In this work, we have studied the commutativity of the product of unbounded normal
and self-adjoint operators. For instance, we showed that if S and T are two strongly
commuting unbounded normal operators, Then S + T is essentially normal.
Moreover, we have investigated the -commutativity for closed linear operators. We
have showed that if S and T are two normal operators such that T is bounded and
TS ST ̸= 0 where 2 C. Then ST is normal if and only if j j = 1:
We have also presented a sufficient conditions to ensure the self-adjointness of the
product of two linear operators. For example, it is shown that if S and T are two selfadjoint
operators where only T is bounded, S is positive and if one of the operators TS
or ST is normal, then both TS and ST are self-adjoint and ST = TS. Some examples
accompany our results.
38

In this work, we have studied the commutativity of the product of unbounded normal
and self-adjoint operators. For instance, we showed that if S and T are two strongly
commuting unbounded normal operators, Then S + T is essentially normal.
Moreover, we have investigated the -commutativity for closed linear operators. We
have showed that if S and T are two normal operators such that T is bounded and
TS ST ̸= 0 where 2 C. Then ST is normal if and only if j j = 1:
We have also presented a sufficient conditions to ensure the self-adjointness of the
product of two linear operators. For example, it is shown that if S and T are two selfadjoint
operators where only T is bounded, S is positive and if one of the operators TS
or ST is normal, then both TS and ST are self-adjoint and ST = TS. Some examples
accompany our results.
38