Closed Ideals of Operators between the Classical Sequence Spaces

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 Zsák, 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.

18 pages

References in corpus (1)

• Closed ideals in $\mathcal{L}(X)$ and $\mathcal{L}(X^{\ast })$ when $X$ contains certain copies of ${\ell }_p$ and $c_0$

Cited by in corpus (5)

• Surjective homomorphisms from algebras of operators on long sequence spaces are automatically injective
• Quotient algebra of compact-by-approximable operators on Banach spaces failing the approximation property
• When are full representations of algebras of operators on Banach spaces automatically faithful?
• Banach spaces for which the space of operators has $2^{\mathfrak{c}}$ closed ideals
• The cardinality of the sublattice of closed ideals of operators between certain classical sequence spaces