On quasisimilarity, almost similarity and metric equivalence of some operators in hilbert spaces
Abstract
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
equ ivalence 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 e 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
Sponsorhip
University of NairobiPublisher
School of mathematics