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

Abstract

The existence of the second (according to the module) eigenvalue λ2 of a completely continuous nonnegative operator A is proved under the conditions that A acts in the space Lp(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 λ1 and λ2 from each other and from other eigenvalues is proved.