Tensor product of quotient Hilbert modules

Abstract

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,….,Z₂] and the doubly commutativity property of module multiplication operators by the coordinate functions. More precisely, for a reproducing kernel Hilbert module over ℂⁿ of analytic functions on the polydisc in ℂⁿ which satisfies certain conditions, we characterize the quotient modules Q of H such that Q is of the form Q₁ ⊗···⊗Q_n, for some onevariable quotient modules {Q₁, ..., Q_n}. For ℋ the Hardy module over polydisc, H²(Dⁿ), 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. Moreover, this provides further insight into the wandering subspaces and ranks of submodules of the Hardy module. Our results include the case of weighted Bergman modules over the unit polydisc in ℂⁿ.