In their recent works L. Zsido’
and P. A. Fillmore have extended Weyl’s version of the classical Weyl-von Neumann
theorem to infinite semi-finite countably decomposable von Neumann factors,
by proving that for every self-adjoint operator A in the factor there is a
diagonal operator B =∑λnEn such that A − B is compact, the En are
one-dimensional projections and {λn} is dense in the essential spectrum of
A. In this paper we extend the Weyl-von Neumann theorem in a different
way.
First we extend the von Neumann version of the theorem to both finite and
infinite factors by proving that A−B can be chosen as a Hilbert-Schmidt operator of
arbitrarily small norm. We have to drop the condition about the λn or the dimension
of the En.
