The Property of Unitary Quasi-Equivalence on Θ –Operators

Abstract

The study of unitary quasi-equivalence was pioneered by Othman in 1996. Two operators, de- noted as F and K, both bounded and linearly defined on a Hilbert space H, are considered unitary quasi-equivalent if there exists a unitary operator U such that F∗F = PK∗KP∗. Previous research has established that the θ -operator preserves almost similarity and unitary equivalence of an operator. However, the study of unitary quasi equivalence on θ -operators has not been estab- lished. An operator F, linear and bounded in Hilbert space H, is said to be a θ -operator if F∗F and F + F∗ commute. The class of all theta-operators in B(H) is denoted by θ and defined as θ = {F ∈ B(H) : [F∗F, F∗ + F] = 0}. This study aims to determine whether the θ -operator pre- serves the unitary quasi-equivalence property. The significance of this research lies in its potential contribution to the fields of functional analysis and operator theory. Understanding operator equiva- lence is crucial in the broader context of mathematical physics and quantum mechanics, where such equivalences often form the basis of theoretical models and computational methods. By investigating the structural properties and equivalence relations of theta-operators, this study aims to provide valu- able insights. To achieve this goal, the study will leverage knowledge from functional analysis and operator theory, while also reviewing previous work on other equivalence operators such as almost similarity and unitary equivalence.