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

TL;DR

This paper extends the Weyl-von Neumann paradigm to antilinear skew-self-adjoint operators on a separable Hilbert space by proving that any such operator $A$ with $\dim N(A)$ even or infinite can be decomposed as $A=K+D$ where $K$ lies in the antilinear Schatten $p$-class and $D$ is a block skew-diagonal operator, with $\\|K\\|_p<\\epsilon$ for any $\\epsilon>0$. The construction leverages a polar-decomposition $A=\kappa|A|=|A|\kappa$ using an anti-conjugation $\kappa$, and builds finite-rank, even-rank projections to reduce $A$ while preserving $e$igens via a block decomposition into $2\times2$ skew blocks. Consequently, a Weyl-von Neumann type result is obtained for complex skew-symmetric operators relative to a conjugation $\tau$, in the form $T=K+UDU^{tr}$ with $K$ a Schatten $p$-class operator and $D$ a block skew-symmetric operator, up to a unitary change of basis. Moreover, the result extends without kernel-dimension restrictions when $N(T)=N(T^*)$, offering a broad analogue of the Weyl-von Neumann theorem in the skew-symmetric setting, with strong ties to polar-factorizations and spectral constructions for $|A|$.

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

Theorems & Definitions (32)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Definition 1.4
• Definition 1.5
• Definition 1.6
• Definition 1.7
• Definition 1.8
• Definition 1.9
• Lemma 1.10