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On the abstract Cauchy problem for operators in locally convex spaces

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Abstract

We provide a complete solution of the abstract Cauchy problem for operator valued Laplace distributions or hyperfunctions on complete ultrabornological locally convex spaces (like spaces of smooth functions and distributions). This extends results of Komatsu for operators on Banach spaces. Concrete examples are provided. The crucial tools for our solution are a general notion of a resolvent for operators on locally convex spaces and the theory of Laplace transform for Laplace hyperfunctions valued in a complete locally convex space X developed earlier by the authors.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, A. Mickiewicz University, Umultowska 87, 61-614, Poznań, Poland

    Paweł Domański

  2. Institute of Mathematics, Polish Academy of Sciences (Poznań branch), Umultowska 87, 61-614, Poznań, Poland

    Paweł Domański

  3. Department of Mathematics, University of Oldenburg, 26111, Oldenburg, Germany

    Michael Langenbruch

Authors
  1. Paweł Domański
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  2. Michael Langenbruch
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Correspondence to Michael Langenbruch.

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Domański, P., Langenbruch, M. On the abstract Cauchy problem for operators in locally convex spaces. RACSAM 106, 247–273 (2012). https://doi.org/10.1007/s13398-011-0052-4

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  • DOI: https://doi.org/10.1007/s13398-011-0052-4

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