Invariant subspace problem and compact operators on non-Archimedean Banach spaces

• M. Babahmed ^[1] ; A. El asri ^[1]
  1. [1] Department of Mathematics, University of Moulay Ismail Faculty of Sciences, Meknes, Morocco
• Localización: Extracta mathematicae, ISSN-e 0213-8743, Vol. 35, Nº 2, 2020, págs. 205-219
• Idioma: inglés
• DOI: 10.17398/2605-5686.35.2.205
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• Resumen

  • In this paper, the invariant Subspace Problem is studied for the class of non-Archimedean compact operators on an infinite-dimensional Banach space E over a nontrivial complete non-Archimedean valued field K. Our first main result (Theorem 9) asserts that if K is locally compact, then each compact operator on E possessing a quasi null vector admits a nontrivial hyperinvariant closed subspace. In the second one (Theorem 17), we prove that each bounded operator on E which contains a cyclic quasi null vector can be written as the sum of a triangular operator and a compact shift operator, each one of them possesses a nontrivial invariant closed subspace. Finally, we conclude that if K is algebraically closed, then every compact operator on E either has a nontrivial invariant closed subspace or is a sum of upper triangular operator and shift operator, each of them is compact and has a nontrivial invariant closed subspace.

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