Reference for a Proof of Weyl-Von-Neumann Theorem I'm looking for a reference for the proof of the Weyl Von Neumann theorem, however there seems to be two (or the two might be the same).
There's the one which is stated in Conways, A Course in Functional Analysis in Section 38 which states (roughly) that for any hermitian operator $A$ on a Hilbert space there is a diagonalizable self adjoint operator $D$ such that $A$ and $D$ are close in lots of norms.
But as far as I can tell this isn't the only one, (or it's implied in a bunch of web pages) that either a different theorem or a corollary of this theorem is that if $A$ and $B$ have the same essential spectrum then (up to a unitary transformation) they are linked by a compact operator.
Can anyone give me a reference for the proof of the second statement in a textbook/webpage? The only place I think I could find it at the minute is in the original paper 'Charakterisierung des Spektrums eines Integraloperators', but I don't know any german.

Try Davidson's $C^*$-algebra by Examples, which gives a full development of BDF theory generalizing from Weyl-von Neumann.
