Let
\(H^{2}_{m}\) be the Drury–Arveson (DA) module which is the reproducing kernel Hilbert space with the kernel function
\((z, w) \in\mathbb{B}^{m} \times\mathbb{B}^{m} \rightarrow (1 - \sum_{i=1}^{m}z_{i} \bar{w}_{i})^{-1}\). We investigate for which multipliers
\(\theta: \mathbb{B}^{m} \rightarrow \mathcal{L}(\mathcal{E}, \mathcal {E}_{*})\) with ran?
M θ closed, the quotient module
\(\mathcal{H}_{\theta}\), given by
$\cdots\longrightarrow H^2_m \otimes\mathcal{E} \stackrel{M_{\theta }}{\longrightarrow}H^2_m \otimes\mathcal{E}_* \stackrel{\pi_{\theta}}{\longrightarrow}\mathcal{H}_{\theta}\longrightarrow0,$
is similar to
\(H^{2}_{m} \otimes \mathcal {F}\) for some Hilbert space
\(\mathcal{F}\). Here
M θ is the corresponding multiplication operator in
\(\mathcal{L}(H^{2}_{m} \otimes\mathcal{E}, H^{2}_{m} \otimes\mathcal{E}_{*})\) for Hilbert spaces
\(\mathcal{E}\) and
\(\mathcal{E}_{*}\) and
\(\mathcal {H}_{\theta}\) is the quotient module
\((H^{2}_{m} \otimes\mathcal{E}_{*})/ M_{\theta}(H^{2}_{m} \otimes\mathcal{E})\), and
π θ is the quotient map. We show that a necessary condition is the existence of a multiplier
ψ in
\(\mathcal{M}(\mathcal{E}_{*}, \mathcal{E})\) such that
$\theta\psi\theta= \theta.$
Moreover, we show that the converse is equivalent to a structure theorem for complemented submodules of
\(H^{2}_{m} \otimes\mathcal{E}\) for a Hilbert space
\(\mathcal {E}\), which is valid for the case of
m=1. The latter result generalizes a known theorem on similarity to the unilateral shift, but the above statement is new. Further, we show that a
finite resolution of DA-modules of arbitrary multiplicity using partially isometric module maps must be trivial. Finally, we discuss the analogous questions when the underlying operator
m-tuple (or algebra) is not necessarily commuting (or commutative). In this case the converse to the similarity result is always valid.
