| dc.description.abstract | 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. | en |
