Weyl-von Neumann-Berg theorem for quaternionic operators

TL;DR

This paper extends the Weyl-von Neumann-Berg theorem to quaternionic right linear operators, showing that such operators can be approximated by a sum of a diagonal and a compact operator within any desired precision.

Contribution

It proves the Weyl-von Neumann-Berg theorem for quaternionic right linear operators, including unbounded operators, which was not previously established.

Findings

Any quaternionic normal operator can be approximated by a diagonal plus compact operator

The theorem holds for unbounded operators in quaternionic Hilbert spaces

Provides a foundation for spectral approximation in quaternionic operator theory

Abstract

We prove the Weyl-von Neumann-Berg theorem for quaternionic right linear operators (not necessarily bounded) in a quaternionic Hilbert space: Let $N$ be a right linear normal (need not be bounded) operator in a quaternionic separable Hilbert space $H$. Then for a given $\epsilon>0$ there exists a compact operator $K$ with $\|K\|<\epsilon$ and a diagonal operator $D$ on $H$ such that $N=D+K$.