# Journal of Operator Theory

Volume 76, Issue 1, Summer 2016 pp. 159-169.

Conditions implying commutativity of unbounded self-adjoint operators and related topics

**Authors**:
Karl Gustafson (1) and Mohammed Hichem Mortad (2)

**Author institution:** (1) Department of Mathematics, Univ. of Colorado at Boulder, Campus
Box 395 Boulder, CO 80309-0395, U.S.A.

(2) Department of Mathematics, University of Oran 1 (Ahmed
Benbella), B.P. 1524, El Menouar, Oran 31000, Algeria and
B.P. 7085, Seddikia Oran,
31013 Algeria

**Summary: ** Let $B$ be a bounded self-adjoint operator and let $A$ be a
nonnegative self-adjoint unbounded operator. It is shown that if
$BA$ is normal, it must be self-adjoint and so must be $AB$.
Commutativity is necessary and sufficient for this result. If $AB$
is normal, it must be self-adjoint and $BA$ is essentially
self-adjoint. Although the two problems seem to be alike, two
different and quite interesting approaches are used to tackle them.

**DOI: **http://dx.doi.org/10.7900/jot.2015oct16.2076

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

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