Closed Ideals of Operators between the Classical Sequence Spaces

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.