On Some Relations with Square Normal, Class Q\(^∗\) and n-normal Operators

The relationship between different classes of operators in Hilbert spaces is an important concern in operator theory. Many authors have extended the properties of one operator to another to enhance their applicability and enable their comparison on a common domain. For instance, every normal operator is square normal but the converse is not true. In this paper, we looked at two classes of operators in Hilbert spaces; Square normal and class Q∗, and their relations with 2-normal operators. We investigated the conditions under which a 2-normal operator is class Q∗ and also showed that square normal operators and 3N operators are independent. Furthermore, square normal operators are not necessarily class Q∗ operators. To achieve this, the relationship and independence among different classes of operators were applied.