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 ${\displaystyle ⨍\mapsto {\Gamma }_{\phi }⨍\prime \left(S\right)}$ to be completely bounded on ${\displaystyle H^{\infty }B\left(H\right)}$; 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 ${\displaystyle (1-r){\left||⨍\prime \left(r{e}^{i\theta }\right)|\right|}^2\_\left\{B\left(H\right)\right\}rdrd\theta }$ and ${\displaystyle (1-r){\left||⨍\left(r{e}^{i\theta }\right)|\right|}_{B\left(H\right)}rdrd\theta }$ are Carleson measures, then ⨍ multiplies ${\displaystyle \left(H^1{c}^1\right)\prime }$ to itself. Such ⨍ form an algebra A, and when φ'∈ BMO(B(H)), the map ${\displaystyle ⨍\mapsto {\Gamma }_{\phi }⨍\prime \left(S\right)}$ is bounded ${\displaystyle A\to B(H^2\left(H\right),L^2\left(H\right)\ominus H^2\left(H\right))}$. Thus we construct a functional calculus for operators of Foiaş-Williams-Peller type.