Polar Decomposition of a Semiseparable Matrix in Pontryagin Spaces

Given a nonsingular signature $J_{2}$ and a causal left invertible operator $G$ by a minimal, u.e.s realization, we state necessary and sufficient conditions in state-space terms under which the indefinite spectral factorization problem $GJ_{2}G^{\ast}=G_{o}J_{x}G_{o}^{\ast}$ has solution for a left-outer $G_{o}$ and another invertible signature $J_{x}$. We also state a single-pass numerically stable algorithm that finds $J_{x}$ and a minimal, u.e.s realization for $G_{o}$.