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# Boundary triplets and Krein type resolvent formula for symmetric operators with unequal defect numbers

### Abstract

Let $H$ be a Hilbert space and let $A$ be a symmetric operator in $H$ with arbitrary (not necessarily equal) deficiency indices $n_\pm (A)$. We introduce a new concept of a $D$-boundary triplet for $A^*$, which may be considered as a natural generalization of the known concept of a boundary triplet (boundary value space) for an operator with equal deficiency indices. With a $D$-triplet for $A^*$ we associate two Weyl functions $M_+(\cdot)$ and $M_-(\cdot)$. It is proved that the functions $M_\pm(\cdot)$ posses a number of properties similar to those of the known Weyl functions ($Q$-functions) for the case $n_+(A)=n_-(A)$. We show that every $D$-triplet for $A^*$ gives rise to Krein type formulas for generalized resolvents of the operator $A$ with arbitrary deficiency indices. The resolvent formulas describe the set of all generalized resolvents by means of two pairs of operator functions which belongs to the Nevanlinna type class $\bar R(H_0,H_1)$. This class has been earlier introduced by the author.

### Article Information

 Title Boundary triplets and Krein type resolvent formula for symmetric operators with unequal defect numbers Source Methods Funct. Anal. Topology, Vol. 12 (2006), no. 3, 258-280 MathSciNet MR2261580 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, Boundary triplets and Krein type resolvent formula for symmetric operators with unequal defect numbers, Methods Funct. Anal. Topology 12 (2006), no. 3, 258-280.

### BibTex

@article {MFAT363,
TITLE = {Boundary triplets and Krein type resolvent formula for symmetric operators with unequal defect numbers},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {12},
YEAR = {2006},
NUMBER = {3},
PAGES = {258-280},
ISSN = {1029-3531},
MRNUMBER = {MR2261580},
URL = {http://mfat.imath.kiev.ua/article/?id=363},
}