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The essential normality of $ N_{\underline{\eta}}$-type quotient module of Hardy module on the polydisc


Author: Penghui Wang
Journal: Proc. Amer. Math. Soc. 142 (2014), 151-156
MSC (2010): Primary 47A13, 46H25
DOI: https://doi.org/10.1090/S0002-9939-2013-11890-3
Published electronically: September 4, 2013
MathSciNet review: 3119190
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Abstract: The purpose of this note is to study the essential normality of $ N_{\underline {\eta }}$-type quotient modules, generated by $ \{\eta _i(z_i)-\eta _{i+1}(z_{i+1})\,\vert\, i=1,\cdots ,n-1\}$, of the Hardy module over the polydisc for inner functions $ \eta _i$. In 1988, Clark studied the structure of $ N_{\underline {\eta }}$, for which $ \eta _i$ are finite Blaschke products, and as a consequence, $ N_{\underline {\eta }}$ is essentially normal if $ \eta _i$ are finite Blaschke products. In this note, we will prove that to obtain the essential normality of $ N_{\underline \eta }$, the condition that $ \eta _i$ are finite Blaschke products is necessary.


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Additional Information

Penghui Wang
Affiliation: School of Mathematics, Shandong University, Jinan 250100, People’s Republic of China
Email: phwang@sdu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-2013-11890-3
Received by editor(s): February 21, 2012
Published electronically: September 4, 2013
Additional Notes: This work was supported by the NSFC 11101240, the Excellent Young Scientist Foundation of Shandong Province, and the Independent Innovation Foundations of Shandong University.
Communicated by: Richard Rochberg
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.