Outer factorization of operator valued weight functions on the torus

An exact criterion is derived for an operator valued weight function $W(e^{is},e^{it})$ on the torus to have a factorization $W(e^{is},e^{it}) = Φ(e^{is},e^{it})*Φ(e^{is},e^{it})$, where the operator valued Fourier coefficients of Φ vanish outside of the