Characterization and commutation relations on square normal and class Q∗ operators in Hilbert spaces

Abstract

Many researchers have widely studied operators in Hilbert spaces due to their wide application in areas like computer programming, financial mathematics and quantum physics. The study of operators in Hilbert space has been categorized according to their properties, the relation between different classes and their spectral properties. Researchers have studied various operators in Hilbert spaces, examining their algebraic properties, commutation relations, independence and inclusions. The classical normal operators has played an important role in the development, study and generalization of these classes of operators in Hilbert spaces. This study focused on the extension of properties of normal operators to two classes of operators, the square normal operators and class Q∗ operators. The aim was to determine their characterization, algebraic properties and relationship with other operators in Hilbert space. By use of the relationships with normal operators, this study has established that for any square normal operator T ∈ B(H), then T∗, T−1 and any other operator unitarily equivalent to T are square normal operators. Furthermore, it has been shown that for square normal operators T,S and scalar λ ∈ C, then (λT), (λ + T), (T + S) and (TS) are square normal operators provided some certain conditions are met. The study shows that class Q∗ operators are not convex and establishes that if two class Q∗ operators T and S commute, then their sum T + S is in class Q∗ and their product TS is in class Q∗ if TS∗ = S∗T and T∗S = ST∗. The research also found that 2-normal operators are both square normal and class Q∗ operators while the class of 3-normal and square normal operators are independent. Furthermore, the study observed that while class Q∗ operators are square normal, the converse is not necessarily true. These findings contribute to the existing knowledge in operators theory and functional analysis and offer potential applications in practical domains such as computer science, finance, and quantum mechanics.