Spectral Rigidity of Commutators: Dynamics, Resonance, and Nilpotency

Abstract

Let $A, T \in M_n(\mathbb{C})$ and let $\Delta_A(T) = AT - TA$ denote the inner derivation induced by $A$. We determine when $T$ and $\Delta_A(T)$ are nilpotent under the second-order relation $$ \Delta_A^2(T) + \alpha\, \Delta_A(T) + \beta\, T = 0, \qquad \alpha, \beta \in \mathbb{R}, $$ according to the location of the roots of $z^2 + \alpha z + \beta$. If the roots have nonzero real parts of the same sign, then both $T$ and $\Delta_A(T)$ 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, $\Delta_A(T)$ 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 $\Delta_A(T)$ are invertible. Finally, for a relation $$ \sum_{k=0}^{m} c_k\,\Delta_A^k(T)=0 $$ of arbitrary order, every iterated commutator $\Delta_A^j(T)$ is nilpotent whenever the roots of the associated polynomial lie in an open half-plane whose boundary passes through the origin.