  In the present paper we give results on the closedness and the
self-adjointness of the sum of two unbounded operators. We present a new
approach to these fundamental questions in operator theory. We also prove a new
version of the Fuglede theorem where the operators involved are all unbounded.
This new Fuglede theorem allows us to prove (under extra conditions) that the
sum of two unbounded normal operators remains normal. Also a result on the
normality of the unbounded product of two normal operators is obtained as a
consequence of this new "Fuglede theorem". Some interesting examples are also
given.
