Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bounded Toeplitz Products on the Hardy Space

Ryan O'Loughlin

TL;DR

The work tackles the long-standing problem of characterizing when the product of two Toeplitz operators is bounded on the Hardy space. By introducing admissible Toeplitz pairs and leveraging the two-weighted Riesz projection, the authors prove that T_u T_v is bounded on $H^2$ if and only if the symbol product $uv$ is essentially bounded, under precise pole/analyticity assumptions. The paper further connects this operator-theoretic result to harmonic-analytic criteria, including Poisson-extension conditions and Carleson measures for associated range spaces, and shows Sarason-type conjectures hold for admissible pairs. This establishes a concrete, function-theoretic criterion linking Toeplitz products to weighted projection operators and Carleson-measure characterizations, bridging operator theory and harmonic analysis with potential implications for related function spaces.

Abstract

A Toeplitz operator on the Hardy space of the unit circle is bounded if and only if its symbol is bounded. For two Toeplitz operators, there are no known function-theoretic conditions for their symbols, which are equivalent to the product of the Toeplitz operators being bounded. In this paper, we provide a solution to this problem, by showing under certain assumptions that the product of two Toeplitz operators is bounded if and only if the product of their symbols is bounded.

Bounded Toeplitz Products on the Hardy Space

TL;DR

The work tackles the long-standing problem of characterizing when the product of two Toeplitz operators is bounded on the Hardy space. By introducing admissible Toeplitz pairs and leveraging the two-weighted Riesz projection, the authors prove that T_u T_v is bounded on if and only if the symbol product is essentially bounded, under precise pole/analyticity assumptions. The paper further connects this operator-theoretic result to harmonic-analytic criteria, including Poisson-extension conditions and Carleson measures for associated range spaces, and shows Sarason-type conjectures hold for admissible pairs. This establishes a concrete, function-theoretic criterion linking Toeplitz products to weighted projection operators and Carleson-measure characterizations, bridging operator theory and harmonic analysis with potential implications for related function spaces.

Abstract

A Toeplitz operator on the Hardy space of the unit circle is bounded if and only if its symbol is bounded. For two Toeplitz operators, there are no known function-theoretic conditions for their symbols, which are equivalent to the product of the Toeplitz operators being bounded. In this paper, we provide a solution to this problem, by showing under certain assumptions that the product of two Toeplitz operators is bounded if and only if the product of their symbols is bounded.
Paper Structure (8 sections, 11 theorems, 25 equations)

This paper contains 8 sections, 11 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hardy Space Theory
  4. Toeplitz operators
  5. Poles of $H(\mathbb{}T)$ Functions
  6. Boundedness of Products of Toeplitz Operators
  7. The Two-Weighted Riesz Projection
  8. Range Spaces

Key Result

Lemma 2.3

Let $u \in H(\mathbb{T})$ and $v \in L^2(\mathbb{T})$. Then $T_u T_{{v}}$ is a bounded map from $H^2(\mathbb{T})$ to $H^2(\mathbb{T})$ if and only if $\operatorname{ran} \Tilde{T} \subseteq H^{2}(\mathbb{T})$, where $\Tilde{T}$ is given by Ttilde.

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Example 3.4
  • Proposition 3.5
  • ...and 19 more