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Boundary Relations and Generalized Resolvents of Symmetric Operators in Krein Spaces

Abstract.

The classical Krein-Naimark formula establishes a one-to-one correspondence between the generalized resolvents of a closed symmetric operator in a Hilbert space and the class of Nevanlinna families in a parameter space. Recently it was shown by V.A. Derkach, S. Hassi, M.M. Malamud and H.S.V. de Snoo that these parameter families can be interpreted as so-called Weyl families of boundary relations, and a new proof of the Krein-Naimark formula in the Hilbert space setting was given with the help of a coupling method. The main objective of this paper is to adapt the notion of boundary relations and their Weyl families to the Krein space case and to prove some variants of the Krein-Naimark formula in an indefinite setting.

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Correspondence to Jussi Behrndt.

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Behrndt, J., Kreusler, HC. Boundary Relations and Generalized Resolvents of Symmetric Operators in Krein Spaces. Integr. equ. oper. theory 59, 309–327 (2007). https://doi.org/10.1007/s00020-007-1529-6

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Mathematics Subject Classification (2000).

  • Primary 47B50, 47A20, 47B25
  • Secondary 46C20, 47A06

Keywords.

  • Symmetric operator
  • self-adjoint extension
  • Krein-Naimark formula
  • generalized resolvent
  • boundary relation
  • boundary triplet
  • (locally) definitizable operator
  • Krein space