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Please use this identifier to cite or link to this item: http://dspace.bits-pilani.ac.in:8080/jspui/handle/123456789/11481
Title: Classification of Drury-Arveson-type Hilbert modules associated with certain directed graphs
Authors: Trivedi, Shailesh
Keywords: Mathematics
Drury-Arveson-type
Issue Date: Sep-2017
Publisher: ARXIV
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.
URI: https://arxiv.org/abs/1709.02922
http://dspace.bits-pilani.ac.in:8080/xmlui/handle/123456789/11481
Appears in Collections:Department of Mathematics

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