Kernels of Bounded Operators on the Classical Transfinite Banach Sequence Spaces

Abstract

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