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

TL;DR

This paper extends the Gantmakher-Krein theorem to 2-totally nonnegative operators in ideal spaces, relating spectra of tensor and exterior squares of operators to the original operator's spectrum.

Contribution

It proves a spectral representation theorem for tensor squares of non-negative operators and establishes conditions for the existence of positive or complex eigenvalues.

Findings

Spectral characterization of tensor squares of operators.

Existence of positive eigenvalues under nonnegativity conditions.

Conditions for complex eigenvalues in exterior squares.

Abstract

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.