Kernels of Bounded Operators on the Classical Transfinite Banach Sequence Spaces

Every closed subspace of each of the Banach spaces $X={\mathrm{\ell }}_p(\mathrm{\Gamma })$ and $X=c_0(\mathrm{\Gamma })$, where $\mathrm{\Gamma }$ is a set and $1<p<\mathrm{\infty }$, is the kernel of a bounded operator $X\to X$. On the other hand, whenever $\mathrm{\Gamma }$ is an uncountable set, ${\mathrm{\ell }}_1(\mathrm{\Gamma })$ contains a closed subspace that is not the kernel of any bounded operator ${\mathrm{\ell }}_1(\mathrm{\Gamma })\to {\mathrm{\ell }}_1(\mathrm{\Gamma })$.