{
  "abstract": "We study the set $Γ$ of graphs of closed, densely defined operators in a Hilbert space $H$, regarded as a subset of the Grassmann manifold $P(H\\times H)$ of orthogonal projections in $H\\times H$. We show that the subset $Γ^b$ of graphs of bounded operators is the open unit ball of $P(H\\times H)$ centered at the graph of the zero operator $P_0$ (which projects onto $H\\times\\{0\\}$). This ball is diffeomorphic to $B(H)$ via the map $T\\mapsto P_T$ ($=$ the projection onto the graph ${Gr(T)}$ of $T$). We show that graphs of unbounded closed operators lie at the boundary of $Γ$. We also study the existence and characteristics of minimal geodesics of $P(H\\times H)$ joining two graphs $Gr(A)$, $Gr(B)$. If $A,B$ are selfadjoint, such a geodesic always exists, and we construct explicitly a distinguished exponent using the five-space decomposition associated to the pair of subspaces $Gr(A)$, $Gr(B)$. An explicit low-dimensional example shows that the geodesic joining two graphs need not remain inside $Γ$, i.e., does not consist entirely of graphs. We also relate graphs of compact operators to the restricted Grassmannian, and study the problem of common complements for pairs $Gr(S)$, $Gr(T)$, giving positive results when one operator is bounded or under lower boundedness conditions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30120",
  "author_total": 3,
  "authors": [
    {
      "name": "Esteban Andruchow"
    },
    {
      "name": "Lazaro Recht"
    },
    {
      "name": "Alejandro Varela"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/88ryn481kzr70.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30120",
  "fulltext_url": null,
  "id": "88ryn481kzr70",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30120",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.OA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2608.30120",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/88ryn481kzr70.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30120",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30120"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Graphs of operators as points in the Grassmann manifold",
  "title_text": "Graphs of operators as points in the Grassmann manifold",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/88ryn481kzr70",
  "venue": "arXiv",
  "versions": []
}