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Gantmacher-Kreĭn theorem for 2 nonnegative operators in spaces of functions

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  • O. Y. Kushel
  • P. P. Zabreiko

Abstract

The existence of the second (according to the module) eigenvalue λ 2 of a completely continuous nonnegative operator A is proved under the conditions that A acts in the space L p ( Ω ) or C ( Ω ) and its exterior square A ∧ A is also nonnegative. For the case when the operators A and A ∧ A are indecomposable, the simplicity of the first and second eigenvalues is proved, and the interrelation between the indices of imprimitivity of A and A ∧ A is examined. For the case when A and A ∧ A are primitive, the difference (according to the module) of λ 1 and λ 2 from each other and from other eigenvalues is proved.

Suggested Citation

  • O. Y. Kushel & P. P. Zabreiko, 2006. "Gantmacher-Kreĭn theorem for 2 nonnegative operators in spaces of functions," Abstract and Applied Analysis, Hindawi, vol. 2006, pages 1-15, May.
    • Handle: RePEc:hin:jnlaaa:048132
    DOI: 10.1155/AAA/2006/48132
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