Multiplier varieties and multiplier algebras of CNP Dirichlet series kernels

TL;DR

This paper characterizes the multiplier varieties of CNP Dirichlet series kernels and classifies their multiplier algebras, revealing a rigidity phenomenon where the algebra determines the kernel.

Contribution

It explicitly determines multiplier varieties for CNP Dirichlet series kernels and establishes conditions for algebraic and isometric isomorphisms of their multiplier algebras.

Findings

Multiplier varieties are described via polynomial equations from arithmetic data.

Multiplier algebras are classified up to isomorphism and isometry.

A rigidity phenomenon shows the algebra determines the kernel up to equivalence.

Abstract

We investigate isometric and algebraic isomorphism problems for multiplier algebras associated with Dirichlet series kernels that possess the complete Nevanlinna-Pick (CNP) property. A central aspect of our work is the explicit determination of the multiplier variety associated with each CNP Dirichlet series kernel, via polynomial equations derived from the arithmetic structure of the associated weight and frequency data. This description of multiplier varieties enables us to classify when the multiplier algebras of a signifincant class of CNP Dirichlet series kernels are isomorphic, or isometrically isomorphic. In this setting, a striking rigidity phenomenon emerges whereby the multiplier algebra determines the kernel up to natural equivalence. The results established for CNP Dirichlet series kernels also extend to classical CNP kernels, yielding new results for the associated multiplier algebras even in the classical setting. As an application, we resolve an open problem posed by McCarthy and Shalit ([19]).