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Some Open Problems and Conjectures Associated with the Invariant Subspace Problem

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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

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Authors and Affiliations

  1. Department of Economics, Krannert School of Management, Purdue University, 403 West State Street, W. Lafayette, IN, 47907-2056, USA

    C. D. Aliprantis

  2. Department of Mathematics, Northern Illinois University, DeKalb, IL, 60115, USA

    G. Sirotkin

  3. Department of Mathematics, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    V. G. Troitsky

Authors
  1. Y. A. Abramovich
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  2. C. D. Aliprantis
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  3. G. Sirotkin
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  4. V. G. Troitsky
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Correspondence to C. D. Aliprantis.

Additional information

Mathematics Subject Classification 1991: 47A15, 47C05, 47L20, 46B99

Y.A. Abramovich: 1945–2003

The research of Aliprantis is supported by the NSF Grants EIA-0075506, SES-0128039 and DMI-0122214 and the DOD Grant ACI-0325846

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Abramovich, Y.A., Aliprantis, C.D., Sirotkin, G. et al. Some Open Problems and Conjectures Associated with the Invariant Subspace Problem. Positivity 9, 273–286 (2005). https://doi.org/10.1007/s11117-004-3786-9

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