For given two unitary and self-adjoint operators on a Hilbert space, a
spectral mapping theorem was proved in \cite{HiSeSu}. In this paper, as an
application of the spectral mapping theorem, we investigate the spectrum of a
one-dimensional split-step quantum walk. We give a criterion for when there is
no eigenvalues around $\pm 1$ in terms of a discriminant operator. We also
provide a criterion for when eigenvalues $\pm 1$ exist in terms of birth
eigenspaces. Moreover, we prove that eigenvectors from the birth eigenspaces
decay exponentially at spatial infinity and that the birth eigenspaces are
robust against perturbations.