de Branges-Rovnyak spaces: basics and theory

For $S$ a contractive analytic operator-valued function on the unit disk ${\mathbb D}$, de Branges and Rovnyak associate a Hilbert space of analytic functions ${\mathcal H}(S)$ and related extension space ${\mathcal D(S)}$ consisting of pairs of analytic functions on the unit disk ${\mathbb D}$. This survey describes three equivalent formulations (the original geometric de Branges-Rovnyak definition, the Toeplitz operator characterization, and the characterization as a reproducing kernel Hilbert space) of the de Branges-Rovnyak space ${\mathcal H}(S)$, as well as its role as the underlying Hilbert space for the modeling of completely non-isometric Hilbert-space contraction operators. Also examined is the extension of these ideas to handle the modeling of the more general class of completely nonunitary contraction operators, where the more general two-component de Branges-Rovnyak model space ${\mathcal D}(S)$ and associated overlapping spaces play key roles. Connections with other function theory problems and applications are also discussed. More recent applications to a variety of subsequent applications are given in a companion survey article

Operator theory and function theory in Drury-Arveson space and its quotients

The Drury-Arveson space $H^2_d$, also known as symmetric Fock space or the $d$-shift space, is a Hilbert function space that has a natural $d$-tuple of operators acting on it, which gives it the structure of a Hilbert module. This survey aims to introduce the Drury-Arveson space, to give a panoramic view of the main operator theoretic and function theoretic aspects of this space, and to describe the universal role that it plays in multivariable operator theory and in Pick interpolation theory.Comment: Final version (to appear in Handbook of Operator Theory); 42 page

Bellman Functions And Two Weight Inequalities For Haar Multipliers

. We are going to give necessary and sufficient conditions for two weight norm inequalities for Haar multipliers operators and for square functions. We also give sufficient conditions for two weight norm inequalities for the Hilbert transform. 0. Introduction Weighted norm inequalities for singular integral operators appear naturally in many areas of analysis, probability, operator theory ect. The one-weight case is now pretty well understood, and the answers are given by the famous Helson--Szego theorem and the Hunt--Muckenhoupt--Wheden Theorem. The fist one state that the Hilbert Transform H is bounded in the weighted space L 2 (w) if and only if w can be represented as w = expfu + Hvg, where u; v 2 L 1 , kuk1 ! ß=2. The Hunt--Muckenhoupt--Wheden Theorem states that the Hilbert transform H is bounded in L p (w) if and only if the weight w satisfies the so-called Muckenhoupt A p condition sup I i 1 jIj Z I w j \Delta i 1 jIj Z I w \Gamma1=(p\Gamma1) j p\Gam..

Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators in nonhomogeneous spaces

Abstract. In the paper we consider Calderón-Zygmund operators in nonhomogeneous spaces. We are going to prove the analogs of classical results for homogeneous spaces. Namely, we prove that a Calderón-Zygmund operator is of weak type if it is bounded in L 2. We also prove several versions of Cotlar’s inequality for maximal singular operator. One version of Cotlar’s inequality (a simpler one) is proved in Euclidean setting, another one in a more abstract setting when Besicovich covering lemma is not available. We obtain also the weak type of maximal singular operator from these inequalities. Let µ be a measure on C satisfying the Ahlfors condition µ(B(x, r)) ≤ r for every x ∈ C, r> 0 (as usual, B(x, r): = {y ∈ C: |x − y | < r}). Let K(x, y) be a Calderon-Zygmund kernel, i.e. 1) |K(x, y) | ≤ an

Cauchy Integral And Calderón-Zygmund Operators On Nonhomogeneous Spaces

this paper is to consider the boundedness of singular integral operators with Calder'on-Zygmund kernels in