Boundary relations and their Weyl families

Vladimir Derkach*, Seppo Hassi, Mark Malamud, Hendrik de Snoo

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

100 Citations (Scopus)


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 h, let H be an auxiliary Hilbert space, let


and let JH be defined analogously. A unitary relation G from the Krein space (h(2), J(h)) to the Kre. in space (H-2, J(H)) is called a boundary relation for the adjoint S* if ker Gamma = S. The corresponding Weyl family M(lambda) is de. ned as the family of images of the defect subspaces (n) over cap (lambda), lambda is an element of C \ 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 H and the class of unitary relations Gamma : ( H-2, J(H)) -> (H-2, J(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 H-valued maximal dissipative (for lambda is an element of 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.

Original languageEnglish
Pages (from-to)5351-5400
Number of pages50
JournalTransactions of the american mathematical society
Issue number12
Publication statusPublished - 2006


  • symmetric operator
  • selfadjoint extension
  • Krein space
  • unitary relation
  • boundary triplet
  • boundary relation
  • Weyl function
  • Weyl family
  • Nevanlinna family


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