Abstract and Applied Analysis
Volume 2006 (2006), Article ID 48132, 15 pages
Gantmacher-Kreĭn theorem for 2 nonnegative operators in spaces of functions
Mechanics and Mathematics Faculty, Belarusian State University, Pr. Independence 4, Minsk 220050, Belarus
Received 26 June 2005; Accepted 1 July 2005
Copyright © 2006 O. Y. Kushel and P. P. Zabreiko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The existence of the second (according to the module) eigenvalue
of a completely continuous nonnegative operator is proved under the conditions that acts in the space or and its exterior square is also nonnegative. For the case when the operators and are indecomposable, the simplicity of the first and second eigenvalues is proved, and the interrelation between the indices of imprimitivity of and is examined. For the case when and are primitive, the difference (according to the module) of and from each other and from other eigenvalues is proved.