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Outer factorization of operator valued weight functions on the torus

Ray Cheng — 1994

Studia Mathematica

An exact criterion is derived for an operator valued weight function W ( e i s , e i t ) on the torus to have a factorization W ( e i s , e i t ) = Φ ( e i s , e i t ) * Φ ( e i s , e i t ) , where the operator valued Fourier coefficients of Φ vanish outside of the Helson-Lowdenslager halfplane Λ = ( m , n ) 2 : m 1 ( 0 , n ) : n 0 , and Φ is “outer” in a related sense. The criterion is expressed in terms of a regularity condition on the weighted space L 2 ( W ) 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...

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