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A canonical decomposition for linear operators and linear relations

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Abstract

An arbitrary linear relation (multivalued operator) acting from one Hilbert space to another Hilbert space is shown to be the sum of a closable operator and a singular relation whose closure is the Cartesian product of closed subspaces. This decomposition can be seen as an analog of the Lebesgue decomposition of a measure into a regular part and a singular part. The two parts of a relation are characterized metrically and in terms of Stone’s characteristic projection onto the closure of the linear relation.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700, 65101, Vaasa, Finland

    S. Hassi

  2. Department of Applied Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117, Budapest, Hungary

    Z. Sebestyén

  3. Department of Mathematics and Computing Science, University of Groningen, P.O. Box 800, 9700 AV, Groningen, Nederland

    H. S. V. De Snoo

  4. Institute of Mathematics, Uniwersytet Jagielloński, ul. Reymonta 4, 30059, Kraków, Poland

    F. H. Szafraniec

Authors
  1. S. Hassi
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  2. Z. Sebestyén
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  3. H. S. V. De Snoo
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  4. F. H. Szafraniec
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Additional information

The research was supported by bilateral agreements between the Eötvös Loránd University in Budapest, the University of Groningen, and the Jagiellonian University in Krakow, and by the Research Institute for Technology of the University of Vaasa. The fourth author was also supported by the KBN grant 2 PO3A 037 024.

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Hassi, S., Sebestyén, Z., De Snoo, H.S.V. et al. A canonical decomposition for linear operators and linear relations. Acta Math Hung 115, 281–307 (2007). https://doi.org/10.1007/s10474-007-5247-y

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  • DOI: https://doi.org/10.1007/s10474-007-5247-y

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