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Maximal functions and multipliers for subkernels of Toeplitz operators

M. Cristina Câmara, C. Carteiro, C. Diogo

Abstract

We study the relations between maximal functions in a Toeplitz kernel and those in a subkernel of the same Toeplitz operator, as well as the question of how multipliers between Toeplitz kernels act on subkernels. We use those relations to obtain model space representations, in particular isometric model space representations and Hayashi's representation, for some important classes of Toeplitz kernels.

Maximal functions and multipliers for subkernels of Toeplitz operators

Abstract

We study the relations between maximal functions in a Toeplitz kernel and those in a subkernel of the same Toeplitz operator, as well as the question of how multipliers between Toeplitz kernels act on subkernels. We use those relations to obtain model space representations, in particular isometric model space representations and Hayashi's representation, for some important classes of Toeplitz kernels.
Paper Structure (6 sections, 38 theorems, 105 equations, 1 figure)

This paper contains 6 sections, 38 theorems, 105 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Toeplitz subkernels
  3. Maximal functions in subkernels
  4. Multipliers on subkernels
  5. Toeplitz kernels contained in model spaces
  6. Toeplitz kernels with generalised Frostman shift symbols

Key Result

Theorem 2.1

Let $w\in\mathcal{N}^+$. Then the following are equivalent:

Figures (1)

  • Figure 1: $\ker T_{\overline{\theta} -h}$ contained in $K_\gamma$, and the multiplier $m_p^\gamma$ acting on both spaces

Theorems & Definitions (61)

  • Theorem 2.1: CP18_multipliers
  • Corollary 2.2: CP18_multipliers
  • Corollary 2.3: CP18_multipliers
  • Proposition 2.4: CP18_multipliers
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • ...and 51 more