Essentially Reductive Weighted Shift Hilbert Modules

TL;DR

This paper explores the essential normality of weighted shift Hilbert modules, connecting it to longstanding conjectures and extending results to specific classes of ideals and modules.

Contribution

It introduces new reductions and generalizations for the essential normality problem in Hilbert modules, linking it to the Arveson conjecture.

Findings

Established results for two tuples and one-dimensional zero varieties.

Reduced quasi-homogeneous ideal problems to homogeneous cases.

Connected essential reductivity to a generalized Arveson problem.

Abstract

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 a 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.