The main objective of this FDP is to give an intriguing picture about the spectral theory of operators and their applications in to science and engineering to the faculties and research scholars. As we all know that the concept of eigenvalues and eigenvectors associated with a matrix are very powerful tool to know about characteristic of the underlying matrix. In order to deal with the operators defined in the infinite dimensional spaces the concept of spectrum is very helpful. The spectrum of a linear operator on a Banach space generalizes the concept of an eigenvalue of a matrix. In Banach spaces spectral theoretic methods play an equally important role as the eigenvalue theory in finite dimensions. These methods are used everywhere in analysis and its applications. In some circumstances, the collection of operators with certain properties having nice algebraic structures. We call them as Banach algebras. The nature of the spectrum of operators can also be studied with the spectral theory of Banach algebra elements. In the current era, the theory of Graphs is a popular research area. There are matrices which are associated with the given graphs too. Using the technical tools developed in the theory of concept of eigenvalues and eigenvectors, one can study the nature of the graph. This program will be fruitful for educators who are either new to teaching the eigenvalues, spectrum and their applications or who wish to enhance their existing knowledge and skills in the area of the spectrum.

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