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# Selfadjoint extensions of relations whose domain and range are orthogonal

### Abstract

The selfadjoint extensions of a closed linear relation $R$ from a Hilbert space $\mathfrak H_1$ to a Hilbert space $\mathfrak H_2$ are considered in the Hilbert space $\mathfrak H_1\oplus\mathfrak H_2$ that contains the graph of $R$. They will be described by $2 \times 2$ blocks of linear relations and by means of boundary triplets associated with a closed symmetric relation $S$ in $\mathfrak H_1 \oplus \mathfrak H_2$ that is induced by $R$. Such a relation is characterized by the orthogonality property ${\rm dom\,} S \perp {\rm ran\,} S$ and it is nonnegative. All nonnegative selfadjoint extensions $A$, in particular the Friedrichs and Krein-von Neumann extensions, are parametrized via an explicit block formula. In particular, it is shown that $A$ belongs to the class of extremal extensions of $S$ if and only if ${\rm dom\,} A \perp {\rm ran\,} A$. In addition, using asymptotic properties of an associated Weyl function, it is shown that there is a natural correspondence between semibounded selfadjoint extensions of $S$ and semibounded parameters describing them if and only if the operator part of $R$ is bounded.

### Article Information

 Title Selfadjoint extensions of relations whose domain and range are orthogonal Source Methods Funct. Anal. Topology, Vol. 26 (2020), no. 1, 39-62 DOI 10.31392/MFAT-npu26_1.2020.03 MathSciNet MR4113580 Milestones Received 11/10/2019; Revised 24/10/2019 Copyright The Author(s) 2020 (CC BY-SA)

### Authors Information

S. Hassi
Department of Mathematics and Statistics, University of Vaasa, P.O. Box 700, 65101 Vaasa, Finland

J.-Ph. Labrousse
63 Avenue Cap de Croix, 06100 Nice, France

H.S.V. de Snoo
Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, P.O. Box 407, 9700 AK Groningen, Nederland

### Citation Example

S. Hassi, J.-Ph. Labrousse, and H.S.V. de Snoo, Selfadjoint extensions of relations whose domain and range are orthogonal, Methods Funct. Anal. Topology 26 (2020), no. 1, 39-62.

### BibTex

@article {MFAT1287,
AUTHOR = {S. Hassi and J.-Ph. Labrousse and H.S.V. de Snoo},
TITLE = {Selfadjoint extensions of relations whose domain and range are orthogonal},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {26},
YEAR = {2020},
NUMBER = {1},
PAGES = {39-62},
ISSN = {1029-3531},
MRNUMBER = {MR4113580},
DOI = {10.31392/MFAT-npu26_1.2020.03},
URL = {http://mfat.imath.kiev.ua/article/?id=1287},
}

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