Restricted diagonalization of finite spectrum normal operators and a theorem of Arveson

TL;DR

This paper extends Arveson's theorem on finite spectrum normal operators by linking it to essential codimension and trace invariance, providing a new proof and generalization using Hilbert-Schmidt perturbations.

Contribution

It introduces a novel connection between Arveson's obstruction and essential codimension, and offers a new approach to prove and generalize Arveson's theorem.

Findings

Arveson's theorem follows from trace invariance of normal operators.

Established a link between Arveson's obstruction and essential codimension.

Provided a new proof using unitary diagonalization via Hilbert-Schmidt perturbations.

Abstract

Kadison characterized the diagonals of projections and observed the presence of an integer. Arveson later recognized this integer as a Fredholm index obstruction applicable to any normal operator with finite spectrum coincident with its essential spectrum whose elements are the vertices of a convex polygon. Recently, in joint work with Kaftal, the author linked the Kadison integer to essential codimension of projections. This paper provides an analogous link between Arveson's obstruction and essential codimension as well as a new approach to Arveson's theorem which also allows for generalization to any finite spectrum normal operator. In fact, we prove that Arveson's theorem is a corollary of a trace invariance property of arbitrary normal operators. An essential ingredient is a formulation of Arveson's theorem in terms of diagonalization by a unitary which is a Hilbert-Schmidt perturbation of the identity.