The Hadamard Product in the Space of Lorch Analytic Mappings

Abstract

For a complex Banach algebra $E$, let ${\mathrm{H}}_L(E)$ be the space of the mappings from $E$ into $E$ that are analytic in the sense of Lorch, endowed with the Hadamard product and with the topology $\tau_b$ of uniform convergence on the bounded subsets of $E$. We study topological and algebraic properties of ${\mathrm{H}}_L(E)$ in connection with the topological and algebraic properties of the underlying space $E$. We also study algebraic and topological properties of the space of the sequences $(a_n)_n \subset E$ such that $\underset{n\to\infty}{\lim}\Vert a_n\Vert^\frac{1}{n}=0.$