الفهرس 
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Abstract
In this thesis we are motivated by the study of the spectrum and fine spectrum of particular types of bounded linear operators, which are represented by infinite lower and upper triangular double-band matrices as linear transformations over some sequence spaces. Given an operator, it is sometimes useful to break up the spectrum into various subdivisions, which are called fine spectra. During our study three methods for subdividing the spectrum are considered.This thesis consists of five chapters.
Chapter 1 is an introductory chapter. It starts by presenting a literature review to get a starting point for our considered problem. Next, the motivation that drove us into this thesis are presented, followed by the formulation of the problem in the form of questions, which was addressed in this research. Then, some classical methods of functional analysis, which are needed in our work, are given, followed by our publications in this thesis, and then we highlight the structure of this thesis.
In Chapter 2, we introduce some notations and recall several mathematical notations. These basic notations, together with some definitions will allow the reader to understand all the results and their proofs which will be stated in the sequel. Some details about matrix transformations in sequence spaces, with some related theorems, which show the necessary and sufficient conditions on an infinite matrix, to generate a bounded linear operator on certain sequence space into itself are mentioned. Also, some concepts of spectral theory are given such as eigenvalues, resolvent operator, resolvent set, spectrum, point spectrum, continuous spectrum residual spectrum, approximate point spectrum, defect spectrum and compression spectrum of a bounded linear operator. Moreover, the classification of Taylor and Halberg of the spectrum has also been shown.
In Chapters 3 and 4, we determine the spectrum and fine spectrum of the generalized difference operator on certain sequence spaces. Some illustrative examples are given to show that our results can be applied in many situations and some results in this chapter improve the corresponding results in the existing literature.
Chapter 5 contains the results regarding the fine structure of spectra of infinite upper triangular double-band matrices as operators on certain sequence spaces. Also, we presented some illustrative examples. Moreover, some obtained results are used to study the eigenvalue problem associated with certain infinite matrices. We will recall in this part some previous known results in the current literature and we will show that our new results improve and generalize many of these results.