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

This paper examines the local asymptotic behavior of the distance \( \left\lVert T\left(t+\xi \left(t\right)\right)-T\left(t\right)\right\lVert \) near the origin, where \( {\left(T\left(t\right)\right)}_{t\gt 0} \) is a semigroup on a commutative Banach algebra and \( \xi ∶\left]0,+\infty \right[\rightarrow \left]0,+\infty \right[ \) is a continuous function satisfying \( \operatorname{\underset{t\rightarrow 0^{+}}{\lim⁡\inf}\ }{\frac{\xi \left(t\right)}{t}\gt 0} \). Our first main result establishes that if the closed subalgebra A generated by the semigroup \( {\left(T\left(t\right)\right)}_{t\gt 0} \) is nonunital, then \( \operatorname{\underset{t\rightarrow 0^{+}}{\lim⁡\sup}}{\left\lVert T\left(t+\xi \left(t\right)\right)-T\left(t\right)\right\lVert \gt 0} \). As an application, we derive an exact asymptotic formula for the disk algebra \( A(\bar{\mathbb{D}}) \). Considering the semigroup \( {\left(T\left(t\right)\right)}_{t\gt 0} \) on \( A(\bar{\mathbb{D}}) \), given by \( \left(T\left(t\right)f\right)\left(z\right)=f\left(e^{-t}z\right) \) for \( f\in A(\bar{\mathbb{D}}),\ z\in \mathbb{D} \), and assuming that \( \lim_{t\rightarrow 0^{+}}{\frac{\xi \left(t\right)}{t}=\mathrm{\gamma }\ } \)finite, we prove that \( \operatorname{\underset{t\rightarrow 0^{+}}{\lim⁡\sup}}{\left\lVert T\left(t+\xi \left(t\right)\right)-T\left(t\right)\right\lVert =\frac{\gamma }{{\left(1+\gamma \right)}^{1+\frac{1}{\gamma }}}} \). Furthermore, the closed subalgebra generated by this semigroup is nonunital.