Laustsen, Niels Jakob and White, Jared T (2018) Subspaces that can and cannot be the kernel of a bounded operator on a Banach space. In: Proceedings of the 24th International Conference on Banach algebras and Applications :. UNSPECIFIED, CAN. (In Press)
AclosedsubspacewhichisnotakernelAcceptedNov2018.pdf - Accepted Version
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Abstract
Given a Banach space E, we ask which closed subspaces may be realised as the kernel of a bounded operator E→E. We prove some positive results which imply in particular that when E is separable every closed subspace is a kernel. Moreover, we show that there exists a Banach space E which contains a closed subspace that cannot be realised as the kernel of any bounded operator on E. This implies that the Banach algebra B(E) of bounded operators on E fails to be weak*-topologically left Noetherian in the sense of (JT White, Left Ideals of Banach Algebras and Dual Banach Algebras, preprint, 2018). The Banach space E that we use is the dual of one of Wark’s non-separable, reflexive Banach spaces with few operators.