{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29574",
  "author_total": 1,
  "authors": [
    {
      "name": "Hranislav Stanković"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fjenp3kxwm9aj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29574",
  "fulltext_url": null,
  "id": "fjenp3kxwm9aj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29574",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7204877283",
  "pdf_url": "https://arxiv.org/pdf/2608.29574",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fjenp3kxwm9aj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29574",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29574"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Spectral Rigidity of Commutators: Dynamics, Resonance, and Nilpotency",
  "title_text": "Spectral Rigidity of Commutators: Dynamics, Resonance, and Nilpotency",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/fjenp3kxwm9aj",
  "venue": "arXiv",
  "versions": []
}