1. Interpolation and duality in algebras of multipliers on the ball
- Author
-
Davidson, Kenneth R. and Hartz, Michael
- Subjects
Mathematics - Functional Analysis ,Mathematics - Complex Variables ,Applied Mathematics ,General Mathematics ,FOS: Mathematics ,Mathematics - Operator Algebras ,Complex Variables (math.CV) ,Operator Algebras (math.OA) ,Functional Analysis (math.FA) ,46E22 (Primary) 47L30, 47L50, 46J15 (Secondary) - 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., 44 pages; minor changes
- Published
- 2022