Infinite-dimensional elliptic coordinates (Q1566486)

Let \(A\) be a symmetric operator in \(H= \mathbb{R}^n(\mathbb{C}^n)\) and \(x\in H\). By definition the elliptic coordinate of a vector \(x\) is given by the set \(\{\lambda_i\}\) which are roots of \[ \textstyle{{1\over 2}}((A-\lambda E)^{-1}x, x)= 1.\tag{1} \] In the case of an orthonormal basis \(\{e_k\}^n_{k=1}\), such that \(Ae_k= \lambda_k e_k\), where \(\lambda_k\) are the eigenvalues of \(A\), with \(\lambda_k< \lambda_{k+1}\), then (1) is equivalent to \[ \sum^n_{k=1} {|(x, e_k)|^2\over \lambda_k- \lambda}= 2. \] The goal of this paper is to extend the theory of elliptic coordinates to the infinite-dimensional case in which \(A\) or \(A^{-1}\) is a compact operator in a separable Hilbert space.