Journal of Operator Theory

Volume 55, Issue 1, Winter 2006 pp. 117-133.

Essentially reductive Hilbert modules

Authors: Ronald G. Douglas
Author institution: Department of Mathematics, Texas A\&M University, College Station, TX 77843, USA

Summary: Consider a Hilbert space obtained as the completion of the polynomials ${\bb C}[\bm{z}]$ in $m$-variables for which the monomials are orthogonal. If the commuting weighted shifts defined by the coordinate functions are essentially normal, then the same is true for their restrictions to invariant subspaces spanned by monomials. This generalizes the result of Arveson \cite{Arv03} in which the Hilbert space is the $m$-shift Hardy space $H^2_m$. He establishes his result for the case of finite multiplicity and shows the self-commutators lie in the Schatten $p$-class for $p>m$. We establish our result at the same level of generality. We also discuss the $K$-homology invariant defined in these cases.

Keywords: essentially reductive Hilbert modules, multivariate operator theory, essentially normal operators, commuting weighted shifts

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