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# Parametrization of scale-invariant self-adjoint extensions of scale-invariant symmetric operators

### Abstract

On a Hilbert space $\frak H$, we consider a symmetric scale-invariant operator with equal defect numbers. It is assumed that the operator has at least one scale-invariant self-adjoint extension in $\frak H$. We prove that there is a one-to-one correspondence between (generalized) resolvents of scale-invariant extensions and solutions of some functional equation. Two examples of Dirac-type operators are considered.

Key words: Symmetric operator, scale-invariant operator, self-adjoint extension, generalized resolvents.

### Article Information

 Title Parametrization of scale-invariant self-adjoint extensions of scale-invariant symmetric operators Source Methods Funct. Anal. Topology, Vol. 24 (2018), no. 1, 1-15 MathSciNet MR3783813 Milestones Received 11/08/2017; Revised 03/11/2017 Copyright The Author(s) 2018 (CC BY-SA)

### Authors Information

Miron B. Bekker
Department of Mathematics, University of Pittsburgh at Johnstown, Johnstown, PA, USA

Martin J. Bohner
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, USA

Alexander P. Ugolʹnikov
Department of Mathematics, Odessa National Academy of Food Technologies, Odessa, Ukraine

Hristo Voulov
Department of Mathematics and Statistics,University of Missouri-Kansas City, Kansas City, MO, USA

### Citation Example

Miron B. Bekker, Martin J. Bohner, Alexander P. Ugolʹnikov, and Hristo Voulov, Parametrization of scale-invariant self-adjoint extensions of scale-invariant symmetric operators, Methods Funct. Anal. Topology 24 (2018), no. 1, 1-15.

### BibTex

@article {MFAT1020,
AUTHOR = {Miron B. Bekker and Martin J. Bohner and Alexander P. Ugolʹnikov and Hristo Voulov},
TITLE = {Parametrization of scale-invariant self-adjoint extensions of scale-invariant symmetric operators},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {24},
YEAR = {2018},
NUMBER = {1},
PAGES = {1-15},
ISSN = {1029-3531},
MRNUMBER = {MR3783813},
URL = {http://mfat.imath.kiev.ua/article/?id=1020},
}

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