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Abstract and Applied Analysis
Volume 2006 (2006), Article ID 48132, 15 pages
http://dx.doi.org/10.1155/AAA/2006/48132

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.

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 AA is also nonnegative. For the case when the operators A and AA are indecomposable, the simplicity of the first and second eigenvalues is proved, and the interrelation between the indices of imprimitivity of A and AA is examined. For the case when A and AA are primitive, the difference (according to the module) of λ1 and λ2 from each other and from other eigenvalues is proved.