Spectral Rigidity of Commutators: Dynamics, Resonance, and Nilpotency
Abstract
Let A, T ∈ Mn(ℂ) and let ΔA(T) = AT − TA denote the inner derivation induced by A. We determine when T and ΔA(T) are nilpotent under the second-order relation ΔA2(T) + α ΔA(T) + β T = 0, α, β∈ ℝ, according to the location of the roots of z2 + αz + β. If the roots have nonzero real parts of the same sign, then both T and ΔA(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, ΔA(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 z1/z2=−p/q in lowest terms, every solution is nilpotent when p+q>n, while for p+q≤ n an explicit cyclic construction yields solutions for which both T and ΔA(T) are invertible. Finally, for a relation ∑k=0m ck ΔAk(T)=0 of arbitrary order, every iterated commutator ΔAj(T) is nilpotent whenever the roots of the associated polynomial lie in an open half-plane whose boundary passes through the origin.
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BibTeX
@misc{stankovic2026spectral,
title = {{Spectral Rigidity of Commutators: Dynamics, Resonance, and Nilpotency}},
author = {Hranislav Stanković},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29574},
eprint = {2608.29574},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29574},
}
RIS
TY - UNPB
TI - Spectral Rigidity of Commutators: Dynamics, Resonance, and Nilpotency
AU - Hranislav Stanković
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29574
UR - https://arxiv.org/abs/2608.29574
AB - Let A, T ∈ M_n(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, α, β∈ 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≤ n an explicit cyclic construction yields solutions for which both T and Δ_A(T) are invertible. Finally, for a relation ∑_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.
ER -