Calculating spectral factors of low rank matrix valued functions on the unit circle

We consider the problem of finding an analytic spectral factor g of a given rank k with k<n, positive semidefinite n×n matrix valued function f on the unit circle of the complex plane. The values of g are k×n matrices. We present an operator equation that must be satisfied by a function g to be a spectral factor. The condition is also sufficient. We also prove that Newton´s method can be applied successfully to calculate spectral factors g in the Wiener algebra, thus the method works for all rational and many instances of non-rational data function f. A numerical example is given