J-spectral factorization and equalizing vectors

For the Wiener class of matrix-valued functions we provide necessary and sufficient conditions for the existence of a J-spectral factorization. One of these conditions is in terms of equalizing vectors. The second one states that the existence of a J-spectral factorization is equivalent to the invertibility of the Toeplitz operator associated to the matrix to be factorized. Our proofs are simple and only use standard results of general factorization theory (Clancey and Gohberg, Factorization of Matrix Functions and Singular Integral Operators, Operator Theory: Advances and Applications, Vol. 3, Birkhäuser, Basel, 1981). Note that we do not use a state space representation of the system. However, we make the connection with the known results for the Pritchard–Salamon class of systems where an equivalent condition with the solvability of an algebraic Riccati equation is given.