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Gantmacher-Krein Theorem for 2-totally Nonnegative Operators in Ideal Spaces

  • Olga Y. Kushel
  • Petr P. Zabreiko
Part of the Operator Theory: Advances and Applications book series (OT, volume 202)

Abstract

The tensor and exterior squares of a completely continuous nonnegative linear operator A acting in the ideal space X(Ω) are studied. The theorem representing the point spectrum (except, probably, zero) of the tensor square (AA) M in the terms of the spectrum of the initial operator A is proved. The existence of the second (according to the module) positive eigenvalue λ2, or a pair of complex conjugate eigenvalues of a completely continuous non-negative operator A is proved under the additional condition, that its exterior square (AA) M is also nonnegative.

Keywords

Total positivity ideal spaces tensor products exterior products point spectrum 

Mathematics Subject Classification (2000)

Primary 47B65 Secondary 47A80 47B38 46E30 

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© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • Olga Y. Kushel
    • 1
  • Petr P. Zabreiko
    • 1
  1. 1.Department of Mechanics and MathematicsBelorussian State UniversityMinskBelarus

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