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Peer-reviewedCC BY 4.0Mathematics

Asymptotic Behavior of Distance near the Origin Between Two Elements of the Semigroup in a Banach Algebra

  • Mohamed Agti1,2
  • Abdelkader Mokhtari3,4
4 affiliations
  1. 1Department of Mathematics, University of Biskra, BP 145 RP, Biskra 07000, Algeria
  2. 2Laboratory of Applied Mathematics, Ouargla 30000, Algeria
  3. 3Department of Mathematics, University of Laghouat, Laghouat 03000, Algeria
  4. 4Pure and Applied Mathematical Laboratory, Laghouat 03000, Algeria
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Abstract

This paper examines the local asymptotic behavior of the distance ‖T(t + ξ(t)) − T(t)‖ near the origin, where (T(t))t>0 is a semigroup on a commutative Banach algebra and ξ : ]0, +∞[ → ]0, +∞[ is a continuous function satisfying lim inft→0+ ξtt>0. Our first main result establishes that if the closed subalgebra A generated by the semigroup (T(t))t>0 is non-unital, then lim supt→0+Tt+ξt−Tt>0. As an application, we derive an exact asymptotic formula for the disk algebra AD¯. Considering the semigroup (T(t))t>0 on AD¯, given by (T(t)f)(z) = f(e−tz) for f∈AD¯, z∈D, and assuming that limt→0+ξtt=γ is finite, we prove that lim supt→0+Tt+ξt−Tt=γ1+γ1+1γ. Furthermore, the closed subalgebra generated by this semigroup is non-unital.

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Cite

BibTeXRIS
BibTeX
@article{agti2026asymptotic,
  title = {{Asymptotic Behavior of Distance near the Origin Between Two Elements of the Semigroup in a Banach Algebra}},
  author = {Mohamed Agti and Abdelkader Mokhtari},
  journal = {Mathematics},
  year = {2026},
  month = oct,
  doi = {10.3390/math14203682},
  url = {https://www.mdpi.com/2227-7390/14/20/3682},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Asymptotic Behavior of Distance near the Origin Between Two Elements of the Semigroup in a Banach Algebra
AU  - Mohamed Agti
AU  - Abdelkader Mokhtari
PY  - 2026
DA  - 2026/10/10
JO  - Mathematics
DO  - 10.3390/math14203682
UR  - https://www.mdpi.com/2227-7390/14/20/3682
AB  - This paper examines the local asymptotic behavior of the distance ‖T(t + ξ(t)) − T(t)‖ near the origin, where (T(t))t>0 is a semigroup on a commutative Banach algebra and ξ : ]0, +∞[ → ]0, +∞[ is a continuous function satisfying lim inft→0+ ξtt>0. Our first main result establishes that if the closed subalgebra A generated by the semigroup (T(t))t>0 is non-unital, then lim supt→0+Tt+ξt−Tt>0. As an application, we derive an exact asymptotic formula for the disk algebra AD¯. Considering the semigroup (T(t))t>0 on AD¯, given by (T(t)f)(z) = f(e−tz) for f∈AD¯, z∈D, and assuming that limt→0+ξtt=γ is finite, we prove that lim supt→0+Tt+ξt−Tt=γ1+γ1+1γ. Furthermore, the closed subalgebra generated by this semigroup is non-unital.
SN  - 2227-7390
ER  -