Multipliers and operator space structure of weak product spaces

TL;DR

This paper characterizes multipliers of weak product spaces in certain reproducing kernel Hilbert spaces, revealing their operator space structure and conditions under which multipliers coincide, with applications to classical function spaces.

Contribution

It introduces a natural operator space structure on weak product spaces and characterizes their multipliers for complete Nevanlinna-Pick spaces, especially when the column-row property holds.

Findings

Multipliers of $ ext{H} ext{H} ext{H}$ spaces are characterized.

Operator space structure on weak product spaces is established.

Results apply to Dirichlet and Drury-Arveson 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.