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Hardy spaces for non-compactly causal symmetric spaces and the most continuous spectrum

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Abstract.

Let G/H be a semisimple symmetric space. Then the space L 2(G/H) can be decomposed into a finite sum of series of representations induced from parabolic subgroups of G. The most continuous part of the spectrum of L 2(G/H) is the part induced from the smallest possible parabolic subgroup. In this paper we introduce Hardy spaces canonically related to this part of the spectrum for a class of non-compactly causal symmetric spaces. The Hardy space is a reproducing Hilbert space of holomorphic functions on a bounded symmetric domain of tube type, containing G/H as a boundary component. A boundary value map is constructed and we show that it induces a G-isomorphism onto a multiplicity free subspace of full spectrum in the most continuous part L mc 2(G/H) of L 2(G/H). We also relate our Hardy space to the classical Hardy space on the bounded symmetric domain.

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Authors and Affiliations

  1. Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA

    Simon Gindikin

  2. The Ohio State, University, Department of Mathematics, 231 West 18th Avenue, Columbus, OH 43210–1174, USA

    Bernhard Krötz

  3. Louisiana State University, Department of Mathematics, Baton Rouge, LA 70803, USA

    Gestur Ólafsson

Authors
  1. Simon Gindikin
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  2. Bernhard Krötz
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  3. Gestur Ólafsson
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Correspondence to Simon Gindikin.

Additional information

Supported in part by NSF-grant DMS-0070816 and the MSRI

Supported in part by NSF-grant DMS-0097314 and the MSRI

Supported in part by NSF-grant DMS-0070607 and the MSRI

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Gindikin, S., Krötz, B. & Ólafsson, G. Hardy spaces for non-compactly causal symmetric spaces and the most continuous spectrum. Math. Ann. 327, 25–66 (2003). https://doi.org/10.1007/s00208-003-0409-x

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  • DOI: https://doi.org/10.1007/s00208-003-0409-x

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