The Weyl-von Neumann theorem for antilinear skew-self-adjoint operators

Abstract

In this article, we prove the Weyl-von Neumann theorem for antilinear skew-self-adjoint operators. More specifically, we prove the following: Let $A$ be an antilinear skew-self-adjoint operator on a separable Hilbert space $H$ whose kernel is either even dimensional or infinite dimensional. Let $1<p<\infty$. Then for every $\epsilon>0$ there exists an antilinear skew block diagonal operator $D$ and an antilinear Schatten $p$-class operator $K$ such that $A=K+D$ with $\|K\|_{p}<\epsilon$. As a consequence of this, we prove the Weyl-von Neumann theorem for complex skew-symmetric operators: Let $\tau$ be a conjugation on $H$ and let $T$ be a $\tau$-skew-symmetric bounded linear operator with $\dim N(T)=\infty$ or $\dim N(T)$ is even. Let $1<p<\infty$. Then for every $\epsilon>0$, there exists a $\tau$-skew-symmetric Schatten $p$-class operator $K$, a skew-symmetric block diagonal operator $D$ and a unitary operator $U$ such that $T=K+UDU^{tr}$ and $\|K\|_{p}<\epsilon$, where $U^{tr}$ is the transpose of $U$ with respect to an orthonormal basis ${\{e_n:n\in \mathbb N}\}$ such that $\tau(e_n)=e_n$ for each $n\in \mathbb N$. Furthermore, the above result holds even without any assumption on the dimension of $N(T)$, provided that $N(T)=N(T^*)$.