Generalized Toeplitz plus Hankel operators: kernel structure and defect numbers

TL;DR

This paper investigates the structure and defect numbers of generalized Toeplitz plus Hankel operators on Hardy spaces, providing explicit kernel and cokernel descriptions under certain conditions.

Contribution

It introduces a condition relating generating functions and characterizes the kernel and cokernel structures of these operators, extending previous results to a broader class.

Findings

Explicit formulas for defect numbers

Kernel and cokernel structures characterized

Conditions for operator invertibility established

Abstract

Generalized Toeplitz plus Hankel operators $T(a)+H_{\alpha}(b)$ generated by functions $a,b$ and a linear fractional Carleman shift $\alpha$ changing the orientation of the unit circle $\mathbb{T}$ are considered on the Hardy spaces $H^p(\mathbb{T})$, $1<p<\infty$. If the functions $a,b\in L^\infty(\mathbb{T})$ and satisfy the condition $$ a(t) a(\alpha(t))=b(t) b(\alpha(t)),\quad t\in \mathbb{T}, $$ the defect numbers of the operators $T(a)+H_{\alpha}(b)$ are established and an explicit description of the structure of the kernels and cokernels of the operators mentioned is given.