert Stahl’s proof of the BMV conjecture. Sb. Math. 206(1), 87–92 (2015)
Hall, B.C.: Lie Groups, Lie Algebras and Representations. Springer, New York (2003)
Katsnelson, V.: The function \(\cosh \sqrt{at^2+b}\) is exponentially convex. arXiv:1502.04201
Mehta, M.L., Kumar, K.: On an integral representation of the function \(\text{ Tr }\,[{\rm e}^{A-\lambda B}]\). J. Phys. A Math. Gen. 9(2), 197–206 (1976)
Stahl H.: Proof of the BMV conjecture, pp. 1–56. arXiv:1107.4875v1. 25 July 2011 (2011)
Stahl H.: Proof of the BMV conjecture, pp. 1–25. arXiv:1107.4875v3. 17 August 2012 (2012)
Stahl, H.: Proof of the BMV conjecture. Acta Math. 211, 255–290 (2013)
Widder, D.V.: Laplace Transform. Princeton University Press, Princeton (1946)
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Communicated by Dmitry Kaliuzhnyi-Verbovetskyi.
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Katsnelson, V. On the BMV Conjecture for 2 \(\times \) 2 Matrices and the Exponential Convexity of the Function \({\cosh (\sqrt{at^2+b})}\) . Complex Anal. Oper. Theory 11, 843–855 (2017). https://doi.org/10.1007/s11785-015-0513-4
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DOI: https://doi.org/10.1007/s11785-015-0513-4
Keywords
- BMV conjecture
- Absolutely monotonic functions
- Exponentially convex functions
- Positive definite functions