Dual J-spectral factorization

Abstract

In this paper, we identify a class of J-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 J-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 J-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 J-spectral factorization is that one can use efficient convex optimization methods instead of invariant subspace methods involving hamiltonian matrices.