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Restricted Toeplitz and Hankel Operators

Priyanka Aroda, Arup Chattopadhyay, Supratim Jana

Abstract

We restrict the classical Toeplitz and Hankel operators on the Beurling subspace $ηH^2$ having range contained in the model space $K_θ$, and characterize their compactness. Moreover, we also obtain their algebraic characterizations, parallel to those of classical Toeplitz and Hankel operators. Additionally, we define the small and big truncated Toeplitz operators and obtain necessary and sufficient conditions for their being zero, finite rank, or compact by employing some of our main results.

Restricted Toeplitz and Hankel Operators

Abstract

We restrict the classical Toeplitz and Hankel operators on the Beurling subspace having range contained in the model space , and characterize their compactness. Moreover, we also obtain their algebraic characterizations, parallel to those of classical Toeplitz and Hankel operators. Additionally, we define the small and big truncated Toeplitz operators and obtain necessary and sufficient conditions for their being zero, finite rank, or compact by employing some of our main results.
Paper Structure (8 sections, 30 theorems, 59 equations)

This paper contains 8 sections, 30 theorems, 59 equations.

Table of Contents

  1. Abstract
  2. Keywords
  3. Mathematics Subject Classification (2020)
  4. Introduction and Preliminaries
  5. Compactness of RTOs and RHOs
  6. Characterizations
  7. Small and Big Truncated Toeplitz operators
  8. Remarks

Key Result

Lemma 2.1

Let $\eta,\theta$ be two inner functions and consider the associated restricted Toeplitz operator (RTO) $\mathcal{T}_\phi:\eta H^2 \rightarrow K_\theta$ for $\phi\in L^\infty$. Then, the RTO $\mathcal{T}_\phi$ can be viewed as the action of $H_{\breve{\theta}}H_{\phi\eta\bar{\theta}}$ on $H^2.$

Theorems & Definitions (56)

  • Definition 1.1
  • Definition 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • ...and 46 more