Accepted author manuscript, 339 KB, PDF document
Research output: Contribution in Book/Report/Proceedings - With ISBN/ISSN › Conference contribution/Paper › peer-review
Research output: Contribution in Book/Report/Proceedings - With ISBN/ISSN › Conference contribution/Paper › peer-review
}
TY - GEN
T1 - Subspaces that can and cannot be the kernel of a bounded operator on a Banach space
AU - Laustsen, Niels Jakob
AU - White, Jared T
N1 - This paper has been independently refereed prior to acceptance
PY - 2018/11/17
Y1 - 2018/11/17
N2 - 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.
AB - 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.
KW - Banach space
KW - bounded operator
KW - kernel
KW - dual Banach algebra
KW - weak-closed ideal
KW - Noetherian
M3 - Conference contribution/Paper
BT - Proceedings of the 24th International Conference on Banach algebras and Applications
A2 - Filali, Mahmoud
T2 - 24th Conference on Banach Algebras and Applications
Y2 - 11 July 2019 through 18 July 2019
ER -