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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.

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

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