  We prove that the spaces $\mathcal L(\ell_p,\mathrm{c}_0)$, $\mathcal
L(\ell_p,\ell_\infty)$ and $\mathcal L(\ell_1,\ell_q)$ of operators with
$1<p,q<\infty$ have continuum many closed ideals. This extends and improves
earlier works by Schlumprecht and Zs\'ak, by Wallis, and by Sirotkin and
Wallis. Several open problems remain. Key to our construction of closed ideals
are matrices with the Restricted Isometry Property that come from Compressed
Sensing.
