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

The existence of the second (according to the module) eigenvalueλ2of a completely continuous nonnegative operatorAis proved under the conditions thatAacts in the spaceLp(Ω)orC(Ω)and its exterior squareA∧Ais also nonnegative. For the case when the operatorsAandA∧Aare indecomposable, the simplicity of the first and second eigenvalues is proved, and the interrelation between the indices of imprimitivity ofAandA∧Ais examined. For the case whenAandA∧Aare primitive, the difference (according to the module) ofλ1andλ2from each other and from other eigenvalues is proved.

Nonlinear eigenvalue–eigenvector problems for STP matrices

Let Ai, i = 1, …, m, be a set of Ni × Ni−1 strictly totally positive (STP) matrices, with N0 = Nm = N. For a vector x = (x1, …, xN) ∈ RN and arbitrary p > 0, set We consider the eigenvalue-eigenvector problem where p1 … pm−1 = r. We prove an analogue of the classical Gantmacher-Krein theorem for the eigenvalue-eigenvector structure of STP matrices in the case where pi ≥ 1 for each i, plus various extensions thereof.