Abstract
We introduce the notions of kernel map and kernel set of a bounded linear operator on a Hilbert space relative to a subspace lattice. The characterization of the kernel maps and kernel sets of finite rank operators leads to showing that every norm closed Lie module of a continuous nest algebra is decomposable. The continuity of the nest cannot be lifted, in general.
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Acknowledgements
The present version of the manuscript owes much to the referee’s suggestions. In particular, the authors wish to thank the referee for thoroughly reviewing the manuscript and suggesting the statement and proof of (2.10). Gabriel Matos was supported by a Calouste Gulbenkian Foundation Grant through the “Novos Talentos em Matemática” program. Lina Oliveira was partially supported by FCT/Portugal Grants UID/MAT/04459/2013 and EXCL/MAT-GEO/0222/2012.
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Communicated by Deguang Han.
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Matos, G., Oliveira, L. Kernel maps and operator decomposition. Banach J. Math. Anal. 14, 361–379 (2020). https://doi.org/10.1007/s43037-019-00012-6
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DOI: https://doi.org/10.1007/s43037-019-00012-6