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

Abstract

In this article, we prove the Weyl--von Neumann theorem for bounded antilinear skew-self-adjoint operators. More specifically, we prove the following: Let $A$ be a bounded 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 block skew-diagonal operator $D$ and an antilinear skew-self-adjoint Schatten $p$-class operator $K$ such that $A=K+D$ with $\Vert K{\Vert }_p<\epsilon$. As a consequence, we prove the Weyl--von Neumann theorem for complex skew-symmetric operators: Let $\tau$ be a conjugation on $H$ and let $T$ be a bounded linear operator $\tau$-skew-symmetric 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 $T=K+UD{U}^{tr}$ and $\Vert K{\Vert }_p<\epsilon$, 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 $N(T)=N(T^{\ast })$.

15 pages. Submitted to a journal. Comments are welcome