Essential normality and the decomposability of homogeneous submodules

TL;DR

This paper introduces the concept of essential decomposability to prove the essential normality of a broad class of homogeneous submodules in the finite rank d-shift Hilbert module, supporting Arveson's conjecture.

Contribution

It defines essential decomposability and demonstrates its use in establishing the essential normality of many homogeneous submodules, including new classes.

Findings

Many homogeneous submodules are essentially decomposable.

Essential decomposability implies essential normality.

Supports Arveson's conjecture for all homogeneous submodules.

Abstract

We establish the essential normality of a large new class of homogeneous submodules of the finite rank d-shift Hilbert module. The main idea is a notion of essential decomposability that determines when an arbitrary submodule can be decomposed into the sum of essentially normal submodules. We prove that every essentially decomposable submodule is essentially normal, and using ideas from convex geometry, we introduce methods for establishing that a submodule is essentially decomposable. It turns out that many homogeneous submodules of the finite rank d-shift Hilbert module 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-shift Hilbert module is essentially normal.