Eigenvalues of integral operators on \(L_ 2(I)\) given by analytic kernels (Q1316474)

The authors consider symmetric Hilbert-Schmidt operators on \(L^ 2(I)\), where \(I\) is an interval in \(\mathbb{R}\). Let \(\{\lambda_ n\}\), \(n\geq 0\), be the corresponding sequence of eigenvalues. It is shown that if \(\ell^ n \lambda_ n \to 0\) as \(n \to \infty\) for \(\ell=1,2,\dots,\) then the operator is unitarily equivalent to an operator with analytic kernel. Conversely, if the operator has an analytic kernel, then \(\ell^{n(\varepsilon)} \lambda_ n \to 0\) as \(n \to \infty\) for \(\ell=1,2,\dots\), where \(n(\varepsilon)=n^{1-\varepsilon}\) and \(\varepsilon>0\). The results are sharp.