On a special case of the Herbert Stahl theorem

TL;DR

This paper provides a matrix-based proof for the BMV conjecture in the special case where matrix A has rank one, avoiding complex analysis and relying on the Lie product formula.

Contribution

It offers a new, purely matrix-analytic proof of the BMV conjecture for rank-one matrices, complementing Stahl's complex-analytic approach.

Findings

Proves the BMV conjecture for rank-one matrices using matrix methods.

Avoids complex analysis by employing the Lie product formula.

Simplifies the proof for a special case of the conjecture.

Abstract

The BMV conjecture states that for $n\times n$ Hermitian matrices $A$ and $B$ the function $f_{A,B}(t)=trace{\, } e^{tA+B}$ 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 the special case $rank\,A=1$. This proof is based on the Lie product formula for the exponential of the sum of two matrices and does not require complex analysis.