  We consider bounded Hankel operators $H_{\psi}$ acting on the Hardy space
$H^2$ to $L^2\ominus H^2$ and obtain results on the Schmidt subspaces
$E^+_s(H_\psi)$ of such operators defined as the kernels of $
H_{\psi}^{\ast}H_{\psi}-s^2I$ where $s>0$. These spaces have been recently
studied in \cite{GP} and \cite{GP1} in the context of anti-linear Hankel
operators. We also discuss the range of the Hankel operators with symbols being
the complex conjugates of functions in the unit ball of $H^{\infty}$.
