On a Weyl-von Neumann -type Theorem for Antilinear Self-adjoint Operators

TL;DR

This paper establishes a Weyl-von Neumann type theorem for antilinear self-adjoint operators, showing they can be approximated by diagonalizable plus compact operators, with implications for spectral theory in mathematical physics.

Contribution

It extends the Weyl-von Neumann theorem to antilinear self-adjoint operators, introducing spectral integral representations and analyzing conjugations.

Findings

Antilinear self-adjoint operators can be decomposed into diagonalizable and compact parts.

A spectral integral representation for these operators is constructed.

The theorem applies to operators relevant in mathematical physics.

Abstract

Antilinear operators on a complex Hilbert space arise in various contexts in mathematical physics. In this paper, an analogue of the Weyl--von Neumann theorem for antilinear self-adjoint operators is proved, i.e. that an antilinear self-adjoint operator is the sum of a diagonalizable operator and of a compact operator with arbitrarily small Schatten $p$-norm. In doing so, we discuss conjugations and their properties. A spectral integral representation for antilinear self-adjoint operators is constructed.