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 $\mathcal{\mathscr{H}}$, 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.