Tensor product of quotient Hilbert modules

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:424

Article

Chattopadhyay, Arup¹  Das, B. Krishna¹  Sarkar, Jaydeb¹ 

[1] Indian Stat Inst, Stat & Math Unit, Bangalore 560059, Karnataka, India

关键词: Hilbert modules;    Hardy and weighted Bergman spaces over polydisc;    Doubly commuting quotient modules;    Essential normality;    Wandering subspace;    Rank;   

DOI  :  10.1016/j.jmaa.2014.11.038

来源: Elsevier

【 摘 要 】

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[z(1),..., z(n)] and the doubly commutativity property of module multiplication operators by the coordinate functions. More precisely, for a reproducing kernel Hilbert module H over C[z(1),..., z(n)] of analytic functions on the polydisc in C-n which satisfies certain conditions, we characterize the quotient modules Q of H such that Q is of the form Q(1 circle times)...circle times Q(n), for some one-variable quotient modules {Q(1),..,Q(n)}. For H the Hardy module over polydisc, H-2 (D-n), 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 polyclisc. 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 C-n. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free