GANTMACHER-KREĬN THEOREM FOR 2 NONNEGATIVE OPERATORS IN SPACES OF FUNCTIONS

The existence of the second (according to the module) eigenvalue {\lambda}_{2} of a completely continuous nonnegative operator A is proved under the conditions that A acts in the space L_{p} (\Omega) or C (\Omega) and its exterior square A \wedge A is also nonnegative. For the case when the operators A and A \wedge 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 \wedge A is examined. For the case when A and A \wedge A are primitive, the difference (according to the module) of \lambda_{1} and \lambda_{2} from each other and from other eigenvalues is proved.