2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/165667We 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.23 pagesOperator AlgebrasFunctional Analysis46E22, 46M20, 47A13, 47B32text