Multiplier varieties and multiplier algebras of CNP Dirichlet series kernels
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 all normalized CNP Dirichlet series kernels in terms of weight and frequency data. A central aspect of our work is the explicit determination of the multiplier varieties associated with CNP Dirichlet series kernels. We show that these varieties are defined by explicit polynomial equations derived from the arithmetic structure of the weight and frequency data associated with the kernel. This explicit description of multiplier varieties enables us to classify when the multiplier algebras of certain CNP Dirichlet series kernels are (isometrically) isomorphic. As an application, we resolve an open question posed by McCarthy and Shalit ([18]) in the negative.