Asymptotic Behavior of Distance near the Origin Between Two Elements of the Semigroup in a Banach Algebra
Year: 2026
Type: article
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.
Source: Mathematics
DOI: 10.3390/math14203682
Authors: Mohamed Agti, Abdelkader Mokhtari
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,137
OpenAlex ID: W7207545887