Holomorphic functions on complex Banach lattices

Boyd, Christopher; Ryan, Raymond; Snigireva, Nina

doi:10.1090/tran/9443

Holomorphic functions on complex Banach lattices
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by Christopher Boyd, Raymond A. Ryan and Nina Snigireva;

Trans. Amer. Math. Soc.

DOI: https://doi.org/10.1090/tran/9443

Published electronically: May 1, 2025

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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.

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Holomorphic functions on complex Banach lattices
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Abstract: