Interpolation and duality in algebras of multipliers on the ball

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.