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.

denote the trace of a matrix.

Hermitian matrices and

is positive semidefinite, define

can be represented as the Laplace transform of a non-negative Borel measure

, for some non-negative measure