On Some Properties of Square Normal Operators

The study of operators in Hilbert spaces is an important concept due to its wide application in areas like computer programming, financial mathematics and quantum physics. This paper focused on a class of square normal operators in a Hilbert space. Let H be a complex Hilbert space and B(H) be a bounded linear operator acting on H. Then an operator T in B(H) is a square normal if T2(T*)2 = (T*)2T2. This paper studied the commutation relations and properties of this class of operators and showed that for any square normal operator T, then T* and T-1 if it exists is square normal. Furthermore, the sum T + S and product TS of two square normal operators which commute with the adjoint of each other is square normal. To achieve this, the properties of normal operators and other operators related to normal operators were extended to square normal operators.