A New Proof of the Weyl-von Neumann-Berg Theorem
Longxiang Fan, Shichang Song
TL;DR
This work provides a self-contained proof of the Weyl-von Neumann-Berg theorem by replacing Halmos' reliance on the Alexandroff–Hausdorff theorem with a topological lemma: every compact subset $Λ$ of the plane is the continuous image of a compact subset of $ℝ$. Using the spectral theorem, a bounded normal operator $A$ is realized as $φ(B)$ with Hermitian $B$ and a continuous $φ$ on a compact $K$ with $φ(K)=Λ$, enabling a reduction to a diagonal plus compact form via a Weyl–von Neumann decomposition $B=D+C$. The proof then employs polynomial (Weierstrass) approximation to transfer the decomposition from $B$ to $D$, showing the compact remainder $L=φ(B)-φ(D)$ yields $A=φ(D)+L$, i.e., a diagonal plus a compact operator. Collectively, the paper provides a self-contained, streamlined route from general bounded normal operators to the classical diagonal-plus-compact decomposition, avoiding stronger topological prerequisites and clarifying the underlying construction.
Abstract
We give a new proof of the Weyl-von Neumann-Berg theorem. Our proof improves Halmos' proof in 1972 by observing the fact that every compact set in the complex plane is the continuous image of a compact set in the real line.