#### Vol. 14, No. 6, 2021

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Multipliers and operator space structure of weak product spaces

### Raphaël Clouâtre and Michael Hartz

Vol. 14 (2021), No. 6, 1905–1924
##### Abstract

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 $\mathsc{ℋ}$, we characterize all multipliers of the weak product space $\mathsc{ℋ}\odot \mathsc{ℋ}$. In particular, we show that if $\mathsc{ℋ}$ has the so-called column-row property, then the multipliers of $\mathsc{ℋ}$ and of $\mathsc{ℋ}\odot \mathsc{ℋ}$ 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 $\mathsc{ℋ}\odot \mathsc{ℋ}$, which enables the use of dilations of completely bounded maps.

##### Keywords
complete Nevanlinna–Pick space, weak product, multiplier, Hankel operator, completely bounded map, dilation
##### Mathematical Subject Classification 2010
Primary: 46E22
Secondary: 46L07, 47A20