Eigenvalue decay rates for positive integral operators
Positive definite integral operators in \(L^2(\Omega,\mathbb C)\) with a metric space \(\Omega\) are considered. In the first part it is shown that if the kernel is continuous with a square-integrable diagonal then it has an series representation in terms of eigenfunctions with a convergent series of singular values (Mercer's theorem); under the hypothesis that a series representation exists, the latter is shown also without the hypothesis of positive definiteness. In the second part, similar results for the Schatten class are established where, roughly speaking, the continuity hypothesis is replaced by two types of hypothesis: It is required that large balls in \(\Omega\) can be decomposed into (at most) a certain number of pieces satisfying diameter and measure estimates (with appropriate powers), and after a decomposition of \(\Omega\) into a finite number of pieces, the values \(k(x,y)\) of the kernel on the side-diagonal depend in a certain sense Hölder continuous from the average of the diagonal values \(k(x,x)\) and \(k(y,y)\). Actually, the results about the Schatten \(p\)-class are only a corollary, since a more precise decay estimate for the eigenvalues is established.
series representation in terms of eigenfunctions
scientific article; zbMATH DE number 6236079
scientific article