Multiplier varieties and multiplier algebras of CNP Dirichlet series kernels
Hamidul Ahmed, B. Krishna Das, Chaman Kumar Sahu
TL;DR
The paper advances the understanding of multiplier algebras for CNP Dirichlet-series kernels by giving an explicit, data-driven description of the associated multiplier varieties and establishing rigidity results: algebraic isomorphisms between multiplier algebras are automatically isometric, and isomorphism types are determined by a notion of similar pattern in weight and frequency data. It provides a full characterization of normalized CNP Dirichlet kernels, connects them to Drury–Arveson spaces via smallest multiplier varieties, and delivers a complete classification of multiplier algebras for a broad class of kernels, including negative answers to open questions posed by McCarthy and Shalit. The work combines Dirichlet-series uniqueness, polynomial descriptions of multiplier varieties, and biholomorphic equivalence to reduce isomorphism problems to arithmetic data comparisons, offering both explicit examples and a structural framework for future classifications.
Abstract
We investigate isometric and algebraic isomorphism problems for multiplier algebras associated with Hilbert spaces of Dirichlet series whose kernels possess the complete Nevanlinna-Pick (CNP) property. We begin by providing a complete characterization of the set of all normalized CNP Dirichlet series kernels by their weight and frequency data. A central aspect of our work is the explicit determination of the multiplier variety associated with each CNP Dirichlet series kernel. We show that these varieties are defined by polynomial equations derived from the arithmetic structure of the weight and frequency data associated with the kernel. This description of multiplier varieties enables us to classify when the multiplier algebras of a significant class of CNP Dirichlet series kernels are isomorphic, or isometrically isomorphic. Surprisingly, in this setting, every algebraic isomorphism between multiplier algebras is automatically isometric, revealing a striking rigidity phenomenon whereby the structure of the multiplier algebra uniquely determines the kernel up to a natural equivalence of the underlying weight and frequency data. As an application, we resolve an open problem posed by McCarthy and Shalit ([19]).