In the theory of reproducing kernel Hilbert spaces, weak product spaces generalize the notion of the Hardy space $H^1$. For complete Nevanlinna–Pick spaces $\mathcal{\mathscr{H}}$, we characterize all multipliers of the weak product space $\mathcal{\mathscr{H}}\odot \mathcal{\mathscr{H}}$. In particular, we show that if $\mathcal{\mathscr{H}}$ has the so-called column-row property, then the multipliers of $\mathcal{\mathscr{H}}$ and of $\mathcal{\mathscr{H}}\odot \mathcal{\mathscr{H}}$ coincide. This result applies in particular to the classical Dirichlet space and to the Drury–Arveson space on a finite-dimensional ball. As a key device, we exhibit a natural operator space structure on $\mathcal{\mathscr{H}}\odot \mathcal{\mathscr{H}}$, which enables the use of dilations of completely bounded maps.

中文翻译：

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弱乘积空间的乘数和算子空间结构

在再生核 Hilbert 空间的理论中，弱积空间推广了 Hardy 空间的概念 $H^1$. 对于完整的 Nevanlinna–Pick 空间$\mathcal{\mathscr{H}}$，我们刻画了弱乘积空间的所有乘数 $\mathcal{\mathscr{H}}\odot \mathcal{\mathscr{H}}$. 特别地，我们表明，如果$\mathcal{\mathscr{H}}$ 具有所谓的列行属性，那么乘数 $\mathcal{\mathscr{H}}$ 和 $\mathcal{\mathscr{H}}\odot \mathcal{\mathscr{H}}$重合。这个结果特别适用于经典 Dirichlet 空间和有限维球上的 Drury-Arveson 空间。作为关键设备，我们展示了一个自然的算子空间结构$\mathcal{\mathscr{H}}\odot \mathcal{\mathscr{H}}$，这使得可以使用完全有界地图的扩张。