In this paper, we identify a class of T-spectral factorization problems that can be cast in the form of ordinary spectral factorization problems. As a basic tool, we introduce the concept of Popov and dual Popov functions and study the relations between their T-spectral factorizations. Inertia relationships between the two Popov functions show that the dual Popov function can be (positive or negative) semidefinite even when the (original) Popov function is not. In such cases, we can obtain the T-spectral factorization of the original Popov function through ordinary spectral factorization of the dual Popov function. The most important advantage of ordinary spectral factorization over T-spectral factorization is that one can use efficient convex optimization methods instead of invariant subspace methods involving hamiltonian matrices.