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

Given a directed Cartesian product $\mathscr T$ of locally finite, leafless, rooted directed trees $\mathscr T_1, \ldots, \mathscr T_d$ of finite joint branching index, one may associate with $\mathscr T$ the Drury-Arveson-type $\mathbb C[z_1, \ldots, z_d]$-Hilbert module $\mathscr H_{\mathfrak c_a}(\mathscr T)$ of vector-valued holomorphic functions on the open unit ball $\mathbb B^d$ in $\mathbb C^d$, where $a >0.$ In case all directed trees under consideration are without branching vertices, $\mathscr H_{\mathfrak c_a}(\mathscr T)$ turns out to be the classical Drury-Arveson-type Hilbert module $\mathscr H_{a}$ associated with the reproducing kernel $\frac{1}{(1 - \langle{z}, {w}\rangle)^a}$ defined on $\mathbb B^d$. Unlike the case of $d=1$, the above association does not yield a reproducing kernel Hilbert module if we relax the assumption that $\mathscr T$ has finite joint branching index. The main result of this paper classifies all directed Cartesian product $\mathscr T$ for which the Hilbert modules $\mathscr H_{\mathfrak c_a}(\mathscr T)$ 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 $\mathscr H_{\mathfrak c_a}(\mathscr T).$ Further, a careful analysis of these Hilbert modules allows us to prove that the cardinality of the $k^{\tiny \mbox{th}}$ generation $(k =0, 1, \ldots)$ of $\mathscr T_1, \ldots, \mathscr T_d$ are complete invariants for $\mathscr H_{\mathfrak c_a}(\cdot)$ provided $ad \neq 1$. 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 $\mathscr M_{z}$ on $\mathscr H_{\mathfrak c_a}(\mathscr T)$ with orthogonal direct sum of tensor products of certain hyperplanes.