Holomorphic functions on complex Banach lattices

TL;DR

This paper explores the properties of holomorphic functions and regular polynomials on complex Banach lattices, extending classical concepts like the Bohr radius and analyzing convergence behaviors in finite and infinite dimensions.

Contribution

It introduces a new framework for understanding holomorphic functions on Banach lattices, including an extension of the Bohr radius concept and convergence analysis.

Findings

The theory of power series with regular terms aligns more with several complex variables than classical Banach space theory.

The Bohr radius provides a lower bound for the ratio of convergence radii in Banach lattices.

In finite dimensions, the Taylor series and monomial expansion convergence radii coincide, but differ significantly on ll_p spaces.

Abstract

We introduce and study the algebraic, analytic and lattice properties of regular homogeneous polynomials and holomorphic functions on complex Banach lattices. We show that the theory of power series with regular terms is closer to the theory of functions of several complex variables than the theory of holomorphic functions on Banach spaces. We extend the concept of the Bohr radius to Banach lattices and show that it provides us with a lower bound for the ratio between the radius of regular convergence and the radius of convergence of a regular holomorphic function. This allows us to show that in finite dimensions the radius of convergence of the Taylor series of a holomorphic function coincides with the radius of convergence of its monomial expansion but that on $\ell_p$ these two radii can be radically different.