Paired and Toeplitz + Hankel operators

TL;DR

This paper provides a comprehensive classification of Toeplitz + Hankel operators on vector-valued Hardy spaces and of paired operators on $L^2(\mathbb{T})$, using algebraic, function-theoretic, and model-space methods. It establishes Brown-Halmos–type equivalences that characterize Toeplitz + Hankel operators, reveals symbol uniqueness properties, and connects scalar/non-injective Hankel cases to Beurling inner-function theory. The work then classifies paired operators on $L^2(\mathbb{T})$ and extends to $\theta$-paired operators on $H^2(\mathbb{T})$ via Beurling-type subspaces, uncovering a close link to truncated Toeplitz operators. By tying these operator classes together through model-space techniques and operator dualities, the paper lays groundwork for future exploration of truncated Toeplitz theory and perturbation questions in Hardy spaces.

Abstract

We present complete classifications of Toeplitz + Hankel operators on vector-valued Hardy spaces and classify paired operators on $L^2(\mathbb{T})$. We also study the latter class through the lens of inner functions on the disc.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Theorem 3.1
• proof
• Theorem 4.1