Some Remarks on Almost Similarity and Other New Equivalence Relations of Operators

In this paper, new results touching on almost similarity and other equivalence relations are presented. We investigate the relationship between almost-similarity relation and other equivalence relations of bounded linear operators on Hilbert spaces. We study the relation between equivalence classes of bounded linear operators with respect to different properties such as being self-adjoint, projections, normal, unitary and having a specific rank. This paper improves on previous works on known operator equivalence relations and classes. We obtain several new results on operator equivalences. Methodology involved the use of properties of orthogonal operators and normality of operators among others. Results show that almost similar projection operators have the same spectrum. Results touching on spectral invariants with respect to some of these equivalence relations are also established.