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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Boundary relations and their Weyl families


Authors: Vladimir Derkach, Seppo Hassi, Mark Malamud and Henk de Snoo
Journal: Trans. Amer. Math. Soc. 358 (2006), 5351-5400
MSC (2000): Primary 47A06, 47A20, 47A56, 47B25; Secondary 47A48, 47B50
Published electronically: July 20, 2006
MathSciNet review: 2238919
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Abstract: The concepts of boundary relations and the corresponding Weyl families are introduced. Let $ S$ be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space $ \mathfrak{H}$, let $ \mathcal{H}$ be an auxiliary Hilbert space, let

$\displaystyle J_\mathfrak{H}=\begin{pmatrix}0&-iI_\mathfrak{H}\\ iI_\mathfrak{H} & 0\end{pmatrix}, $

and let $ J_\mathcal{H}$ be defined analogously. A unitary relation $ \Gamma$ from the Krein space $ (\mathfrak{H}^2,J_\mathfrak{H})$ to the Krein space $ (\mathcal{H}^2,J_\mathcal{H})$ is called a boundary relation for the adjoint $ S^*$ if $ \ker \Gamma=S$. The corresponding Weyl family $ M(\lambda)$ is defined as the family of images of the defect subspaces $ \widehat{\mathfrak{N}}_\lambda$, $ \lambda\in \mathbb{C}\setminus\mathbb{R}$, under $ \Gamma$. Here $ \Gamma$ need not be surjective and is even allowed to be multi-valued. While this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space $ \mathcal{H}$ and the class of unitary relations $ \Gamma:(\mathfrak{H}^2,J_\mathfrak{H})\to(\mathcal{H}^2,J_\mathcal{H})$, it also generalizes the notion of so-called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems. Moreover, these new notions yield, for instance, the following realization theorem: every $ \mathcal{H}$-valued maximal dissipative (for $ \lambda\in\mathbb{C}_+$) holomorphic family of linear relations is the Weyl family of a boundary relation, which is unique up to unitary equivalence if certain minimality conditions are satisfied. Further connections between analytic and spectral theoretical properties of Weyl families and geometric properties of boundary relations are investigated, and some applications are given.


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Additional Information

Vladimir Derkach
Affiliation: Department of Mathematics, Donetsk National University, Universitetskaya str. 24, 83055 Donetsk, Ukraine
Email: derkach@univ.donetsk.ua

Seppo Hassi
Affiliation: Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700, 65101 Vaasa, Finland
Email: sha@uwasa.fi

Mark Malamud
Affiliation: Department of Mathematics, Donetsk National University, Universitetskaya str. 24, 83055 Donetsk, Ukraine
Email: mmm@univ.donetsk.ua

Henk de Snoo
Affiliation: Department of Mathematics and Computing Science, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands
Email: desnoo@math.rug.nl

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04033-5
PII: S 0002-9947(06)04033-5
Keywords: Symmetric operator, selfadjoint extension, Kre\u{\i}n space, unitary relation, boundary triplet, boundary relation, Weyl function, Weyl family, Nevanlinna family
Received by editor(s): September 15, 2004
Published electronically: July 20, 2006
Additional Notes: The present research was supported by the Research Institute for Technology at the University of Vaasa and by the Academy of Finland (projects 203226, 208055)
Article copyright: © Copyright 2006 American Mathematical Society