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Paired kernels and truncated Toeplitz operators

M. Cristina Câmara, Jonathan R. Partington

TL;DR

This work studies paired operators $S_{a,b}$ on $L^2(\mathbb T)$ and their kernels, emphasizing projected kernels ${\rm ker}^+ S_{a,b}$ and connections to Toeplitz kernels via invertibility relations. It develops a framework relating ${\rm ker}^+ S_{a,b}$ and ${\rm ker}^- S_{a,b}$, and proves near-invariance properties that mirror those of Toeplitz kernels. The authors establish detailed inclusion relations for projected kernels under symbol multiplications by inner and outer factors, with precise conditions for strictness and finite-dimensional behavior. These results are then applied to describe kernels of asymmetric truncated Toeplitz operators (ATTO), giving explicit scalar-type kernel descriptions and, in the finite-rank case, closed-form kernel formulas that can be independent of the right-hand symbol under certain size constraints.

Abstract

This paper considers paired operators in the context of the Lebesgue Hilbert space $L^2$ on the unit circle and its subspace, the Hardy space $H^2$. The kernels of such operators, together with their analytic projections, which are generalizations of Toeplitz kernels, are studied. Inclusion relations between such kernels are considered in detail, and the results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators.

Paired kernels and truncated Toeplitz operators

TL;DR

This work studies paired operators on and their kernels, emphasizing projected kernels and connections to Toeplitz kernels via invertibility relations. It develops a framework relating and , and proves near-invariance properties that mirror those of Toeplitz kernels. The authors establish detailed inclusion relations for projected kernels under symbol multiplications by inner and outer factors, with precise conditions for strictness and finite-dimensional behavior. These results are then applied to describe kernels of asymmetric truncated Toeplitz operators (ATTO), giving explicit scalar-type kernel descriptions and, in the finite-rank case, closed-form kernel formulas that can be independent of the right-hand symbol under certain size constraints.

Abstract

This paper considers paired operators in the context of the Lebesgue Hilbert space on the unit circle and its subspace, the Hardy space . The kernels of such operators, together with their analytic projections, which are generalizations of Toeplitz kernels, are studied. Inclusion relations between such kernels are considered in detail, and the results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators.
Paper Structure (4 sections, 24 theorems, 102 equations)

This paper contains 4 sections, 24 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Projected paired kernels and near invariance
  3. Inclusion relations
  4. Kernels of ATTO

Key Result

Proposition 3.1

Let $\theta_1$ and $\theta_2$ be inner functions. Then and, for ${\rm ker}^+ S_{a,b} \ne \{0\}$, the inclusion is proper if and only if $\theta_1 \theta_2 \not\in \mathbb C$; that is, $\theta_1$ or $\theta_2$ is non-constant.

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 2.2
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Corollary 3.5
  • proof
  • ...and 34 more