Weak majorization inequalities for the cubic and quartic coefficients of $e^{(A+B)t}$ versus $e^{At}e^{Bt}$
Teng Zhang
TL;DR
The paper studies spectral comparisons between the cubic and quartic Taylor coefficients of $e^{(A+B)t}$ and those of $e^{At}e^{Bt}$ for Hermitian $A,B$. It combines the Ky Fan variational principle, the Fan--Hoffman inequality, and explicit commutator identities to show that the eigenvalues of the cubic and quartic coefficients satisfy weak majorization by the singular values of the corresponding coefficients of the Golden--Thompson product, i.e., $ lambda(H^k) prec_w sigma(Q_k)$ for $k=3,4$. Key contributions include the exact identities $R_3-H^3= rac{1}{4}[X,[X,H]]$ and $R_4-H^4= rac{1}{2}[X,[X,H^2]]- rac{1}{4}[X,H]^2$, plus a general Ky Fan reduction and a sufficient commutator-decomposition condition that paves the way to higher orders; an explicit $D_5$ formula is also provided. Significantly, the work offers a local, coefficientwise spectral analogue of the Golden--Thompson framework and BCH corrections, giving a structured method to compare higher-order terms in the Lie--Trotter setting.
Abstract
Let $A,B\in\mathbb{H}_n$ and set $H=A+B$. For each integer $k\ge 1$ define $$ Q_k:=\sum_{p=0}^k \binom{k}{p} A^pB^{k-p}, R_k:=\Re\,Q_k=\frac{Q_k+Q_k^*}{2}. $$ Then $H^k=\left.\frac{d^k}{dt^k}e^{Ht}\right|_{t=0}$ and $Q_k=\left.\frac{d^k}{dt^k}(e^{At}e^{Bt})\right|_{t=0}$. We prove that, for $k=3,4,$ $$ λ(H^k)\prec_w σ(Q_k). $$ 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.