2013/10/18 by Arup Chattopadhyay, Chattopadhyay, Arup, B. Krishna Das +3
Mathematics · #30H10 #46E20 #47A13 #47A15 #47A20 #47A45 #47A80 #47B32 #47B38 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA #msc:30H10 #msc:46E20 #msc:47A13 #msc:47A15 #msc:47A20 #msc:47A45 #msc:47A80 #msc:47B32 #msc:47B38
paper · pdf · doi:10.48550/arxiv.1310.5122
22 pages
arxiv created 2013/10/18 · arxiv updated 2013/10/21
In this paper, we present a unified approach to problems of tensor product of quotient modules of Hilbert modules over ℂ[z] and corresponding submodules of reproducing kernel Hilbert modules over ℂ[z1, …, zn] and the doubly commutativity property of module multiplication operators by the coordinate functions. More precisely, for a reproducing kernel Hilbert module \clh over ℂ[z1, …, zn] of analytic functions on the polydisc in ℂn which satisfies certain conditions, we characterize the quotient modules \q of \clh such that \q is of the form \q1 ⊗ ⋯ ⊗ \qn, for some one variable quotient modules \\q1, …, \qn\. For \clh the Hardy module over polydisc H2(\mathbbDn), this reduces to some recent results by Izuchi, Nakazi and Seto and the third author. This is used to obtain a classification of co-doubly commuting submodules for a class of reproducing kernel Hilbert modules over the unit polydisc. These results are applied to compute the cross commutators of co-doubly commuting submodules. This is used to give further insight into the wandering subspaces and ranks of submodules of the Hardy module case. Our results includes the case of weighted Bergman modules over the unit polydisc in ℂn.