# Journal of Operator Theory

Volume 82, Issue 2, Fall 2019 pp. 285-306.

On the absolute value of unbounded operators

**Authors**:
Imene Boucif (1), Souheyb Dehimi (2), Mohammed Hichem
Mortad (3)

**Author institution:**(1) Department of
Mathematics, Universit\'e Oran1, B.P. 1524, El
Menouar, Oran 31000, Algeria

(2) University of Mohamed El Bachir El Ibrahimi, Bordj Bou Arreridj,
Algeria

(3) Department of
Mathematics, University of Oran1, Ahmed Ben Bella, B.P. 1524, El
Menouar, Oran 31000, Algeria

**Summary: **The primary purpose of the present paper is to investigate when
relations of the types $|AB|=|A||B|$, $|A\pm B|\leqslant |A|+|B|$,
$||A|-|B||\leqslant |A\pm B|$ and $|\overline{\mathrm{Re} A}|\leqslant |A|$ (among
others) hold in an unbounded operator setting. As consequences, we
obtain a characterization of (unbounded) self-adjointness as well as
a characterization of invertibility for the class of unbounded
normal operators. Some examples accompany our results.

**DOI: **http://dx.doi.org/10.7900/jot.2018may14.2193

**Keywords: **normal and self-adjoint operators, commutativity, Fuglede theorem

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