Resolutions of Hilbert Modules and Similarity

Abstract

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.