MULTIPLIERS OF HARDY-SPACES, QUADRATIC INTEGRALS AND FOIAS-WILLIAMS-PELLER OPERATORS

We obtain a sufficient condition on a B(H)-valued function phi for the operator f bar right arrow Gamma(phi)f'(S) to be completely bounded o n (HB)-B-infinity(H); the Foias-Williams-Peller operator [GRAPHICS] is then similar to a contraction. We show that if f : D --> B(H) is a bo unded analytic function for which (1 - r)parallel to f'(re(i theta))pa rallel to(B(H))(2)rdrd theta and (1 - r)parallel to f ''(re(i theta))p arallel to(B(H))rdrd theta are Carleson measures, then f multiplies (H (1)c(1))' to itself. Such f form an algebra A, and when phi' is an ele ment of BMO(B(H)), the map f bar right arrow Gamma(phi)f'(S) is bounde d A --> B(H-2(H),L-2(H) - H-2(H)). Thus we construct a functional calc ulus for operators of Foias-Williams-Peller type.