L1-Factorization forC00-Contractions with Isometric Functional Calculus

LetTbe an absolutely continuous contraction acting on a Hilbert space H. Forx, y∈H, definex·Ty∈L1(T) by its Fourier coefficients:x·Ty∧(n)=(T*nx, y) ifn<0. The main technical result of the paper is that the vanishing condition limn→∞(‖xn·Tw‖L1/H10+‖w·Txn‖L1/H10)=0,w∈H implies that limn→∞‖xn·Tw‖L1=0,w∈H. Using known factorization techniques, we exhibit a Borel setσTsuch that for anyf∈L1(σT), there existx, y∈H such thatf=(x·Ty)|σT. In the case whereT∈A∩C00, this leads to a simple proof of the fact that for everyf∈L1(T) there existsx, y∈H such thatf=x·Ty. In this case we also show, using dilation theory in the unit disk, that every strictly positive lower semicontinuous functionϕ∈L1(T) can be written in the formϕ=x·Tx. Examples show that this is the best possible result for the class A∩C00.