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 $\mathrm{Mult}(\mathcal{H})$. This is applied to obtain several definitive results in interpolation. In particular, we establish a sharp peak interpolation result for compact $\mathrm{Mult}(\mathcal{H})$-totally null sets as well as a Pick and peak interpolation theorem. Conversely, we show that a mere interpolation set is $\mathrm{Mult}(\mathcal{H})$-totally null.