Higher-order de Branges–Rovnyak and sub-Bergman spaces

Abstract

The sub-Bergman spaces are de Branges–Rovnyak subspaces of Bergman space A2 defined by the contraction Tb or Tb⁎ for an analytic symbol b. The fact that both Tb and Tb⁎ are 2-hypercontractions on A2 leads to the introduction of a new type of sub-Bergman spaces, which will be called higher-order sub-Bergman spaces. We show these new spaces are different and yet connected in a nice way with the sub-Bergman spaces. The close relationship of these new spaces to the original de Branges–Rovnyak subspaces of the Hardy spaces is also explored. A similar study is conducted on weighted Bergman spaces Aα2 where both Tb and Tb⁎ are [α+2]-hypercontractions.