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.