Properties Of Unitary Quasi-Equivalence On Selected Classes Of Operators In Hilbert Spaces

Abstract

Two operators F and G are considered unitary quasi-equivalent if there exists a unitary operator U satisfying the conditions F ∗F = UG∗GU∗ and FF∗ = UGG∗U ∗ . This concept was introduced in 1996 under the idea of nearly equivalent operators. Since then, various scholars have explored the properties of unitary quasi-equivalence on different operators. For instance, properties of unitary quasi-equivalence on normal, hyponormal, and binomal operators have been investigated. The relationship between unitary quasi-equivalence and other equivalence operators has been established. Specifically, it has been shown that unitary quasi-equivalence implies unitary equivalence. However, the converse is not always true. Partial isometry, co-isometry, isometry, and projection operators have been established to be unitary quasi-equivalence invariants. However, similar properties on the class of w-hyponormal operators, θ-operators, and (p, k)-quasi-hyponormal operators have not been established. This research has therefore, determined the properties of unitary quasi-equivalence on θ-operators using the commutativity concept of an operator. A similar result on w-hyponormal and (p, k)-quasi-hyponormal operators has also been determined in this study using the Aluthge transform and polar decomposition properties. Determining these properties has significant implications in theoretical physics and mathematics. In functional analysis, it contributes to understanding operator algebras and C ∗ -algebras, impacting their representations, spectra, and K-theory. The outcomes of this research will advance knowledge in interpreting equivalence relations of operators in Hilbert spaces and find practical applications in calculations, wave function differentiation, and the study of vibrations, interfacial waves, and stability analysis. The result of this study shows that unitary quasi-equivalence preserves the properties of θ-operators, w-hyponormal operators, and (p, k)-quasihyponormal operators.