Approximation with normal operators with finite spectrum, and an elementary proof of a Brown–Douglas–Fillmore theorem

Peter Friis and Mikael Rørdam

Vol. 199 (2001), No. 2, 347–366

DOI: 10.2140/pjm.2001.199.347

Abstract

We give a short proof of the theorem of Brown, Douglas and Fillmore that an essentially normal operator on a Hilbert space is of the form “normal plus compact” if and only if it has trivial index function. The proof is basically a modification of our short proof of Lin’s theorem on almost commuting self-adjoint matrices that takes into account the index.

Using similar methods we obtain new results, generalizing results of Lin, on approximating normal operators by ones with finite spectrum.

Milestones

Received: 1 November 1998

Published: 1 June 2001