On eqivalence of some operators in hilbert spaces
Sitati, Isaiah Nalianya
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The second chapter of this project enhances the essential aspects to be discussed in the subsequent chapters, on quasisimilarity and almost similarity of operators. In this chapter, we show that unitary equivalence (and similarity) are equivalence relations. A result showing that two similar operators have equal spectra (i.e. point and approximate point spectrum) is also proved. More so, unitary equivalence results for invariant subspaces and normal operators are also proved. For similar normal operators, we state the Fuglede - Putnam -Rosenblum theorem that makes proofs for similar normal operators more simplified. It is also noted that direct sums and summands are preserved under unitary equivalence. We also see that the natural concept of equivalence between Ililbert Space operators is unitary equivalence which is stronger than similarity. Finally, some results on unitary equivalence and the unilateral shift are discussed. In chapter three, we introduce the notion of quasisimilarity of operators which is the same thing as similarity in finite dimensional spaces, but in infinite dimensional spaces, it is a much weaker relation. It is further shown that quasisimilarity is an equivalence relation. We also link invariant subspaces and hyperinvariant subspaces with quasisimilarity where it is seen that similarity preserves nontrivial invariant subspaces while quasisimilarity preserves nontrivial hyperinvariant subspaces. Equality of the spectra of quasisimilar hyponorrnal is also shown and similar results extended to quasisimilar -quasihyponormat operators. Here, quasisimilarity preserves the Fredholm property. The latter extends William's results on the equality of essential spectra to certain quasisimilar seminormal operators on quasinormal operators. The concepts of local spectrum and operators satisfying Dunford's, C condition are introduced, and some results proved. The last section of this chapter characterizes contractions quasisimilar to a unitary operator. The fourth chapter is on almost similarity of operators. It is a new relation in operator theory and was first introduced by A.A.S.Jibril. Just like in the previous chapters, we show that almost similarity is an equivalence relation. Some results on almost similarity and isometries, compact operators, hermitian, normal and projection operator are also shown.Unitary equivalence and characterization of 0 - operators is also analyzed. In addition, we prove that operators that are similar are almost similar. It is also claimed that quasisimilarity implies almost similarity under certain conditions (i.e., if the quasi-affinities are assumed to be unitary operators). Von-Neumann-Wold decomposition for isometries and Nagy-Foias -Langer Decompositon theorems are introduced which are a powerful tool in proving some results on completely non-unitary operators, unitary equivalence and almost similarity.