Multipliers of Hardy spaces, quadratic integrals and Foiaş-Williams-Peller operators

We obtain a sufficient condition on a B(H)-valued function φ for the operator $⨍ ↦ Γ_φ ⨍'(S)$ to be completely bounded on $H^∞ B(H)$; the Foiaş-Williams-Peller operator | S^t Γ_φ | R_φ = | | | 0 S | is then similar to a contraction. We show that if ⨍ : D → B(H) is a bounded analytic function for which $(1-r) ||⨍'(re^{iθ})||^2_{B(H)} rdrdθ$ and $(1-r) ||⨍"(re^{iθ})||_{B(H)} rdrdθ$ are Carleson measures, then ⨍ multiplies $(H^1c^1)'$ to itself. Such ⨍ form an algebra A, and when φ'∈ BMO(B(H)), the map $⨍ ↦ Γ_φ ⨍'(S)$ is bounded $A → B(H^2(H), L^2(H) ⊖ H^2(H))$. Thus we construct a functional calculus for operators of Foiaş-Williams-Peller type.