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Paired kernels and their applications

M. Cristina Câmara, Jonathan R. Partington

Abstract

This paper considers paired operators in the context of the Lebesgue Hilbert space 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. Results on near-invariance properties, representations, and inclusion relations for these kernels are obtained. The existence of a minimal Toeplitz kernel containing any projected paired kernel and, more generally, any nearly $S^*$-invariant subspace of $H^2$, is derived. The results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators.

Paired kernels and their applications

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. Results on near-invariance properties, representations, and inclusion relations for these kernels are obtained. The existence of a minimal Toeplitz kernel containing any projected paired kernel and, more generally, any nearly -invariant subspace of , is derived. The results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators.
Paper Structure (7 sections, 29 theorems, 90 equations)

This paper contains 7 sections, 29 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Projected paired kernels
  3. An example and questions it raises
  4. Near invariance properties
  5. Minimal Toeplitz kernels and representations of projected paired kernels
  6. Inclusion relations
  7. Kernels of finite-rank asymmetric truncated Toeplitz operators

Key Result

Proposition 2.1

The operators and are well-defined and bijective with inverses and and we have $\mathcal{P}^+ \phi = P^+ \phi$, and $\mathcal{M}_{a/b}(\mathcal{P}^+ \phi)=P^- \phi$ for $\phi \in {\rm ker}_{a,b}$.

Theorems & Definitions (48)

  • Proposition 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Proposition 2.4
  • proof
  • Proposition 4.1
  • Corollary 4.2
  • Corollary 4.3
  • Proposition 4.4
  • proof
  • ...and 38 more