On algebras of holomorphic functions of a given type

TL;DR

This paper demonstrates that various spaces of holomorphic functions on Riemann domains over Banach spaces form locally m-convex Fréchet algebras, with a structured spectrum aiding in understanding their envelopes of holomorphy.

Contribution

It establishes the algebraic and spectral structure of specific holomorphic function spaces, including nuclear and Hilbert-Schmidt types, as locally m-convex Fréchet algebras.

Findings

Spaces are locally m-convex Fréchet algebras

Spectrum has a natural analytic structure

Envelope of holomorphy characterized

Abstract

We show that several spaces of holomorphic functions on a Riemann domain over a Banach space, including the nuclear and Hilbert-Schmidt bounded type, are locally $m$-convex Fr\'echet algebras. We prove that the spectrum of these algebras has a natural analytic structure, which we use to characterize the envelope of holomorphy. We also show a Cartan-Thullen type theorem.