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 T-1, ... ,T-d of finite joint branching index, one may associate with T the Drury-Arveson-type C [z(1, ... ,)z(d) ]-Hilbert module H-ca(T) of vector-valued holomorphic functions on the unit ball B-d in C-d, where a > 0. The main result of this paper classifies all directed Cartesian products T for which the Hilbert modules H-ca(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 , T-1, ... ,T-d are complete invariants for H-ca. (.) if ad not equal 1.