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Preprint 33-2006

Boundary Relations and Generalized Resolvents of Symmetric Operators in Krein Spaces

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Author(s) : Jussi Behrndt , Hans-Christian Kreusler

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 33-2006

MSC 2000

47B50 Operators on spaces with an indefinite metric
47A20 Dilations, extensions, compressions
47B25 Symmetric and selfadjoint operators
46C20 Spaces with indefinite inner product
47A06 Linear relations

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 generalize the notion of boundary relations and their Weyl families to the Krein space case and to proof some variants of the Krein-Naimark formula in an indefinite setting.

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

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