论文标题
球上乘数代数的插值和二元性
Interpolation and duality in algebras of multipliers on the ball
论文作者
论文摘要
我们研究了作为在某些复制的内核Hilbert Spaces $ \ Mathcal {h} $上获得的乘数代数$ a（\ MATHCAL {H}）$在球$ \ Mathbb {B} b} _d $ of $ \ MATHBB $ \ MATHBB {c c}^d $上获得的多项式$。我们的结果尤其适用于Drury-Arveson空间，Dirichlet空间和球上的耐寒空间。我们首先在亨金的互补带中获得$ a（\ Mathcal h）$的双重和第二个双空间的完整描述，以及$ \ permatatorName {mult}（\ Mathcal {h}）$的完全单数度量。将其应用于插值中的几个确切结果。特别是，我们为紧凑型$ \ operatorname {mult}（\ Mathcal {h}）$建立了一个尖锐的峰值插值结果 - 完全无效的集合以及选择和峰值插值定理。相反，我们表明仅仅是插值集是$ \ operatatorName {mult}（\ Mathcal {h}）$ - 完全null。
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.
