Outer factorization of operator valued weight functions on the torus

An exact criterion is derived for an operator valued weight function ${\displaystyle W(e^{is},e^{it})}$ on the torus to have a factorization ${\displaystyle W(e^{is},e^{it})=\Phi (e^{is},e^{it})\cdot \Phi (e^{is},e^{it})}$, where the operator valued Fourier coefficients of Φ vanish outside of the Helson-Lowdenslager halfplane ${\displaystyle \Lambda =\{(m,n)\in {\mathbb{Z}}^2:m\ge 1\}\cup \{(0,n):n\ge 0\}}$, and Φ is "outer" in a related sense. The criterion is expressed in terms of a regularity condition on the weighted space ${\displaystyle L^2\left(W\right)}$ of vector valued functions on the torus. A logarithmic integrability test is also provided. The factor Φ is explicitly constructed in terms of Toeplitz operators and other structures associated with W. The corresponding version of Szegö's infimum is given.