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

Abstract

Given a Banach space E, we ask which closed subspaces may be realised as the kernel of a bounded operator E Ñ 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 BpEq 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.Definition 1.1. Let A be a dual Banach algebra. We say that A is weak*-topologically left Noetherian if every weak*-closed left ideal I of A is weak*-topologically finitely-generated, i.e. there exists n P N, and there exist x 1 , . . . , x n P I such that I " Ax 1 `Cx 1 `¨¨¨`Ax n `Cx n w ˚.