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# The Invariant Subspace Problem for Non-Archimedean Banach Spaces

Published:2008-12-01
Printed: Dec 2008
• Wies{\l}aw {\'S}liwa
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## Abstract

It is proved that every infinite-dimensional non-archimedean Banach space of countable type admits a linear continuous operator without a non-trivial closed invariant subspace. This solves a problem stated by A.~C.~M. van Rooij and W.~H. Schikhof in 1992.
 Keywords: invariant subspaces, non-archimedean Banach spaces
 MSC Classifications: 47S10 - Operator theory over fields other than ${\bf R}$, ${\bf C}$ or the quaternions; non-Archimedean operator theory 46S10 - Functional analysis over fields other than ${\bf R}$ or ${\bf C}$ or the quaternions; non-Archimedean functional analysis [See also 12J25, 32P05] 47A15 - Invariant subspaces [See also 47A46]