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| Format: | Preprint |
| Publié: |
2026
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2601.07286 |
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| _version_ | 1866912817818370048 |
|---|---|
| author | Zhang, Teng |
| author_facet | Zhang, Teng |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07286 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weak majorization inequalities for the cubic and quartic coefficients of $e^{(A+B)t}$ versus $e^{At}e^{Bt}$ Zhang, Teng Functional Analysis 15A42, 15A60 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. |
| title | Weak majorization inequalities for the cubic and quartic coefficients of $e^{(A+B)t}$ versus $e^{At}e^{Bt}$ |
| topic | Functional Analysis 15A42, 15A60 |
| url | https://arxiv.org/abs/2601.07286 |