Weyl–von Neumann theorem

Remarks

In words, every bounded self-adjoint operator is the sum of a diagonal operator and a compact self-adjoint operator of arbitrarily small norm. Weyl proved the version with $K$ compact, and von Neumann showed that $K$ can be chosen in the Hilbert–Schmidt class, the Schatten class $S_2$, with Hilbert–Schmidt norm less than $\varepsilon$. The idea of the proof is to use spectral projections of $A$ for small intervals. They give finite-dimensional subspaces that are almost invariant under $A$, and $A$ is diagonalized up to a small error on each of them. Example: the multiplication operator $(Af)(x) = x f(x)$ on $L^2([0,1])$ has no eigenvalues at all, yet by the theorem a compact perturbation of small norm has an orthonormal basis of eigenvectors. The theorem does not extend to trace class perturbations. By the Kato–Rosenblum theorem, if $A$ has a nonzero absolutely continuous part, then so does $A + K$ for every trace class self-adjoint $K$. Then $A + K$ is not diagonal.

Going deeper

For a student

The theorem shows that the point spectrum and the continuous spectrum of a self-adjoint operator are not stable under compact perturbations, while the essential spectrum is, by Weyl's theorem. It is a starting point of the theory of unitary equivalence modulo compact operators, which later led to the Brown–Douglas–Fillmore theory of extensions of C*-algebras.