Commutativity of unbounded normal operators

Abstract

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. 38In 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