Closed Ideals of Operators between the Classical Sequence Spaces

TL;DR

This paper demonstrates that certain spaces of bounded linear operators between classical sequence spaces have a continuum of closed ideals, using matrices with the Restricted Isometry Property from Compressed Sensing.

Contribution

It establishes the existence of continuum many closed ideals in specific operator spaces, extending previous results and employing techniques from Compressed Sensing.

Findings

Spaces $\\mathcal L(\ell_p, c_0)$, $\mathcal L(\ell_p, \ell_\infty)$, and $\mathcal L(\ell_1, \ell_q)$ have continuum many closed ideals.

Uses matrices with the Restricted Isometry Property to construct closed ideals.

Extends and improves earlier work by Schlumprecht, Zsák, Wallis, and Sirotkin.

Abstract

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.