  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].
