In the first chapter we give a brief introduction on some of the developments of the equivalence relations came about. We also define some of the notations and terminologies that will be used in this work.
The second chapter of this project gives results on unitary equivalence and similarity of operators. It acts as the corner stone on which the subsequent chapters are built. In this chapter, we show that unitary equivalence and similarity is an equivalence relation. We also give some results on invariant subspaces and show that similarity preserves nontrivial invariant subspaces. We also give some results on similarity, invertibility of some operators (self-adjoint, A-self adjoint and A-unitary operators etc.).
In chapter three, we discuss quasisimilarity of operators. We also link the nontrivial invariant subspaces and the hyperinvariant subspaces; where it is established that quasisimilarity preserves the nontrivial hyperinvariant subspace. We also show some results on equality of spectra of quasisimilar hyponormal operators. We then extend these results in to discuss class h-operators. We observe that quasisimilar p-hyponormal, p-quasihyponormal and log-hyponormal operators (class h-operators) have the same spectra and essential spectra. Moreover in the same vein, we discuss quasisimilar quasi-class A operators, (p,k)-quasihyponormal operators and class wF(p,r,q) operators. We also confirm that they have equal spectra and essential spectra. Finally, we look at quasisimilar quasihyponormal operators where quasisimilarity preserves the Fredholm property
Chapter four is on almost similarity of operators. We show that almost similarity is an equivalence relation. Some results on some classes of operators and almost similarity relation are given. In addition, we give some results on unitary (similarity) equivalence and almost similarity relation. We give some characterization of ϴ and isometric operators in relation to almost similarity. Finally we discuss some results on direct summands in relation to almost similarity. In this section, Von-Neuman-Wold decomposition for isometries and Nagy -Foias- Langer decomposition theorem are key tools in the development of results.
In chapter five, we give results on metric equivalence of some operators. We also give some results on metric equivalence and the spectral picture of some operators. Eventually, we look at the relationship between metric equivalence and other equivalence relations namely; similarity unitary equivalence and near similarity.
In chapter six, we give a summary of our project. It is shown that quasisimilarity, almost similarity and metric equivalence are equivalence relations. We also indicate conditions under
which unitary equivalence, similarity and quasisimilarity imply almost similarity. We also give the necessary and sufficient condition under which metric equivalence implies unitary equivalence. More over similar operators have equal spectra. Some quasisimilar operators have equal spectra and essential spectra. Regarding the Fredholm it is shown that metric equivalent and quasisimilar quasihyponormal operators preserve the Fredholm property.

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