{
  "abstract": "For an isometry A in M_{n^2 x n}(C) with block rows A_1,...,A_n, consider the two\n\nGram spaces L_A = span{A_i A_j^*} and R_A = span{A_i^* A_j}. The paper studies\n\nthe dimensional asymmetry between them, with emphasis on the question of how\n\nsmall dim R_A can be when dim L_A is maximal.\n\nFor n = 2 a complete classification is given: the minimum is mu(2) = 4, the\n\nrealised pairs (dim R_A, dim L_A) are (1,1), (2,2), (4,1), (4,2) and (4,4), and\n\ndim R_A = 3 is impossible.\n\nFor n = 3 it is proved that dim L_A = 9 implies dim R_A >= 4, and an exact\n\nrational isometry with dim L_A = 9 and dim R_A = 5 is exhibited; hence the\n\nminimum lies in {4,5}. The lower bound is obtained through structural rank\n\nbounds, a presentation lemma for three-dimensional self-adjoint subspaces of\n\nM_3(C), a no-go theorem for the resulting rigid family, and a reduction via the\n\nmultiplicative domain of the ucp map Phi(C) = A^*(C tensor I_3)A, which shows\n\nthat any counterexample to dim R_A >= 5 must satisfy dim R_A = 4 and trivial\n\nmultiplicative domain. The witness with dim R_A = 5 lies in an explicit family\n\nparametrised by orthonormal frames in R^5, within which the bound 5 is sharp.\n\nFor n >= 4 a Weyl-operator construction yields explicit upper bounds that\n\nimprove the trivial bound 1+(n-1)^2 for n >= 5.\n\nThe conjecture dim R_A >= 5 remains open, reduced to the single residual case\n\ndim R_A = 4 with trivial multiplicative domain.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "JOSÉ TORREGROSA JIMÉNEZ",
      "orcid": "0009-0007-3745-542X"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/48jcrvdehgxra.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23558598",
  "fulltext_url": null,
  "id": "48jcrvdehgxra",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23558598",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23558598/files/DIMENSIONAL%20ASYMMETRY%20OF%20LEFT%20AND%20RIGHT%20ADJOINT%20PRODUCT%20SPACES%20OF%20ISOMETRIES.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/48jcrvdehgxra.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T12:20:40Z",
      "native_id": "23558598",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23558598"
    }
  ],
  "subjects": [],
  "title": "DIMENSIONAL ASYMMETRY OF LEFT AND RIGHT ADJOINT PRODUCT SPACES OF ISOMETRIES",
  "title_text": "DIMENSIONAL ASYMMETRY OF LEFT AND RIGHT ADJOINT PRODUCT SPACES OF ISOMETRIES",
  "type": "preprint",
  "updated_at": "2026-11-05T12:20:40Z",
  "url": "https://research.draconitic.com/paper/48jcrvdehgxra",
  "venue": "Zenodo",
  "versions": []
}