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PreprintarXiv licencearXiv

Structure of Generalized bi-circular idempotents and isometric reflections on C1[0,1]

  • Himanshu Kumar
  • Hemant Kumar
  • Abdullah Bin Abu Baker
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Abstract

Let C1[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 P1, P2: C1[0,1] → C1[0,1] be distinct, nonzero idempotent maps, which are not necessarily linear, such that P1P2 = P2P1 = 0 and P1+P2 = I, where I denotes the identity operator. It is proved that λ1P1 + λ2P2 is an isometry on C1[0,1], for some distinct unit modulus complex numbers λ1, λ2, if and only if either λ1 + λ2 = 0, or λ1P1 + λ2P2 is an isometry for all such complex numbers λ1, λ2. In the former case, the collection {P1, P2} 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 C1[0,1], that is, an isometry T such that T2 = I, is characterized, and its relationship with the above class of idempotents maps is also discussed.

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Cite

BibTeXRIS
BibTeX
@misc{kumar2026structure,
  title = {{Structure of Generalized bi-circular idempotents and isometric reflections on $C^1[0,1]$}},
  author = {Himanshu Kumar and Hemant Kumar and Abdullah Bin Abu Baker},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18967},
  eprint = {2609.18967},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18967},
}
RIS
TY  - UNPB
TI  - Structure of Generalized bi-circular idempotents and isometric reflections on C^1[0,1]
AU  - Himanshu Kumar
AU  - Hemant Kumar
AU  - Abdullah Bin Abu Baker
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.18967
UR  - https://arxiv.org/abs/2609.18967
AB  - 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] → 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.
ER  -