Gantmakher–Krein theorem for $2$-completely nonnegative operators in ideal spaces

The exterior square of the ideal space $X(\Omega)$ is studied. The theorem representing the point spectrum of the tensor square of a completely continuous non-negative linear operator $A\colon X(\Omega)\to X(\Omega)$ in the terms of the spectrum of the initial operator 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.