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Range spaces of co-analytic Toeplitz operators

  • Emmanuel Fricain,
    Laboratoire Paul Painlevé, Université Lille 1 , 59 655 Villeneuve d'Ascq Cédex, France
  • Andreas Hartmann,
    Institut de Mathématiques de Bordeaux, Université Bordeaux/Bordeaux INP/CNRS, 351 cours de la Libération 33405 Talence, France
  • William T. Ross,
    Department of Mathematics and Computer Science, University of Richmond, Richmond, VA 23173, USA
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Abstract

In this paper we discuss the range of a co-analytic Toeplitz operator. These range spaces are closely related to de Branges-Rovnyak spaces (in some cases they are equal as sets). In order to understand its structure, we explore when the range space decomposes into the range of an associated analytic Toeplitz operator and an identifiable orthogonal complement. For certain cases, we compute this orthogonal complement in terms of the kernel of a certain Toeplitz operator on the Hardy space where we focus on when this kernel is a model space (backward shift invariant subspace). In the spirit of Ahern-Clark, we also discuss the non-tangential boundary behavior in these range spaces. These results give us further insight into the description of the range of a co-analytic Toeplitz operator as well as its orthogonal decomposition. Our Ahern-Clark type results, which are stated in a general abstract setting, will also have applications to related sub-Hardy Hilbert spaces of analytic functions such as the de Branges-Rovnyak spaces and the harmonically weighted Dirichlet spaces.
Keywords: Toeplitz operator, Hardy space, range space, de Branges-Rovnyak space, boundary behavior, kernel function, non-extreme point, corona pair Toeplitz operator, Hardy space, range space, de Branges-Rovnyak space, boundary behavior, kernel function, non-extreme point, corona pair
MSC Classifications: 30J05, 30H10, 46E22 show english descriptions Inner functions
Hardy spaces
Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32]
30J05 - Inner functions
30H10 - Hardy spaces
46E22 - Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32]
 

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