We obtain a sufficient condition on a B(H)-valued function φ for the operator $⨍\mapsto {\Gamma }_{\phi }{⨍}^{\text{'}}\left(S\right)$ to be completely bounded on $H^{\infty }B\left(H\right)$; the Foiaş-Williams-Peller operator | St Γφ | Rφ = | | | 0 S | is then similar to a contraction. We show that if ⨍ : D → B(H) is a bounded analytic function for which $(1-r)\left|\right|{⨍}^{\text{'}}\left(r{e}^{i\theta }\right){\left|\right|}_{B\left(H\right)}^{2}rdrd\theta$ and $(1-r)\left|\right|⨍"\left(r{e}^{i\theta }\right){\left|\right|}_{B\left(H\right)}rdrd\theta$ are Carleson measures, then ⨍ multiplies ${\left(H^1{c}^1\right)}^{\text{'}}$ to itself. Such ⨍ form an algebra A, and when φ’∈ BMO(B(H)), the map $⨍\mapsto {\Gamma }_{\phi }{⨍}^{\text{'}}\left(S\right)$ is bounded $A\to B(H^2\left(H\right),L^2\left(H\right)\ominus H^2\left(H\right))$. Thus we construct a functional calculus for operators of Foiaş-Williams-Peller type.

Let X be a metric space with doubling measure and L a one-to-one operator of type ω having a bounded H∞ -functional calculus in L2(X) satisfying the reinforced (pL; qL) off-diagonal estimates on balls, where pL ∊ [1; 2) and qL ∊ (2;∞]. Let φ : X × [0;∞) → [0;∞) be a function such that φ (x;·) is an Orlicz function, φ(·;t) ∊ A∞(X) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index l(φ) ∊ (0;1] and φ(·; t) satisfies the uniformly reverse Hölder inequality of order...