An extension of Mercer's theorem via Pontryagin spaces
Results connected with Mercer's theorem describe series representations for the generating kernel of an integral operator. E.g., if \(T\) is a nonnegative compact operator on \(L^2([a,b])\) with continuous kernel \(K\), then \(K\) can be written in terms of an orthonormal basis of the eigenfunctions \(\varphi_n\) and the eigenvalues \(\lambda_n\) of \(T\), \[ K(x,y) = \sum_{n=1}^{\infty} \lambda_n \varphi_n(x) \overline{ \varphi_n(y)}, \quad x,y \in [a,b]. \] The present paper generalizes this idea. Here, the interval \([a,b]\) is replaced by a locally finite measure space \((X,\nu)\) and \(T\) is replaced by the composition of two self-adjoint integral operators \(R\) and \(S\) on \(L^2(X)\) generated by continuous kernels such that \(R\) is compact and positive and \(S\) is continuous and satisfies some spectral decomposition. The proof of this result is based on a Pontryagin space structure on \(L^2(X)\). The negative subspace of this Pontryagin space is given as the spectral subspace of \(S\) corresponding to the (finitely many) negative eigenvalues of \(S\).
scientific article; zbMATH DE number 6153937