Gantmakher-Krein theorem for 2-totally nonnegative operators in ideal spaces

Details Gantmakher-Krein theorem for 2-totally nonnegative operators in ideal spaces Gantmakher-Krein theorem for 2-totally nonnegative operators in ideal spaces

2008-12-04

arxiv.org/abs/0812.0902v1

Mathematics

Spectral Theory

13 pages

Scientific paper

The tensor and exterior squares of a completely continuous non-negative linear operator $A$ acting in the ideal space $X(\Omega)$ are studied. The theorem representing the point spectrum (except, probably, zero) of the tensor square $A \otimes A$ in the terms of the spectrum of the initial operator $A$ is proved. The existence of the second (according to the module) positive eigenvalue $\lambda_2$, or a pair of complex adjoint eigenvalues of a completely continuous non-negative operator $A$ is proved under the additional condition, that its exterior square $A\wedge A$ is also nonnegative.

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