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Journal of Operator Theory

Volume 65, Issue 2, Spring 2011  pp. 379-401.

Essentially reductive weighted shift Hilbert modules

Authors:  Ronald G. Douglas (1) and Jaydeb Sarkar (2)
Author institution: (1) Texas A \& M University, College Station, Texas 77843, U.S.A.
(2) Texas A \& M University, College Station, Texas 77843, U.S.A.

Summary:  We discuss the relation between questions regarding the essential normality of finitely generated essentially spherical isometries and some results and conjectures of Arveson and Guo--Wang on the closure of homogeneous ideals in the $m$-shift space. We establish general results for the case of two tuples and ideals with one dimensional zero variety. Further, we show how to reduce the analogous question for quasi-homogeneous ideals, to those results for homogeneous ones. Finally, we show that the essential reductivity of positive regular Hilbert modules is directly related to a generalization of the Arveson problem.

Keywords:  Hilbert modules, spherical isometries, weighted shifts

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