Eigenvalues of smooth kernels

Suppose is a symmetric square integrable kernel on the unit square [0, 1]2. Then is a compact symmetric operator on the Hilbert space L2[0, 1]. H. Weyl (see [2]) has shown that, if then the eigenvalues of T satisfy as n → ∞. We prove a related result. Let W12[0, 1]2 denote the space of all K(x, t) ε...

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Published in:Mathematical proceedings of the Cambridge Philosophical Society 1984-01, Vol.95 (1), p.135-140
Main Author: Reade, J. B.
Format: Article
Language:English
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Summary:Suppose is a symmetric square integrable kernel on the unit square [0, 1]2. Then is a compact symmetric operator on the Hilbert space L2[0, 1]. H. Weyl (see [2]) has shown that, if then the eigenvalues of 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 .
ISSN:0305-0041
1469-8064
DOI:10.1017/S0305004100061375