  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.
