In the theory of reproducing kernel Hilbert spaces, weak
product spaces generalize the notion of the Hardy space
. For complete
Nevanlinna–Pick spaces
,
we characterize all multipliers of the weak product space
. In particular,
we show that if
has the so-called column-row property, then the multipliers of
and
of
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
,
which enables the use of dilations of completely bounded maps.
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