Characterization of Square Normal 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 normal operators in an attempt to provide practical solutions to complex problems in diverse fields. This paper focuses on a class of square normal operators in a Hilbert space denoted by H, whereB(H) represents bounded linear operators acting on H. An operator T inB(H) is said to be a square normal if T2(T*)2=(T*)2T2. The study investigates the commutation relations and properties unique to this class of operators. A detailed analysis of their relation with other classes of operators is presented shedding more light on the unique features that distinguish square normal operators within the mathematical landscape. Through this paper, we aim to produce a valuable resource for mathematicians and physicists interested in the properties and applications of square normal operators fueling further innovations in functional analysis. To achieve this, we extend the properties of normal operators and other operators related to normal to square normal operators. In this paper, the algebraic properties of square normal operators are presented. The relationship between square normal operators with some operators such as isometry, unitary and n-normal is also given.