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.
Full Text
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},
}
