In 1966 de Branges and Rovnyak introduced a concept of complementation associated to a contraction between Hilbert spaces that generalizes the classical concept of orthogonal complement. When applied to Toeplitz operators on the Hardy space of the disc, H2, this notion turned out to be the starting point of a beautiful subject, with many applications to function theory. The work has been in constant progress for the last few years. We study here the multipliers of some de Branges-Rovnyak spaces...

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.

For subspaces, $X$ and $Y$, of the space, $D$, of all derivatives $M(X,Y)$ denotes the set of all $g\in D$ such that $fg\in Y$ for all $f\in X$. Subspaces of $D$ are defined depending on a parameter $p\in [0,\infty ]$. In Section 6, $M(X,D)$ is determined for each of these subspaces and in Section 7, $M(X,Y)$ is found for $X$ and $Y$ any of these subspaces. In Section 3, $M(X,D)$ is determined for other spaces of functions on $[0,1]$ related to continuity and higher order differentiation.

Let A be a commutative semisimple Banach algebra, Δ(A) its Gelfand spectrum, T a multiplier on A and T̂ its Gelfand transform. We study the following problems. (a) When is δ(T) = inf{|T̂(f)|: f ∈ Δ(A), T̂(f) ≠ 0} > 0? (b) When is the range T(A) of T closed in A and does it have a bounded approximate identity? (c) How to characterize the idempotent multipliers in terms of subsets of Δ(A)?

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...

We characterize Banach lattices under which each b-weakly compact (resp. b-AM-compact, strong type (B)) operator is L-weakly compact (resp. M-weakly compact).