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Positivity

, Volume 9, Issue 3, pp 273–286 | Cite as

Some Open Problems and Conjectures Associated with the Invariant Subspace Problem

  • Y. A. Abramovich
  • C. D. AliprantisEmail author
  • G. Sirotkin
  • V. G. Troitsky
Article

Abstract

There is a subtle difference as far as the invariant subspace problem is concerned for operators acting on real Banach spaces and operators acting on complex Banach spaces. For instance, the classical hyperinvariant subspace theorem of Lomonosov [Funktsional. Anal. nal. i Prilozhen 7(3)(1973), 55–56. (Russian)], while true for complex Banach spaces is false for real Banach spaces. When one starts with a bounded operator on a real Banach space and then considers some “complexification technique” to extend the operator to a complex Banach space, there seems to be no pattern that indicates any connection between the invariant subspaces of the “real” operator and those of its “complexifications.” The purpose of this note is to examine two complexification methods of an operator T acting on a real Banach space and present some questions regarding the invariant subspaces of T and those of its complexifications

Keywords

invariant subspaces complexification algebra of operators 

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Copyright information

© Springer 2005

Authors and Affiliations

  • Y. A. Abramovich
  • C. D. Aliprantis
    • 1
    Email author
  • G. Sirotkin
    • 2
  • V. G. Troitsky
    • 3
  1. 1.Department of Economics, Krannert School of ManagementPurdue UniversityW. LafayetteUSA
  2. 2.Department of MathematicsNorthern Illinois UniversityDeKalbUSA
  3. 3.Department of MathematicsUniversity of AlbertaEdmontonCanada

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