Eigenvalues of Integral-Operators with Positive Definite Kernels Satisfying Integrated Holder

Abstract

For a compact metric space X let μ be a finite Borel measure on X. The authors investigate the asymptotic behavior of eigenvalues of integral operators on L2(X, μ). These integral operators are assumed to have a positive definite kernel which satisfies certain conditions of H¨older continuity. For the eigenvalues _n, n 2 N, which are counted according to their algebraic multiplicities and ordered with respect to decreasing absolute values, the main result of this paper consists of estimates _n = O(n−1(_n(X))_) for n ! 1. Here _n(X) represents the entropy numbers of X, and _ is the exponent in the H¨older continuity condition of the kernel. It is shown that in some respect this estimate is optimal. In the special case where X = _ RN is a bounded Borel set, the above estimate yields _n = O(n−_/N−1) for n ! 1. The article concludes with some non-trivial examples of compact metric spaces with regular entropy behavior.