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

2018/11/06 by Laustsen, Niels Jakob, White, Jared T. · 1 citation
#16P40 #46B26 #46H10 #47A05 #47L10 (Primary) #47L45 (secondary) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1811.02399

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 realized 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. The Banach space E that we use is the dual of Wark's non-separable, reflexive Banach space with few operators.

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