Resolutions of Hilbert Modules and Similarity

TL;DR

This paper characterizes when certain quotient modules of the Drury-Arveson space are similar to tensor products of the space with a Hilbert space, providing new structure theorems and insights into module resolutions and non-commutative cases.

Contribution

It introduces a new necessary and sufficient condition for similarity of quotient modules to tensor products, generalizes a theorem on similarity to the unilateral shift, and explores module resolutions and non-commutative cases.

Findings

A necessary condition involves the existence of a specific multiplier \\psi.

The converse of the similarity condition relates to a structure theorem for complemented submodules.

Finite resolutions of DA-modules with partial isometries are trivial.

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 B^m \times B^m \raro (1 - <z,w>)^{-1}. We investigate for which multipliers \theta : \mathbb{B}^m \raro \cll(\cle, \cle_*) the quotient module \clh_{\theta} is similar to H^2_m \otimes \clf for some Hilbert space \clf, where M_{\theta} is the corresponding multiplication operator in \cll(H^2_m \otimes \cle, H^2_m \otimes \cle_*) for Hilbert spaces \cle and \cle_* and \clh_{\theta} is the quotient module (H^2_m \otimes \cle_*)/ {clos} [M_{\theta}(H^2_m \otimes \cle)]. We show that a necessary condition is the existence of a multiplier $\psi$ in \clm(\cle_*, \cle) 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 \cle for a Hilbert space \cle, 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 tuple or algebra is not necessarily commuting. In this case the converse to the similarity result is always valid.