Multipliers and operator space structure of weak product spaces

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

21 pages