L-1 factorizations for some perturbations of the unilateral shift

We say that T is an element of L(H) (where H is a Hilbert space) factorizes f is an element of L-1 (T) if there exist x,y is an element of H such that f(n) = (T*(n)x,y) if n greater than or equal to 0 and (f) over cap(-n) = ( T(n)x,: y) if n greater than or equal to 1. By virtue of one of Bourgain's results, the unilateral shift S is an element of L(H-2) of multiplicity one factorizes f is an element of L-1 (T) if and only if log \f\ is an element of L-1 (T). We study the absolutely continuous contractions A such that th e operator S + A factorizes all functions in L-1 (T). (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.