Spectral theory of the G-symmetric tridiagonal matrices related to Stahl's counterexample

We recast Stahl's counterexample from the point of view of the spectral theory of the underlying non-symmetric Jacobi matrices. In particular, it is shown that these matrices are self-adjoint and non-negative in a Krein space and have empty resolvent sets. In fact, the technique of Darboux transformations (aka commutation methods) on spectra which is used in the present paper allows us to treat the class of all G-non-negative tridiagonal matrices. We also establish a correspondence between this class of matrices and the class of signed measures with one sign change. Finally, it is proved that the absence of the spurious pole at infinity for Pad\'e approximants is equivalent to the definitizability of the corresponding tridiagonal matrix.