Subspaces that can and cannot be the kernel of a bounded operator on a Banach space

Laustsen, Niels Jakob and White, Jared T. (2025) Subspaces that can and cannot be the kernel of a bounded operator on a Banach space. In: Banach Algebras and Harmonic Analysis : Arens Products, Factorizations, and Bounded Operators. de Gruyter, pp. 241-248. ISBN 9783111643243

Abstract

Given a Banach space E, we ask which closed subspaces may be realized 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 reflexive Banach space E which contains a closed subspace that cannot be realized as the kernel of any bounded operator on E. This implies that the Banach algebra of bounded operators on E fails to be weak ∗ -topologically left Noetherian in the sense of [7]. The Banach space E that we use is the dual of one of Wark’s non-separable, reflexive Banach spaces with few operators.