Classification of Drury-Arveson-type Hilbert modules associated with certain directed graphs

Abstract

Given a directed Cartesian product T of locally finite, leafless, rooted directed trees T1,…,Td of finite joint branching index, one may associate with T the Drury-Arveson-type C[z1,…,zd]-Hilbert module Hca(T) of vector-valued holomorphic functions on the open unit ball Bd in Cd, where a>0. In case all directed trees under consideration are without branching vertices, Hca(T) turns out to be the classical Drury-Arveson-type Hilbert module Ha associated with the reproducing kernel 1(1−⟨z,w⟩)a defined on Bd. Unlike the case of d=1, the above association does not yield a reproducing kernel Hilbert module if we relax the assumption that T has finite joint branching index. The main result of this paper classifies all directed Cartesian product T for which the Hilbert modules Hca(T) are isomorphic in case a is a positive integer. One of the essential tools used to establish this isomorphism is an operator-valued representing measure arising from Hca(T). Further, a careful analysis of these Hilbert modules allows us to prove that the cardinality of the kth generation (k=0,1,…) of T1,…,Td are complete invariants for Hca(⋅) provided ad≠1. Failure of this result in case ad=1 may be attributed to the von Neumann-Wold decomposition for isometries. Along the way, we identify the joint cokernel E of the multiplication d-tuple Mz on Hca(T) with orthogonal direct sum of tensor products of certain hyperplanes.