Envelopes of holomorphy and extension of functions of bounded type

TL;DR

This paper investigates the extension of bounded type holomorphic functions in Banach spaces, characterizing envelopes of holomorphy, and exploring their properties and implications for bounded and unbounded sets.

Contribution

It provides a characterization of envelopes of holomorphy for bounded type functions and addresses the extension behavior for unbounded sets, answering a longstanding question.

Findings

Extensions to the envelope are always of bounded type for bounded balanced sets.

Unbounded sets do not necessarily extend to the envelope of holomorphy of bounded type.

The paper relates domain of holomorphy to polynomial convexity and explores spectrum properties for specific domains.

Abstract

We study the extension of holomorphic functions of bounded type defined on an open subset of a Banach space, to larger domains. For this, we first characterize the envelope of holomorphy of a Riemann domain over a Banach space, with respect to the algebra of bounded type holomorphic functions, in terms of the spectrum of the algebra. We then give a simple description of the envelopes of balanced open sets and relate the concepts of domain of holomorphy and polynomial convexity. We show that for bounded balanced sets, extensions to the envelope are always of bounded type, and that this does not necessarily hold for unbounded sets, answering a question posed by Hirschowitz in 1972. We also consider extensions to open subsets of the bidual, present some Banach-Stone type results and show some properties of the spectrum when the domain is the unit ball of $\ell_p$.