Asymptotic Behavior of Distance near the Origin Between Two Elements of the Semigroup in a Banach Algebra
Mohamed Agti, Abdelkader Mokhtari
Source 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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