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 = (T T ∗
)
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
T S ∈ 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.
