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  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 ,
 () = , 
for some non-negative measure  depending upon  and .

References

Conjectures that have been proved
Theorems in analysis
Theorems in measure theory