{
  "abstract": "It is well known that every hyponormal operator on a complex Hilbert space whose spectrum has planar Lebesgue measure zero is normal. In particular, every compact hyponormal operator on a complex Hilbert space is normal. In this paper, we investigate analogous structural and spectral phenomena for so-called sum-hyponormal $d$-tuples on a complex Hilbert space $\\mathcal H$, namely, commuting $d$-tuples ${\\bf T}=(T_1, \\ldots, T_d)$ satisfying $\\sum_{j=1}^d [T^*_j, T_j] \\geqslant 0$. We show that every sum-hyponormal $d$-tuple of compact operators decomposes as the direct sum of a normal $d$-tuple and a quasinilpotent sum-hyponormal $d$-tuple. As a consequence, every sum-hyponormal $d$-tuple on a finite-dimensional Hilbert space is normal. In contrast to the single-operator case, a sum-hyponormal $d$-tuple need not be normaloid. Nevertheless, we establish a multivariable analogue of Putnam's inequality for sum-hyponormal $d$-tuples with commuting imaginary parts. As a consequence, we prove that ${\\bf T}$ is normal whenever the planar Lebesgue measure of $s(σ({\\bf T}))$ is zero, where $σ({\\bf T})$ denotes the Taylor spectrum of ${\\bf T}$ and ${s}$ is the complex-linear polynomial with alternating coefficients $1$ and $i$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19287",
  "author_total": 3,
  "authors": [
    {
      "name": "Sameer Chavan"
    },
    {
      "name": "Md. Ramiz Reza"
    },
    {
      "name": "Shanola S. Sequeira"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4eccyc556z2zq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19287",
  "fulltext_url": null,
  "id": "4eccyc556z2zq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19287",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7213608824",
  "pdf_url": "https://arxiv.org/pdf/2609.19287",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4eccyc556z2zq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19287",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19287"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sum of self-commutators of commuting operators",
  "title_text": "Sum of self-commutators of commuting operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/4eccyc556z2zq",
  "venue": "arXiv",
  "versions": []
}