Paired kernels and truncated Toeplitz operators

TL;DR

This paper investigates paired operators on the Hardy space, focusing on their kernels and analytic projections, and applies these findings to characterize kernels of finite-rank asymmetric truncated Toeplitz operators.

Contribution

It introduces a detailed study of kernels of paired operators and their analytic projections, extending the understanding of Toeplitz kernels and their inclusion relations.

Findings

Characterization of kernels of paired operators on $L^2$ and $H^2$

Inclusion relations between such kernels are established

Application to kernels of finite-rank asymmetric truncated Toeplitz operators

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.