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.

Authors