On equivalence of some operators in hilbert spaces
Unitarily equivalence is the natural concept of equivalence between Hilbert space operators. Thus, this concept is the building block of other equivalence relations such as similarity, quasisimilarity, or even metric equivalence. In this project, firstly, it is shown that unitary equivalence, similarity, quasisimilarity and almost similarity are equivalence relations. Then, through the Putnam-Fuglede theorem, similarity and hence unitarily equivalence of normal operators were discussed. In addition, it is shown that, reducing subspaces are preserved under unitarily equivalence and that, similarity preserves nontrivial invariant subspaces, while quasisimilarity preserves nontrivial hyperinvariant subspaces. Moreover, several known results, (but which are scattered in different accounts), such as those touching on; equality of spectra of quasisimilar hyponormal operators, direct summands in relation to almost similarity, and metric equivalence of operators preserves Fredholmness were presented. As a consequence, more independent results were struck. For instance, a new equivalence relation, that is, unitary quasi-equivalence relation is introduced, and an observation which shows that, for an A-self-adjoint operator T, such that, T is metrically equivalent to S, then T2 is similar to S2, after demanding self-adjointedness of S, is deduced and proved.