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The usefulness of the notion of spectrum of an operator on a Banach space is the analogy to eigenvalues of operators on finite-dimensional spaces. Naturally, things become more complicated in infinite-dimensional Banach spaces. To calculate the spectrum of an operator, we must determine the space in which it acts. It means that changing spaces or norms, generally speaking, will change spectra. In this thesis, we are concerned with study the spectra of linear operators which can represented by infinite matrices on Banach sequence spaces. Several authors have studied the spectra of bounded linear operators, which are represented by particular infinite matrices over sequence spaces. To our knowledge, the earliest study about spectra of infinite matrices was published by Brown, Halmos and Shields in 1965 for the Cesàro matrix. This study followed by many authors. In this thesis, we are concerned with the spectra of infinite-dimensional Jacobi matrices, that are, symmetric (infinite) tridiagonal matrices with constant entries as linear operators on Banach sequences spaces.
This thesis consists of four chapters followed by a list of references. It is organized as follows:
Chapter one. We give a short survey concerning the spectrum and fine spectrum of linear operators defined by some particular matrices over some sequence spaces. Furthermore, in this chapter, we formulate our problem.
Chapter two. We introduce some basic and fundamental definitions, lemmas, propositions and important theorems that will be used throughout the thesis.
Chapter three. We present the constant Jacobi matrix as a linear transformation on Banach sequence spaces and then we investigate the spectra and its subdivisions (point spectrum, continuous spectrum and residual spectrum) on two sequence spaces bv₀ and h.
Chapter four. We continue to study the spectra and its subdivisions on sequence space cs. Then, we use these results in addition to the results in chapter three and the relation between the spectra of a bounded linear operator and its adjoint to obtain the spectra and its subdivisions on other sequence spaces. On the other hand, we show that there is an equivalence between the spectral problems of both Jacobi and the free Jacobi.