The theory of analytic functions is an important section of nonlinear functional analysis.In many modern investigations topological algebras of analytic functions and spectra of suchalgebras are studied. In this work we investigate the properties of the topological algebras of entire functions,generated by countable sets of homogeneous polynomials on complex Banach spaces.
Let $X$ and $Y$ be complex Banach spaces. Let $\mathbb{A}= \{A_1, A_2, \ldots, A_n, \ldots\}$ and $\mathbb{P}=\{P_1, P_2,$ \ldots, $P_n, \ldots \}$ be sequences of continuous algebraically independent homogeneous polynomials on spaces $X$ and $Y$, respectively, such that $\|A_n\|_1=\|P_n\|_1=1$ and $\deg A_n=\deg P_n=n,$ $n\in \mathbb{N}.$ We consider the subalgebras $H_{b\mathbb{A}}(X)$ and $H_{b\mathbb{P}}(Y)$ of the Fr\'{e}chet algebras $H_b(X)$ and $H_b(Y)$ of entire functions of bounded type, generated by the sets $\mathbb{A}$ and $\mathbb{P}$, respectively. It is easy to see that $H_{b\mathbb{A}}(X)$ and $H_{b\mathbb{P}}(Y)$ are the Fr\'{e}chet algebras as well.
In this paper we investigate conditions of isomorphism of the topological algebras $H_{b\mathbb{A}}(X)$ and $H_{b\mathbb{P}}(Y).$ We also present some applications for algebras of symmetric analytic functions of bounded type. In particular, we consider the subalgebra $H_{bs}(L_{\infty})$ of entire functions of bounded type on $L_{\infty}[0,1]$ which are symmetric, i.e. invariant with respect to measurable bijections of $[0,1]$ that preserve the measure. We prove that$H_{bs}(L_{\infty})$ is isomorphic to the algebra of all entire functions of bounded type, generated by countable set of homogeneous polynomials on complex Banach space $\ell_{\infty}.$
THE POLYNOMIAL NUMERICAL INDEX OF A BANACH SPACE
