Dual Truncated Hankel Operators: Characterization and Properties

TL;DR

This work defines the dual truncated Hankel operator $\mathcal{H}_\phi^\theta$ on the orthogonal complement of a model space and develops a comprehensive operator-theoretic framework around it. By leveraging the dual compressed shift $U_\theta$ and related projections, the authors derive explicit operator equations, intertwining relations, and rank-two commutator identities that mirror classical Hankel/THO results while introducing new structural features. They characterize the solutions to Hankel-related operator equations, obtain parametric representations, and reveal connections to Toeplitz-type behavior, including unique symbol determination, boundedness criteria, and compactness conditions. Further, the paper expands the landscape with the introduction of the O'Toeplitz operator, establishing Brown–Halmos-type results and interplays with traditional Toeplitz operators. Together, these results illuminate both the Toeplitz-like and Hankel-like aspects of DTHO, prescribe when DTHOs behave analogously to classical operators, and introduce a novel operator class (O’Toeplitz) with its own algebraic characterizations and intertwinings. The findings advance model-space operator theory by extending classical operator equations to the dual truncated Hankel context and by clarifying how DTHOs relate to and extend THO, DTTO, and Toeplitz structures.

Abstract

We introduce the notion of the Dual Truncated Hankel Operator (DTHO) and provide several operator equation characterizations using the dual compressed shift operator. These characterizations are similar to classical results concerning Hankel operators and align with recent findings related to Truncated Hankel Operators (THO) \cite{GM}. Additionally, our work addresses comprehensive solutions to various operator equations encountered in studying THO and the classical Hankel operator. We have also established some fundamental operator-theoretic properties of DTHO that apply to general symbols and symbols under specific conditions.

Theorems & Definitions (71)

• Lemma 2.1
• Theorem 2.2
• proof
• Corollary 2.3
• proof
• Theorem 2.4
• proof
• Corollary 2.5
• proof
• Theorem 2.6