Essential normality and the decomposability of homogeneous submodules

Abstract

We establish the essential normality of a large new class of homogeneous submodules of the finite rank d d -shift Hilbert module. The main idea is a notion of essential decomposability that determines when a submodule can be decomposed into the algebraic sum of essentially normal submodules. We prove that every essentially decomposable submodule is essentially normal, and introduce methods for establishing that a submodule is essentially decomposable. It turns out that many submodules have this property. We prove that many of the submodules considered by other authors are essentially decomposable, and in addition establish the essential decomposability of a large new class of homogeneous submodules. Our results support Arveson’s conjecture that every homogeneous submodule of the finite rank d d -shift Hilbert module is essentially normal.