Reference for a Proof of Weyl-Von-Neumann Theorem

Question

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.

Answer 1

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

Answer 2

In Tosio Kato's book Perturbation Theory for Linear Operators (reprint of the 1980s version), subsection X.2.1 is called "A Theorem by Weyl-Von Neumann" which starts the section on perturbations of the continuous spectrum on p. 525. According to Kato, the proof of the first theorem is due to von Neumann, and your mentioned German paper of von Neumann is cited.

In Reed and Simon's Analysis of Operators (Methods of Modern Mathematical Physics IV), you can find "Weyl's Essential Spectrum Theorem" (Theorem XIII.14, p. 112) and "Classical Weyl Theorem" (Example 3 in section XIII.4, p.117).

All of the three theorems are concerned with the stability of the spectrum:

The theorem in Kato is a negative result which, in short, states that a selfadjoint operator $A$ can be perturbed with an arbitrarily small Hilbert-Schmidt operator $H$ such that $A+H$ has only pure point spectrum, i.e. no continuous spectrum at all.

The two theorems in Reed/Simon are concerned with the stability of the essential spectrum under compact perturbations (which include different finite-dimensional, self-adjoint extensions of the same symmetric operator).