交换子的谱刚性：动力学、共振与幂零性
Spectral Rigidity of Commutators: Dynamics, Resonance, and Nilpotency
- Faculty of Electronic Engineering, University of Niš（尼什大学电子工程学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对矩阵的内导子，依据多项式根的位置，探究二阶及任意阶关系下矩阵及其内导子的幂零性，明确了不同根分布对应的幂零判定规则。
AI中文摘要：
设 $A, T \in M_n(\mathbb{C})$，记 $\Delta_A(T) = AT - TA$ 为由 $A$ 诱导的内导子。我们根据多项式 $z^2 + \alpha z + \beta$ 的根的位置，确定在二阶关系 $\Delta_A^2(T) + \alpha\\, \Delta_A(T) + \beta\\, T = 0$（其中 $\alpha, \beta \in \mathbb{R}$）下 $T$ 和 $\Delta_A(T)$ 何时为幂零的。若根的非零实部符号相同，则 $T$ 和 $\Delta_A(T)$ 均为幂零；若根为非零纯虚数，一般无法得出幂零结论；若一根为零另一根非零，则 $\Delta_A(T)$ 为幂零但 $T$ 未必是；二重零根对应 Kleinecke–Shirokov 定理。对于异号实根，答案由算术阈值决定：将 $z_1/z_2=-p/q$ 化为既约分数，当 $p+q>n$ 时所有解均为幂零，当 $p+q\le n$ 时，显式循环构造给出使 $T$ 和 $\Delta_A(T)$ 均可逆的解。最后，对任意阶关系 $\sum_{k=0}^{m} c_k\\,\Delta_A^k(T)=0$，当关联多项式的根位于边界过原点的开半平面内时，每个迭代交换子 $\Delta_A^j(T)$ 均为幂零。
英文摘要：
Let $A, T \in M_n(\mathbb{C})$ and let $Δ_A(T) = AT - TA$ denote the inner derivation induced by $A$. We determine when $T$ and $Δ_A(T)$ are nilpotent under the second-order relation $$ Δ_A^2(T) + α\, Δ_A(T) + β\, T = 0, \qquad α, β\in \mathbb{R}, $$ according to the location of the roots of $z^2 + αz + β$. If the roots have nonzero real parts of the same sign, then both $T$ and $Δ_A(T)$ are nilpotent. If the roots are purely imaginary and nonzero, no nilpotency conclusion holds in general, whereas if one root is zero and the other is nonzero, $Δ_A(T)$ is nilpotent but $T$ need not be. The double zero root gives the Kleinecke--Shirokov theorem. For real roots of opposite signs, the answer is governed by an arithmetic threshold: writing $z_1/z_2=-p/q$ in lowest terms, every solution is nilpotent when $p+q>n$, while for $p+q\le n$ an explicit cyclic construction yields solutions for which both $T$ and $Δ_A(T)$ are invertible. Finally, for a relation $$ \sum_{k=0}^{m} c_k\,Δ_A^k(T)=0 $$ of arbitrary order, every iterated commutator $Δ_A^j(T)$ is nilpotent whenever the roots of the associated polynomial lie in an open half-plane whose boundary passes through the origin.
