Equivalent forms of the Bessis-Moussa-Villani conjecture

Abstract

The BMV conjecture for traces, which states that $Trexp(A-\lambda B)$ is the Laplace transform of a positive measure, is shown to be equivalent to two other statements: (i) The polynomial $\lambda \mapsto Tr(A+\lambda B{)}^p$ has only non-negative coefficients for all $A,B\ge 0$, $p\in N$ and (ii) $λ\mapsto \Tr (A+λB)^{-p}$ is the Laplace transform of a positive measure for $A,B\ge 0$, $p>0$.

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