Journal of Operator Theory

Volume 81, Issue 1, Winter 2019 pp. 21-60.

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

Authors: Sameer Chavan (1), Deepak Kumar Pradhan (2), Shailesh Trivedi (3)
Author institution:(1) Department of Mathematics, Indian Institute of Technology Kanpur, 208016, India
(2) Department of Mathematics, Indian Institute of Technology Kanpur, 208016, India
(3) Department of Mathematics, Indian Institute of Technology Kanpur, 208016, India

Summary: 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 unit ball $\mathbb B^d$ in $\mathbb C^d$, where $a >0.$ The main result of this paper classifies all directed Cartesian products $\mathscr T$ for which the Hilbert modules $\mathscr H_{\mathfrak c_a}(\mathscr T)$ 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 $\mathscr T_1, \ldots, \mathscr T_d$ are complete invariants for $\mathscr H_{\mathfrak c_a}(\cdot)$ if $ad \neq 1$.

DOI: http://dx.doi.org/10.7900/jot.2017sep16.2203
Keywords: Hilbert module, reproducing kernel, representing measure

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