# A Toeplitz-like operator with rational symbol having poles on the unit circle III: the adjoint

## TL;DR

This paper analyzes the adjoint of Toeplitz-like operators with rational symbols having poles on the unit circle, providing conditions for symmetry, selfadjoint extensions, and connections to Sarason's unbounded Toeplitz operators.

## Contribution

It extends previous work by describing the adjoint operator, characterizing symmetry and selfadjointness, and linking to known unbounded Toeplitz operators in specific cases.

## Key findings

- Characterization of the adjoint operator $T__$ for rational symbols with poles on the unit circle.
- Conditions under which $T__$ is symmetric or admits a selfadjoint extension.
- Equivalence of $T__$ with Sarason's unbounded Toeplitz operator when $$ has poles only on $$ and is proper.

## Abstract

This paper contains a further analysis of the Toeplitz-like operators $T_\omega$ on $H^p$ with rational symbol $\omega$ having poles on the unit circle that were previously studied in [5.6]. Here the adjoint operator $T_\omega^*$ is described. In the case where $p=2$ and $\omega$ has poles only on the unit circle $\mathbb{T}$, a description is given for when $T_\omega^*$ is symmetric and when $T_\omega^*$ admits a selfadjoint extension. Also in the case where $p=2$, $\omega$ has only poles on $\mathbb{T}$ and in addition $\omega$ is proper, it is shown that $T_\omega^*$ coincides with the unbounded Toeplitz operator defined by Sarason in [10].