  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.
