On Spectra and Almost Similarity of Operators in Hilbert Spaces
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Date
2019Author
Matheka, Joseph Mutuku
Type
ThesisLanguage
enMetadata
Show full item recordAbstract
This project is on spectra and almost similarity of operators in Hilbert spaces.
In chapter one we discuss the meaning and the structure of a Hilbert space. Here the
linear structure, the norm, the inner product structure and convergence of sequences in a
set of vectors are discussed to yield the meaning of a Hilbert space.
In chapter two, transformation of elements in a Hilbert space is discussed. The nature of
transformations are also discussed in this chapter i.e. the preservation of linear structure,
boundedness and the norm. The Banach algebra of bounded linear operators is also
established. We use the linear operator to de ne invariant subspaces of a Hilbert space.
We also de ne the spectra of operators on Hilbert spaces. The structure and the subsets of
the spectrum are discussed in this chapter. We also discuss the spectrum of some classes
of operators.
The third chapter is on similarity and quasi-similarity of operators. We show that unitary
equivalence, similarity and quasi-similarity of operators are equivalence relations. Also
unitary equivalence implies similarity and similarity implies quasi-similarity. Unitary
equivalent and Similar operators have equal spectra in general. Quasi-similar operators on
a nite dimensional Hilbert space have equal spectra but on in nite dimensional Hilbert
spaces, quasi similar operators have equal spectra if the operators are hypo-normal.
The fourth chapter is on almost similarity of operators. We discuss the relationship of
cartesian and polar decomposition of operators with almost similarity of operators. We
show that almost similarity of operators is an equivalence relation. Almost similar
operators which are Hermitian or projections have equal spectra. This project is on spectra and almost similarity of operators in Hilbert spaces.
In chapter one we discuss the meaning and the structure of a Hilbert space. Here the
linear structure, the norm, the inner product structure and convergence of sequences in a
set of vectors are discussed to yield the meaning of a Hilbert space.
In chapter two, transformation of elements in a Hilbert space is discussed. The nature of
transformations are also discussed in this chapter i.e. the preservation of linear structure,
boundedness and the norm. The Banach algebra of bounded linear operators is also
established. We use the linear operator to de ne invariant subspaces of a Hilbert space.
We also de ne the spectra of operators on Hilbert spaces. The structure and the subsets of
the spectrum are discussed in this chapter. We also discuss the spectrum of some classes
of operators.
The third chapter is on similarity and quasi-similarity of operators. We show that unitary
equivalence, similarity and quasi-similarity of operators are equivalence relations. Also
unitary equivalence implies similarity and similarity implies quasi-similarity. Unitary
equivalent and Similar operators have equal spectra in general. Quasi-similar operators on
a nite dimensional Hilbert space have equal spectra but on in nite dimensional Hilbert
spaces, quasi similar operators have equal spectra if the operators are hypo-normal.
The fourth chapter is on almost similarity of operators. We discuss the relationship of
cartesian and polar decomposition of operators with almost similarity of operators. We
show that almost similarity of operators is an equivalence relation. Almost similar
operators which are Hermitian or projections have equal spectra.
Publisher
University of Nairobi
Rights
Attribution-NonCommercial-NoDerivs 3.0 United StatesUsage Rights
http://creativecommons.org/licenses/by-nc-nd/3.0/us/Collections
- Faculty of Education (FEd) [5981]
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