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

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 unit ball Bd in Cd, where a>0. The main result of this paper classifies all directed Cartesian products T for which the Hilbert modules Hca(T) are isomorphic in case a is an integer. Indeed, a careful analysis of these Hilbert modules allows us to prove that the cardinality of generations of T1,…,Td are complete invariants for Hca(⋅) if ad≠1.