s-numbers of integral operators with Hölder continuous kernels over metric compacta (Q1118139)

Given a compact metric space X, sharp estimates are derived, in terms of the entropy of X, for the Kolmogorov and the approximation numbers of kernel operators defined by continuous kernels and by kernels which are \(\alpha\)-Hölder continuous in the first variable. Fundamental for this is to give precise estimates for these numbers for operators of the form \(I_{\alpha}S\) where S is an operator from a Hilbert space into \(C^{\alpha}(X)\), the space of all \(\alpha\)-Hölder continuous functions on X, and \(I_{\alpha}\) denotes the inclusion map of \(C^{\alpha}(X)\) into C(X), the usual Banach space of continuous functions on X. There are several interesting relations to the covering dimension of X, its Hausdorff dimension and its metric dimension, which are discussed at the end of the paper.