Eigenvalues of smooth kernels

Abstract

Suppose is a symmetric square integrable kernel on the unit square [0, 1]2. Thenis a compact symmetric operator on the Hilbert space L2[0, 1]. H. Weyl (see [2]) has shown that, if then the eigenvaluesof T satisfy as n → ∞. We prove a related result. Let W12[0, 1]2 denote the space of all K(x, t) ε L2[0, 1]2 which are absolutely continuous in x for each t and absolutely continuous in t for each x, and the partial derivatives ∂K/∂x(x, t), ∂K/∂t(x, t) are both in L2[0, 1]2. We slow that the eigenvalues of any satisfy .