{
  "abstract": "In this paper, we investigate structural properties of the parallel sum of set-valued operators in a real Hilbert space, with particular emphasis on monotonicity, maximal monotonicity, and paramonotonicity. We examine the decomposition-invariance property of the parallel sum of two paramonotone operators and provide a direct proof based on monotonicity and the equality of admissible decompositions. This approach establishes the required orthogonality relations without invoking the Attouch--Th{é}ra Duality duality framework. We also revisit the classical maximal-monotonicity property of the parallel sum and present an alternative graph-based derivation involving an auxiliary shifted operator. Furthermore, we study the parallel sum of a set-valued operator and the normal cone operator associated with a closed linear subspace. Explicit characterizations of its values, domain, and range are presented, and a decomposition-independent representation is obtained under paramonotonicity through the orthogonality of the underlying subspaces. These results provide a unified and self-contained treatment of selected properties of parallel sums and clarify the role of paramonotonicity in their structural behavior.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05523",
  "author_total": 1,
  "authors": [
    {
      "name": "Salihah Thabet Alwadani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e8rdqk8qkep2t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05523",
  "fulltext_url": null,
  "id": "e8rdqk8qkep2t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05523",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05523",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e8rdqk8qkep2t.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05523",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05523"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On Parallel Sums of Set-Valued Operators: Monotonicity, Paramonotonicity, and Decomposition Invariance",
  "title_text": "On Parallel Sums of Set-Valued Operators: Monotonicity, Paramonotonicity, and Decomposition Invariance",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:29Z",
  "url": "https://research.draconitic.com/paper/e8rdqk8qkep2t",
  "venue": "arXiv",
  "versions": []
}