On lower estimates for the eigenvalues of positive operators

Abstract

A closer lower estimate for the eigenvalues of positive operators can be found by the following method∗∗. Suppose that on some set in a Hilbert space the symmetric and positive operator A with a discrete eigenspectrum is given, and let α i , i = 1, 2, 3, … be the eigenvalues of this operator. We choose the operator B possessing the same properties and with eigen-values β i , such that B ⩽ A. Then the eigenvalues of these operators satisfy the inequalities β i ⩽ α i i = 1, 2, 3, … To obtain more exact lower estimates for the eigenvalues of the operator A a sequence of symmetric positive operators A n , n = 1, 2, 3, … with discrete eigenspectra λ i ( n) , i, n = 1, 2, 3, …, is constructed such that B ⩽ A n ⩽ A where A n−1 ⩽ A n → A as n → ∞. Then, if for some n, iβ i ≠ λ i ( n) , λ i ( n) can be considered as a closer (compared with β i ) lower estimate for the eigenvalue α i of the operator A. The present article is concerned with the practical application of this method. The explanation of the method follows the book [4].