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Range Spaces of Co-Analytic Toeplitz Operators

Part of: Linear function spaces and their duals Spaces and algebras of analytic functions Function theory on the disc

Published online by Cambridge University Press:  20 November 2018

Emmanuel Fricain ,
Andreas Hartmann  and
William T. Ross
Emmanuel Fricain
Affiliation:
Laboratoire Paul Painlevé, Université Lille 1, 59 655 Villeneuve d’Ascq Cédex, France, e-mail: emmanuel.fricain@math.univ-lille1.fr
Andreas Hartmann
Affiliation:
Institut de Mathématiques de Bordeaux, Université Bordeaux/Bordeaux INP/CNRS, 351 cours de la Libération 33405 Talence, France, e-mail: Andreas.Hartmann@math.u-bordeaux.fr
William T. Ross
Affiliation:
Department of Mathematics and Computer Science, University of Richmond, Richmond, VA 23173, USA, e-mail: wross@richmond.edu
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Abstract

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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 operatorHardy spacerange spacede Branges–Rovnyak spaceboundary behaviorkernel functionnon-extreme pointcorona pair

MSC classification

Secondary: 30J05: Inner functions 30H10: Hardy spaces 46E22: Hilbert spaces with reproducing kernels (=
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 6 , 01 December 2018 , pp. 1261 - 1283
Copyright
Copyright © Canadian Mathematical Society 2018

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