The cardinality of the sublattice of closed ideals of operators between certain classical sequence spaces

Abstract

Theorem A and Theorem B of [1] state that for $1<p<\infty$ the lattice of closed ideals of $\mathcal{L}({\ell }_p,c_0)$, $\mathcal{L}({\ell }_p,{\ell }_{\infty })$ and of $\mathcal{L}({\ell }_1,{\ell }_p)$ are at least of cardinality $2^{\ddot{I}}‰$. Here we show that the cardinality of the lattice of closed ideals of $\mathcal{L}({\ell }_p,c_0)$, $\mathcal{L}({\ell }_p,{\ell }_{\infty })$ and of $\mathcal{L}({\ell }_1,{\ell }_p)$, is at least $2^{2^{\ddot{I}}‰}$, and thus equal to it.