Kernels of Bounded Operators on the Classical Transfinite Banach Sequence Spaces

Every closed subspace of each of the Banach spaces $X = \ell_p(\Gamma)$ and $X=c_0(\Gamma)$, where $\Gamma$ is a set and $1<p<\infty$, is the kernel of a bounded operator $X\to X$. On the other hand, whenever $\Gamma$ is an uncountable set, $\ell_1(\Gamma)$ contains a closed subspace that is not the kernel of any bounded operator $\ell_1(\Gamma)\to\ell_1(\Gamma)$.