On Spectra and Almost Similarity of Operators in Hilbert Spaces

Abstract

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.