A further result on analyticity of some kernels (Q1340783)

Let \(H_ n\) \((n\in \mathbb{N})\) be the Hermite functions and define the class \(\Gamma_ R^{1/2}= \{(\lambda_ n)_{n\in \mathbb{N}}\): \(\sup R^{\sqrt {n}} | \lambda_ n|< +\infty\}\) for some \(R>1\). The author proves that the series \(\sum_{n=1}^ \infty \lambda_ n H_ n (x) H_ n(y)\) is an analytic function in \(R^ 2\). A corollary states that if the sequence \((\lambda_ n)_{n\in N}\) converges geometrically, the series is analytic in \(R^ 2\). The theorem is then applied to the study of kernels of integral operators. Let \(T: L^ 2 (R)\to L^ 2 (R)\) be a selfadjoint integral operator whose kernel belongs to \(L^ 2 (R)\) and has its eigenvalues in the \(\Gamma_ R^{1/2}\). Then the operator \(T\) is unitarily equivalent to an integral operator \(T_ G\) given by a kernel \(G\) which is analytic and belongs to the Schwartz space \(S(R)\).