{
  "abstract": "Let $C^1[0,1]$ be the complex linear space of all continuously differentiable complex-valued functions $f$ on the unit interval $[0,1]$ with respect to the norm $ \\|f\\|_σ = |f(0)| +\\|f'\\|_\\infty$. Let $P_1, P_2: C^1[0,1] \\rightarrow C^1[0,1]$ be distinct, nonzero idempotent maps, which are not necessarily linear, such that $P_1P_2 = P_2P_1 = 0$ and $P_1+P_2 = I$, where $I$ denotes the identity operator. It is proved that $λ_1P_1 + λ_2P_2$ is an isometry on $C^1[0,1]$, for some distinct unit modulus complex numbers $λ_1, λ_2$, if and only if either $λ_1 + λ_2 = 0$, or $λ_1P_1 + λ_2P_2$ is an isometry for all such complex numbers $λ_1, λ_2$. In the former case, the collection $\\{P_1, P_2\\}$ is called a family of generalized bi-circular idempotents; in the latter case, it is called a family of bi-circular idempotents. The structure of isometric reflection on $C^1[0,1]$, that is, an isometry $T$ such that $T^2 = I$, is characterized, and its relationship with the above class of idempotents maps is also discussed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18967",
  "author_total": 3,
  "authors": [
    {
      "name": "Himanshu Kumar"
    },
    {
      "name": "Hemant Kumar"
    },
    {
      "name": "Abdullah Bin Abu Baker"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3hhd2mv0vevwa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18967",
  "fulltext_url": null,
  "id": "3hhd2mv0vevwa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18967",
  "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",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7213523858",
  "pdf_url": "https://arxiv.org/pdf/2609.18967",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3hhd2mv0vevwa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18967",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18967"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Structure of Generalized bi-circular idempotents and isometric reflections on $C^1[0,1]$",
  "title_text": "Structure of Generalized bi-circular idempotents and isometric reflections on C^1[0,1]",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/3hhd2mv0vevwa",
  "venue": "arXiv",
  "versions": []
}