Interpolation and duality in algebras of multipliers on the ball

TL;DR

This paper explores the structure and duality of multiplier algebras on the ball, providing new interpolation theorems and characterizations of null sets in various reproducing kernel Hilbert spaces.

Contribution

It offers a complete duality description and new interpolation results for multiplier algebras on the ball, including sharp peak interpolation and characterization of null sets.

Findings

Complete dual and second dual space descriptions in terms of Henkin and singular measures.

Sharp peak interpolation results for null sets.

New Pick and peak interpolation theorems.

Abstract

We study the multiplier algebras $A(\mathcal{H})$ obtained as the closure of the polynomials on certain reproducing kernel Hilbert spaces $\mathcal{H}$ on the ball $\mathbb{B}_d$ of $\mathbb{C}^d$. Our results apply, in particular, to the Drury-Arveson space, the Dirichlet space and the Hardy space on the ball. We first obtain a complete description of the dual and second dual spaces of $A(\mathcal H)$ in terms of the complementary bands of Henkin and totally singular measures for $\operatorname{Mult}(\mathcal{H})$. This is applied to obtain several definitive results in interpolation. In particular, we establish a sharp peak interpolation result for compact $\operatorname{Mult}(\mathcal{H})$-totally null sets as well as a Pick and peak interpolation theorem. Conversely, we show that a mere interpolation set is $\operatorname{Mult}(\mathcal{H})$-totally null.