Some remarks on the notions of boundary systems and boundary triple(t)s

Marcus Waurick, Sven-Ake Wegner

Research output: Contribution to journalArticle

Abstract

In this note we show that if a boundary system in the sense of (Schubert et al. 2015) gives rise to any skew‐self‐adjoint extension, then it induces a boundary triplet and the classification of all extensions given by (Schubert et al. 2015) coincides with the skew‐symmetric version of the classical characterization due to (Gorbachuk et al. 1991). On the other hand we show that for every skew‐symmetric operator there is a natural boundary system which leads to an explicit description of at least one maximal dissipative extension. This is in particular also valid in the case that no boundary triplet exists for this operator.
LanguageEnglish
Number of pages9
JournalMathematische Nachrichten
Early online date4 Jul 2018
DOIs
Publication statusE-pub ahead of print - 4 Jul 2018

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Skew
Self-adjoint Extension
Symmetric Operator
Valid
Operator

Keywords

  • boundary system
  • boundary triple
  • boundary triplet
  • deficiency index
  • extension problem

Cite this

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Some remarks on the notions of boundary systems and boundary triple(t)s. / Waurick, Marcus; Wegner, Sven-Ake.

In: Mathematische Nachrichten, 04.07.2018.

Research output: Contribution to journalArticle

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AU - Wegner, Sven-Ake

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N2 - In this note we show that if a boundary system in the sense of (Schubert et al. 2015) gives rise to any skew‐self‐adjoint extension, then it induces a boundary triplet and the classification of all extensions given by (Schubert et al. 2015) coincides with the skew‐symmetric version of the classical characterization due to (Gorbachuk et al. 1991). On the other hand we show that for every skew‐symmetric operator there is a natural boundary system which leads to an explicit description of at least one maximal dissipative extension. This is in particular also valid in the case that no boundary triplet exists for this operator.

AB - In this note we show that if a boundary system in the sense of (Schubert et al. 2015) gives rise to any skew‐self‐adjoint extension, then it induces a boundary triplet and the classification of all extensions given by (Schubert et al. 2015) coincides with the skew‐symmetric version of the classical characterization due to (Gorbachuk et al. 1991). On the other hand we show that for every skew‐symmetric operator there is a natural boundary system which leads to an explicit description of at least one maximal dissipative extension. This is in particular also valid in the case that no boundary triplet exists for this operator.

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KW - boundary triple

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KW - deficiency index

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