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When is the Optimal Domain of a Positive Linear Operator a Weighted L 1-space?

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Vector Measures, Integration and Related Topics

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 201))

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Abstract

Let E be a reflexive Banach function space. Let T be a positive order continuous operator with values in E. Then the optimal domain [T, E] is (isomorphic to) a weighted L 1-space if and only if the operator T is an integral operator with kernel T(x, y), the adjoint operator T′ is a Carleman integral operator and there exists 0≤g∈E′ such that ϕ(y)=∥T ′y (·)∥E≤T′g(y) a.e. on Y. In this case [T, E]=L 1(Y, ϕdυ).

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of South Carolina, Columbia, SC, 29212, USA

    Anton R. Schep

Authors
  1. Anton R. Schep
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Editor information

Editors and Affiliations

  1. Facultad de Matemáticas, Universidad de Sevilla, 41080, Sevilla, Spain

    Guillermo P. Curbera

  2. Fachbereich 6: Mathematik, Universität Siegen, 57068, Siegen, Germany

    Gerd Mockenhaupt

  3. Mathematisch-Geographische Fakultät, Katholische Universität Eichstätt-Ingolstadt, 85072, Eichstätt, Germany

    Werner J. Ricker

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Schep, A.R. (2009). When is the Optimal Domain of a Positive Linear Operator a Weighted L 1-space?. In: Curbera, G.P., Mockenhaupt, G., Ricker, W.J. (eds) Vector Measures, Integration and Related Topics. Operator Theory: Advances and Applications, vol 201. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0211-2_33

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