A Factorization problem and the problem of predicting non-stationary vector-valued stochastic processes

In this paper a prediction theory is developed under the general idea that the infinite dimensional covariance matrix is a self-adjoint element in a symmetric Banach algebra. The usual Wiener's spectral factorization method for solving stationary Wiener-Hopf equations has been extended to this algebra. Finally, a theorem for factoring a positive definite covariance matrix into upper and lower triangular factors with similar inverses has been proved. © 1969 Springer-Verlag.