Multiplicative linear functionals on reproducing kernel Hilbert spaces

TL;DR

This paper characterizes multiplicative linear functionals on certain reproducing kernel Hilbert spaces over the unit ball, using kernel function actions and properties of complete Nevanlinna--Pick kernels.

Contribution

It provides explicit, verifiable characterizations of these functionals based on kernel structures, diverging from traditional proof methods.

Findings

Characterizations are easy to verify.

Focus on kernels that are powers, Schur products, or tensor products of CNP kernels.

Proofs rely on structural properties of CNP kernels.

Abstract

This note characterizes multiplicative linear functionals on reproducing kernel Hilbert spaces of functions on the Euclidean unit ball in complex d-dimensional space, in terms of their action on kernel functions. The kernels considered are either positive integral powers of a complete Nevanlinna--Pick (CNP) kernel, or Schur products of two CNP kernels, or tensor products of two CNP kernels. The characterizations are easy to verify, and the proofs rely on structural properties of CNP kernels rather than the traditional routes seen in the context of generalizations of the Gleason--Kahane--Zelazko theorem.