Weak majorization inequalities for the cubic and quartic coefficients of $e^{(A+B)t}$ versus $e^{At}{e}^{Bt}$

Abstract

Let $A,B\in {\mathbb{H}}_n$ and set $H=A+B$. For each integer $k\ge 1$ define $$Q_k:=\sum _{p=0}^k\left(\genfrac{}{}{0ex}{}{k}{p}\right)A^p{B}^{k-p},R_k:=\Re Q_k=\frac{Q_k+Q_k^{\ast }}{2}.$$ Then $H^k={\frac{d^k}{d{t}^k}{e}^{Ht}|}_{t=0}$ and $Q_k={\frac{d^k}{d{t}^k}(e^{At}{e}^{Bt})|}_{t=0}$. We prove that, for $k=3,4,$ $$\lambda (H^k){\prec }_w\sigma (Q_k).$$ Equivalently, the eigenvalues of the cubic and quartic Taylor coefficients of $e^{(A+B)t}$ are weakly majorized by the singular values of the corresponding coefficients of the Golden--Thompson product $e^{At}{e}^{Bt}$. Our argument combines Ky Fan variational principles with explicit commutator identitiesfor $R_k-H^k$ at orders $k=3,4$, reducing the problem to the positivity of certain double-commutator trace forms tested against Ky Fan maximizing projections. We also record a general sufficient condition for higher orders based on commutator decompositions.

arXiv comment10 pages. All comments are welcome!