The spectra of Banach algebras of holomorphic functions on polydisk type domains

TL;DR

This paper characterizes the spectra of certain Banach algebras of holomorphic functions on polydisk domains in infinite-dimensional Banach spaces, extending known results and confirming the Cluster Value Theorem in this context.

Contribution

It generalizes the isometric isomorphism between algebras of bounded holomorphic functions on polydisk domains to all infinite-dimensional Banach spaces with a Schauder basis.

Findings

Existence of polydisk type domains with isometric algebra isomorphism

Extension of the Cluster Value Theorem to these domains

Analysis of the spectrum's algebraic and analytic structure

Abstract

R.M. Aron et al. proved that the Cluster Value Theorem in the infinite dimensional Banach space setting holds for the Banach algebra $\mathcal{H}^\infty (B_{c_0})$. On the other hand, B.J. Cole and T.W. Gamelin showed that $\mathcal{H}^\infty (\ell_2 \cap B_{c_0})$ is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$ in the sense of an algebra. Motivated by this work, we are interested in a class of open subsets $U$ of a Banach space $X$ for which $\mathcal{H}^\infty (U)$ is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$. We prove that there exist polydisk type domains $U$ of any infinite dimensional Banach space $X$ with a Schauder basis such that $\mathcal{H}^\infty (U)$ is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$, which generalizes the result by Cole and Gamelin. Furthermore, we study the analytic and algebraic structure of the spectrum of $\mathcal{H}^\infty (U)$ and show that the Cluster Value Theorem is true for $\mathcal{H}^\infty (U)$.