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Fine Spectrum of the Generalized Difference Operator over the Class of Convergent Series

DOI: 10.1155/2013/630436

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Abstract:

We have determined the spectra of the generalized difference operator over the class of convergent series. 1. Preliminaries and Background Spectral theory is one of the thrust areas of research in mathematical sciences. Due to its usefulness and application-oriented scope, its importance is not only confined to mathematics but also the theory finds its applications in other fields like aeronautics, electrical engineering, quantum mechanics, structural mechanics and probability theory, ecology, and some others. Throughout , we denote the set of all bounded linear operators on into itself. If , where is a Banach space, then the adjoint operator of is a bounded linear operator on the dual of defined by = for all and . Let be a linear operator defined on , where denotes the domain of and is a complex normed linear space. For we associate the operator denoted by defined on the same domain , where is a complex number and is the identity operator on . The inverse operator is denoted by and known as the resolvent operator of . The resolvent set of is the set of all the regular values of , such that exists, bounded and is defined on a set which is dense in . Its complement in the complex plane is called the spectrum of , denoted by . Thus the spectrum consists of those values of , for which is not invertible. The spectrum is partitioned into three disjoint sets. The point spectrum is the set such that does not exist. Any is called the eigen value of . The continuous spectrum is the set such that exists, unbounded and the domain of is dense in . The residual spectrum is the set such that exists (and may be bounded or not) and the domain of is not dense in . In finite-dimensional case, continuous spectrum coincides with the residual spectrum, is the empty set and the spectrum consists of only the point spectrum. Throughout , , , , and denote the class of all, bounded, convergent, null, and p-absolutely summable sequence of fuzzy real or complex terms. Let and be two sequence spaces and be an infinite matrix of real or complex numbers , where , . We say that defines a matrix mapping from into , denoted by , if for every sequence the sequence is in , where ( and ), provided the right hand side converges for every and . Spectra of some particular types of matrix operators have been investigated from different aspects by Okutoyi [1], Rhoades [2], Tripathy and Saikia [3], Tripathy and Paul [4, 5], and Dutta and Tripathy [6]. Okutoyi [1] studied the spectrum of the Cesàro operator on . Akhmedov and Ba?ar [7] worked on the spectra of the difference operator over the

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