L1-factorization for C∞-contractions with isometric functional calculus

Let T be an absolutely continous contraction acting on a Hilbert space H. For x, y is an element of H, define x.(T) y is an element of L-1(T) by its Fourier coefficients: x.(T)y(boolean AND)(n) = (T(*n)x, y) if n greater than or equal to 0 and x.(T)y(boolean AND)(n) = (T(-n)x, y) if n < 0. The main technical result of the paper is that the vanishing condition lim(n-->infinity)(\\x(n).(T)w\\(L1/H01)+\\w.(T)x(n)\\(L1/H01)) = 0, w is an element of H implies that lim (n-->proportional to) \\x(n).(T)w\\(L1) = 0, w is an element of H. Using known fractorization techniques, we exhibit a Borel set sigma(T) such that for any f is an element of L-1(sigma(T)), there exist x, y is an element of H such that F = (x.(T) y)(sigma T). In the case where T is an element of A boolean AND C-00, this leads to a simple proof of the fact that for every f is an element of L-1(T) there exist x, y is an element of H such that f = x.(T)y. In this case we also show, using dilation theory in the unit disk, that every strictly positive lower semicontinuous function phi is an element of L-1(T) can be written in the form phi = s.(T)x. Examples show that this is the best possible result for the class A boolean AND C-00. (C) 1998 Academic Press.