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Bounded symbols of Toeplitz operators on Paley-Wiener spaces and a weak factorization theorem

Petr Kulikov

TL;DR

This work extends Rochberg's bounded-symbol result from the Hilbert Paley–Wiener space $PW_a^2$ to the Banach spaces $PW_a^p$ for $1<p<\infty$, and proves a continuous Toeplitz commutator theorem via a symbol-splitting approach. By decomposing symbols into left, central, and right parts and leveraging Hankel/Nehari theory, it constructs bounded symbols for bounded Toeplitz operators and establishes a duality framework between preduals and Toeplitz operators. The paper then derives a weak factorization theorem: for $p,q>1$ with $1/p+1/q=1$, any $h\in PW^1_{2a}$ can be written as $h=\sum_{k\ge0} f_k \overline{g_k}$ with $f_k\in PW_a^p$ and $g_k\in PW_a^q$, with summable product norms. Collectively, these results provide a robust structural theory for Toeplitz operators on Paley–Wiener spaces and connect bounded-symbol phenomena with weak factorization via a predual framework, advancing both operator theory and harmonic analysis on these spaces.

Abstract

A classical result by R. Rochberg says that every bounded Toeplitz operator $T$ on the Hilbert Paley-Wiener space $\mathrm{PW}_a^2$ admits a bounded symbol $\varphi$. We generalize this result to Toeplitz operators on the Banach Paley-Wiener spaces $\mathrm{PW}_a^p$, $1<p<+\infty$. The Toeplitz commutator theorem describes the integral identity that must hold for a bounded operator $T$ on $\mathrm{PW}_a^p$ to be a Toeplitz operator on $\mathrm{PW}_a^p$. We prove this theorem in the continuous case, thus extending the result previously obtained by D. Sarason in the discrete case. Upon combining the results, we establish the weak factorization theorem, namely, for $p,q>1$, $\frac{1}{p}+\frac{1}{q}=1$, any function $h$ belonging to $\mathrm{PW}^1_{2a}$ can be represented as $$h=\sum_{k\geqslant 0}f_k\bar{g}_k,\qquad f_k\in\mathrm{PW}_a^p,\,g_k\in\mathrm{PW}_a^q.$$

Bounded symbols of Toeplitz operators on Paley-Wiener spaces and a weak factorization theorem

TL;DR

This work extends Rochberg's bounded-symbol result from the Hilbert Paley–Wiener space to the Banach spaces for , and proves a continuous Toeplitz commutator theorem via a symbol-splitting approach. By decomposing symbols into left, central, and right parts and leveraging Hankel/Nehari theory, it constructs bounded symbols for bounded Toeplitz operators and establishes a duality framework between preduals and Toeplitz operators. The paper then derives a weak factorization theorem: for with , any can be written as with and , with summable product norms. Collectively, these results provide a robust structural theory for Toeplitz operators on Paley–Wiener spaces and connect bounded-symbol phenomena with weak factorization via a predual framework, advancing both operator theory and harmonic analysis on these spaces.

Abstract

A classical result by R. Rochberg says that every bounded Toeplitz operator on the Hilbert Paley-Wiener space admits a bounded symbol . We generalize this result to Toeplitz operators on the Banach Paley-Wiener spaces , . The Toeplitz commutator theorem describes the integral identity that must hold for a bounded operator on to be a Toeplitz operator on . We prove this theorem in the continuous case, thus extending the result previously obtained by D. Sarason in the discrete case. Upon combining the results, we establish the weak factorization theorem, namely, for , , any function belonging to can be represented as

Paper Structure

This paper contains 18 sections, 26 theorems, 134 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations
  3. Earlier results
  4. Plan of the proof
  5. Toeplitz operators as integral operators. Splitting a symbol
  6. Riesz projector and related operators
  7. Splitting procedure. Norm estimates
  8. Reproducing kernels. Central part of a symbol
  9. Nehari Theorem. Right and left parts of a symbol
  10. Hankel operators on the Hardy space. Nehari Theorem
  11. Analytic Toeplitz operators on $\mathrm{PW}_a^p$ as Hankel operators
  12. Existence of a bounded symbol
  13. Characteristic property of Toeplitz operators
  14. Conjugate kernel
  15. Toeplitz commutator theorem
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $1<p<+\infty$. Let $T_\varphi$ be a Toeplitz operator on $\mathrm{PW}_a^p$ with symbol $\varphi \in \mathcal{S}(\mathbb{R})$. Then $T_\varphi$ admits a bounded symbol $\psi$ such that for a universal constant $c > 0$.

Figures (1)

  • Figure 1: Graphs of functions $\psi_\mathfrak{L}, \psi_\mathfrak{C},\psi_\mathfrak{R}$.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: bessonov, Theorem 2.4
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • ...and 37 more