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# 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.

### 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

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,
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},
}