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Nevanlinna type families of linear relations and the dilation theorem

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Abstract

Let H1 be a subspace in a Hilbert space H0 and let $\widetilde C(H_0,H_1)$ be the set of all closed linear relations from $H_0$ to $H_1$. We introduce a Nevanlinna type class $\widetilde R_+ (H_0,H_1)$ of holomorphic functions with values in $\widetilde C(H_0,H_1)$ and investigate its properties. In particular we prove the existence of a dilation for every function $\tau_+(\cdot)\in \widetilde R_+ (H_0,H_1)$. In what follows these results will be used for the derivation of the Krein type formula for generalized resolvents of a symmetric operator with arbitrary (not necessarily equal) deficiency indices.


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Article Information

TitleNevanlinna type families of linear relations and the dilation theorem
SourceMethods Funct. Anal. Topology, Vol. 12 (2006), no. 1, 38-56
MathSciNet MR2210904
CopyrightThe Author(s) 2006 (CC BY-SA)

Authors Information

Vadim Mogilevskii
Department of Calculus, Lugans'k National Pedagogical University, 2 Oboronna, Lugans'k, 91011, Ukraine


Citation Example

Vadim Mogilevskii, Nevanlinna type families of linear relations and the dilation theorem, Methods Funct. Anal. Topology 12 (2006), no. 1, 38-56.


BibTex

@article {MFAT344,
    AUTHOR = {Mogilevskii, Vadim},
     TITLE = {Nevanlinna type families of linear relations and the dilation theorem},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {12},
      YEAR = {2006},
    NUMBER = {1},
     PAGES = {38-56},
      ISSN = {1029-3531},
       URL = {http://mfat.imath.kiev.ua/article/?id=344},
}


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