A Note on Quasi-Similarity of Operators in Hilbert Spaces

Abstract

This paper reports on the notion of Quasi-similarity of bounded linear operators in Hilbert Spaces, defines a quasi-affinity from one Hilbert Space H to K and discusses results on quasi-affinities. It has been shown that on a finite dimensional Hilbert Space, quasi-similarity is an equivalence relation; it is reflexive, symmetric and transitive. Using the definition of commutants of two operators, an alternative result is given to show that quasi-similarity is an equivalence relation on an infinite dimensional Hilbert Space. The relationship between quasi-similarity and almost similarity equivalence relations in Hilbert Spaces using hermitian and normal operators is established.