On Some Properties of Class Q∗ Operators

Abstract

The study of operators in Hilbert spaces holds significant importance, finding broad applications in diverse fields such as computer programming, financial mathematics and quantum physics. Many authors have extended the concept of normal operators in an attempt to provide practical solutions to complex problems in diverse fields. This paper focuses on a class Q∗ operators in a Hilbert space H. An operator T ∈ B(H) (where B(H) represents bounded linear operators acting on H) is said to be class Q∗ if T ∗2T 2 = (TT ∗)2. By considering the properties of normal operators and other operators related to normal the study investigated the commutation relations and properties unique to class Q∗ operators. The study shows that if two operators T, S ∈ Q∗ are such that the sum (T + S) commutes with (T + S)∗, then (T + S) ∈ Q∗ and the product TS ∈ Q∗ if T and S commute with their adjoint. The results of this research are a valuable resource for mathematicians and physicists interested in the properties and applications of class Q∗ operators fueling further innovations in functional analysis.