Nevanlinna type families of linear relations and the dilation theorem

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.

Full Text

Article Information

Title	Nevanlinna type families of linear relations and the dilation theorem
Source	Methods Funct. Anal. Topology, Vol. 12 (2006), no. 1, 38-56
MathSciNet	MR2210904
Copyright	The 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},
  MRNUMBER = {MR2210904},
       URL = {http://mfat.imath.kiev.ua/article/?id=344},
}