Kernels of bounded operators on Banach spaces.

In a recent paper, Laustsen and White initiated the study of the following question: which closed subspaces of a Banach space X can be realized as the kernel of a bounded linear operator from X to itself? They showed that every separable, closed subspace can be realized as such a kernel, but they also gave an example of a closed subspace of a reflexive Banach space which is not a kernel in this way, thus solving an open problem concerning dual Banach algebras. The purpose of this project is to investigate the above question further, beginning with the classical non-separable sequence spaces l_p(A) and c_0(A) for an uncountable index set A.