Holomorphic Functions and polynomial ideals on Banach spaces

TL;DR

This paper studies the spectrum of algebras of holomorphic functions associated with polynomial ideals on Banach spaces, showing they behave similarly to classical cases and establishing a Banach-Stone type theorem.

Contribution

It extends the understanding of spectra of holomorphic function algebras linked to polynomial ideals, demonstrating their structure as Riemann domains and proving a Banach-Stone type theorem.

Findings

Spectrum $M_{b}$ can be structured as a Riemann domain over $E''.

Extensions of functions are $$-holomorphic of bounded type.

Spectrum behaves like classical $M_b(E)$ under natural conditions.

Abstract

Given $\u$ a multiplicative sequence of polynomial ideals, we consider the associated algebra of holomorphic functions of bounded type, $H_{b\u}(E)$. We prove that, under very natural conditions verified by many usual classes of polynomials, the spectrum $M_{b\u}(E)$ of this algebra "behaves" like the classical case of $M_{b}(E)$ (the spectrum of $H_b(E)$, the algebra of bounded type holomorphic functions). More precisely, we prove that $M_{b\u}(E)$ can be endowed with a structure of Riemann domain over $E"$ and that the extension of each $f\in H_{b\u}(E)$ to the spectrum is an $\u$-holomorphic function of bounded type in each connected component. We also prove a Banach-Stone type theorem for these algebras.