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COPYRIGHT 2006 Hindawi Publishing Corp.
The existence of the second (according to the module) eigenvalue [[lambda].sub.2] of a completely continuous nonnegative operator A is proved under the conditions that A acts in the space [L.sub.p]([OMEGA]) or C([OMEGA]) and its exterior square A [conjunction] A is also nonnegative. For the case when the operators A and A [conjunction] 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 [conjunction] A is examined. For the case when A and A [conjunction] A are primitive, the difference (according to the module) of [[lambda].sub.1] and [[lambda].sub.2] from each other and from other eigenvalues is proved.
1. Introduction
In the monograph [3] the following statement was proved: if the matrix A of a linear operator A in the space [R.sub.n] is primitive along with its associated [A.sup(j)] (1 < j [less than or equal to] k) up to the order k, then the operator A has k positive simple eigenvalues < [[lambda].sub.k] < ... < [[lambda].sub.2] < [[lambda].sub.1], with a positive eigenvector [e.sub.1] corresponding to the maximal eigenvalue [[lambda].sub.1], and an eigenvector [e.sub.j], which has exactly j - 1 changes of sign, corresponding to jth eigenvalue [[lambda].sub.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 Krein have thoroughly studied the linear integral operators
Kx(t) = [[integral].sup.b.sub.a] k(t, s)x(s)ds (1.1)
acting in the space [L.sub.2]([a,b]) with continuous kernels k(t, s), for which the matrices [parallel]k([t.sub.i], [t.sub.j])[[parallel].sup.n.sub.1] (n = 1, 2, ...) for any points [t.sub.1], ..., [t.sub.n] [member of] [a,b], among which at least one is interior, are oscillatory. Such kernels, named in [3] oscillatory, form quite full analogue to oscillatory matrices. In [3], for the integral operators with continuous oscillatory kernels, it was proved that there exists a converging-to-zero sequence of positive simple eigenvalues [[lambda].sub.1] > [[lambda].sub.2] > ... > [[lambda].sub.n] > ... with eigenfunctions [e.sub.n](t) that has exactly n-1 changes of sign, corresponding to the nth eigenvalue [[lambda].sub.n] (see [3, page 211]).
In connection with the formulated Gantmacher-Krein theorem, there arises a natural question on the possibility of spreading the statements about k-completely-nonnegative matrices from [3] onto the integral operators with k-completely-nonnegative kernels, that is, the kernels k(t, s), for which the matrices [parallel]k([t.sub.i], [t.sub.j])[[parallel].sup.n.sub.1] (n = 1, 2, ..., k) for any points [t.sub.1], ..., [t.sub.n] [member of] [a,b], among which at least one is interior, are oscillatory. The answer to this question is positive. Moreover, this statement was actually proved exactly in [3].
However, here arises a question how substantial the condition of continuity of the kernel k(t, s) is in these statements and how substantial the assumption that the problem is regarded in the space of functions, defined exactly on the interval [a,b], is. And of course the natural question arises whether it is possible to obtain similar statements for abstract (not necessarily integral) operators in an arbitrary Banach spaces.
In the present paper we study 2-completely-nonnegative (or otherwise bi-nonnegative) operators in the spaces [L.sub.p]([OMEGA]) (1 [less than or equal to] p[less than or equal to][infinity]) and C([OMEGA]). As the authors believe, the natural machinery for the examination of such operators is a crossway from studying an operator A in one of the spaces [L.sub.p]([OMEGA]) and C([OMEGA]) to the study of the operators A [cross product] A and A [conjunction] A, acting, respectively, in the spaces [L.sub.p] [cross product] [L.sub.p] = [L.sb.p][OMEGA] x [OMEGA]) and [L.sub.p] [conjunction] [L.sub.p] = [L.sup.a.sub.p] ([OMEGA] x [OMEGA]) (the latter is a subspace of the space [L.sub.p] [cross product] [L.sub.p] = [L.sub.p][OMEGA] x [OMEGA]), consisting of antisymmetric functions, i.e., functions x(t, s), for which x(t, s)=-x(s, t)).
2. Tensor and exterior square of the spaces [L.sub.p]([OMEGA]) and C([OMEGA])
Let ([OMEGA], [??], [mu]) be a triple consisting of some set [OMEGA], some [sigma]-algebra [??] of "measurable" subsets and some s-finite and s-additive measure on A. We will be interested in the space [L.sub.p]([OMEGA]) of functions, integrable on [OMEGA] with the power p for 1 [less than or equal to] p < [infinity] or measurable and substantially bounded for p=[infinity], the analogous space [L.sub.p]([OMEGA] x [OMEGA]) of functions, integrable on [OMEGA] x [OMEGA] with the power p for 1 [less than or equal to] p < [infinity] or essentially bounded for p=[infinity] and, finally, the subspace [L.sup.a.sub.p][OMEGA] x [OMEGA]) of the space [L.sub.p][OMEGA] x [OMEGA]) of antisymmetric functions. Henceforth let p be a fixed number from [1, [infinity]].
We start with observing the following facts:
(1) the space [L.sup.a.sub.p](OMEGA]X[OMEGA]) is one of the tensor products of the space [L.sub.p](OMEGA]) by itself, and, respectively,
(2) the space [L.sup.a.sub.p]([OMEGA] x [OMEGA]) is one of the exterior products of the space [L.sub.p]([OMEGA]) by itself.
The first of these statements means the following.
(a) For arbitrary functions [x.sub.1], [x.sub.2][member of] [L.sub.p]([OMEGA]) their [??]-product [x.sub.1][??] [x.xub.2]([t.sub.1], [t.sub.2])=[x.sub.1]([t.sub.1])[x.sub.2]([t.sub.2]) belongs to the space [L.sub.p]([OMEGA] x [OMEGA]), with
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.1)
(b) The linear hull of the set of all [??]-products of functions from [L.sub.p]([OMEGA]), that is, the set of all functions of the form
x([t.sub.1], [t.sub.2]) = [summation over i][x.sup.i.sub.1]([t.sub.1])[x.sup.i.sub.2]([t.sub.2]) (2.2)
is dense in the space [L.sub.p]([OMEGA] x [OMEGA]).
The second statement means the following.
(a) The [conjunction]-product of arbitrary functions [x.sub.1], [x.sub.2] [member of] [L.sub.p]([OMEGA]) with [x.sub.1] [conjunction] [x.sub.2]([t.sub.1], [t.sub.2]) = [x.sub.1]([t.sub.1])[x.sub.2]([t.sub.2])- [x.sub.1]([t.sub.2])[x.sub.2]([t.sub.1]) also belongs to the space [L.sub.p([OMEGA] x [OMEGA]), and it is obvious that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.3)
(b) The linear hull of the set of all [conjunction]-products of the functions from [L.sub.p](OMEGA]) is dense in the space [L.sup.a.sub.p]([OMEGA] x [OMEGA]).
The space [L.sup.a.sub.p]([OMEGA] x [OMEGA]) is isomorphic in the category of Banach spaces to the space [L.sub.p](W), where W is the subset [OMEGA] X [OMEGA], for which the sets W [intersection] [??] and ([OMEGA] x [OMEGA]) \ (W [intersection] [??] have zero measure; here [??] = {([t.sub.2], [t.sub.1]):([t.sub.1], [t.sub.2])...
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