GANTMACHER-KRE
˘
IN THEOREM FOR 2 NONNEGATIVE
OPERATORS IN SPACES OF FUNCTIONS
O. Y. KUSHEL AND P. P. ZABREIKO
Received 26 June 2005; Accepted 1 July 2005
The existence of the second (according to the module) eigenvalue λ
2
of a completely con-
tinuous nonnegative operator A is proved under the conditions that A acts in the space
L
p
(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 eigen-
values 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 λ
1
and λ
2
from each other and from other eigenvalues is proved.
Copyright © 2006 O. Y. Kushel and P. P. Zabreiko. This is an open access article distrib-
uted under the Creative Commons Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly
cited.
1. Introduction
In the monograph [3] the following statement was proved: if the matrix A of a linear
operator A in the space R
n
is primitive along with its associated A
( j )
(1 <j ≤ k) up to the
order k, then the operator A has k positive simple eigenvalues 0 <λ
k
< ··· <λ
2
<λ
1
, with
a positive eigenvector e
1
corresponding to the maximal eigenvalue λ
1
, and an eigenvector e
j
,
which has exactly j − 1 changes of sign, corresponding to j th eigenvalue λ
j
(see [3, page
310, Theorem 9]). Matrices with mentioned features are called henceforth k-completely
nonnegative; in the most important case k = n they are called oscillatory.
Naturally, there arises a problem whether it is possible to extend this statement to
operators in infinite-dimensional spaces, for example, to linear integral operators. This
problem practically has not been studied in full volume. However, in the monograph [3],
Gantmacher and Kre˘ ın have thoroughly studied the linear integral operators
Kx(t ) =
b
a
k(t , s)x(s)ds (1.1)
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2006, Article ID 48132, Pages 1–15
DOI 10.1155/AAA/2006/48132