  In this paper, we prove that, under precise spectral assumptions, some finite
difference approximations of scalar leftgoing transport equations on the
positive half-line with numerical boundary conditions are $\ell^1$-stable but
$\ell^q$-unstable for any $q>1$. The proof relies on the accurate description
of the Green's function for a particular family of finite rank perturbations of
Toeplitz operators whose essential spectrum belongs to the closed unit disk and
with a simple eigenvalue of modulus $1$ embedded into the essential spectrum.
