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

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