  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 each one of them.
