Open Access

# Factorizations of nonnegative symmetric operators

### Abstract

We prove that each closed denselydefined and nonnegative symmetric operator $\dot A$ having disjointnonnegative self-adjoint extensions admits infinitely manyfactorizations of the form $\dot A=\mathcal L\mathcal L_0$, where $\mathcal L_0$ is aclosed nonnegative symmetric operator and $\mathcal L$ its nonnegativeself-adjoint extension. The same factorizations are also establishedfor a non-densely defined nonnegative closed symmetric operator withinfinite deficiency indices while for operator with finitedeficiency indices we prove impossibility of such a kindfactorization. A construction of pairs $\mathcal L_0\subset\mathcal L$ ($\mathcal L_0$ isclosed and densely defined, $\mathcal L=\mathcal L^*\ge 0$) having the property${\rm dom\,}(\mathcal L\mathcal L_0)=\{0\}$ (and, in particular, ${\rm dom\,}(\mathcal L^2_0)=\{0\}$) is given.

Key words: Symmetric operator, divergence form, factorization, Friedrichs extension, Krein-von Neumann extension.

### Article Information

 Title Factorizations of nonnegative symmetric operators Source Methods Funct. Anal. Topology, Vol. 19 (2013), no. 3, 211-226 MathSciNet MR3136728 zbMATH 1289.47044 Milestones Received 01/04/2013 Copyright The Author(s) 2013 (CC BY-SA)

### Authors Information

Yury Arlinskii
Department of Mathematical Analysis, East Ukrainian National University, 20-A Kvartal Molodizhny, Lugans'k, 91034, Ukraine

Yury Kovalev
Department of Mathematical Analysis, East Ukrainian National University, 20-A Kvartal Molodizhny, Lugans'k, 91034, Ukraine

### Citation Example

Yury Arlinskiĭ and Yury Kovalev, Factorizations of nonnegative symmetric operators, Methods Funct. Anal. Topology 19 (2013), no. 3, 211-226.

### BibTex

@article {MFAT688,
AUTHOR = {Arlinskiĭ, Yury and Kovalev, Yury},
TITLE = {Factorizations of nonnegative symmetric operators},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {19},
YEAR = {2013},
NUMBER = {3},
PAGES = {211-226},
ISSN = {1029-3531},
MRNUMBER = {MR3136728},
ZBLNUMBER = {1289.47044},
URL = {http://mfat.imath.kiev.ua/article/?id=688},
}