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

Abstract

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.

Spectra and approximations of a class of sign-symmetric matrices

“…W is also irreducible (see [3], theorem 6). This statement easily follows from the theorem 3 and the given above fact, that the matrix of the exterior square A ∧ A of the operator A in the W -basis {e i ∧ e j } (i,j)∈W \∆ coincides with the W -matrix A (2) W . The method of constructing the set W , for which the corresponding W -matrix A (2) W is positive, by the set J in the definition of J -sign-symmetricity of A (2) is given in [3] (see the proof of theorems 5 and 6):…”

“…. , n}, which satisfies conditions (1) and (2). Let {λ i } n i=1 be the set of all eigenvalues of the matrix A, repeated according to multiplicity.…”

“…Let {λ i } n i=1 be the set of all eigenvalues of the matrix A, repeated according to multiplicity. Then all the possible products of the type {λ i λ j }, where 1 ≤ i < j ≤ n, forms the set of all the possible eigenvalues of the corresponding W -matrix A (2) W , repeated according to multiplicity. In the case W = M theorem 1 turns into the Kronecker theorem (see [1], p. 80, theorem 23) about the eigenvalues of the second compound matrix.…”

“…This is the principal difference between the statement of theorem B and the statements, proved in [1]. The necessary and sufficient condition of the existence of complex eigenvalues on the spectral circle |λ| = ρ(A) in the case, when both the matrices A and A (2) are similar to nonnegative irreducible matrices with diagonal matrices of similarity transformations, was obtained in [3].…”

“…, n} (see, for example, [6]). If the set W satisfies conditions (1) and (2), and, in addition, the inclusion (i, k) ∈ W follows from the inclusions (i, j) ∈ W and (j, k) ∈ W for any i, j, k ∈ {1, . .…”