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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.

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

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