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International Journal of Analysis
Volume 2013 (2013), Article ID 630436, 4 pages
Fine Spectrum of the Generalized Difference Operator over the Class of Convergent Series
Mathematical Sciences Division, Institute of Advanced Study in Science and Technology, Paschim Boragoan, Garchuk, Assam Guwahati 781035, India
Received 14 September 2012; Accepted 16 November 2012
Academic Editor: Alexandre Timonov
Copyright © 2013 Amar Jyoti Dutta and Binod Chandra Tripathy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 , Rhoades , Tripathy and Saikia , Tripathy and Paul [4, 5], and Dutta and Tripathy . Okutoyi  studied the spectrum of the Ces####^~^~^~^~^~^####xe0;ro operator on . Akhmedov and Ba####^~^~^~^~^~^####x15f;ar  worked on the spectra of the difference operator over the sequence space (). Recently Altay and Ba####^~^~^~^~^~^####x15f;ar  and Furkan et al.  determined the spectra of over , and , , respectively.
In this paper, we have analyzed the spectra of the generalized difference operator on of all convergent series. By well-established convention, we define the space of all ####^~^~^~^~^~^####x201c;convergent series####^~^~^~^~^~^####x201d; by .
The generalized difference operator is represented by If is a bounded linear operator with the matrix , then its adjoint operator is defined by transpose of the matrix and is isomorphic to with the norm .
The following lemma helps us to determines the norm and to verify that .
Lemma 1. The matrix gives rise to a bounded linear operator from to itself if and only if the supremum of norms of the columns of is bounded.
Lemma 2. is a bounded linear operator with .
Proof. It can be easily verified that is linear.
Lemma 3 (see ). A linear operator has a dense range if and only if its adjoint operator is one to one.
Lemma 4 (see ). The adjoint operator of####^~^~^~^~^~^####x2009; is onto if and only if####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;has a bounded inverse.
Lemma 5 (see ). Let , where is any Banach space. Then the spectrum of is identical with the spectrum of T. Furthermore and .
Now we shall determine the spectrum, point spectra, continuous spectra, and residual spectra of the operator on the space .
2. Main Result
Theorem 6. .
Proof. We shall prove this theorem by showing that exists and is in for and then show that the operator is not invertible for .
Let be such that . Since is a lower triangular matrix, so exists. Solving####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009; for in terms of , we get the following system of equations:
The th row is given by
Thus the matrix determined by is defined by We observe that Thus .
Now we show that the operator is not invertible for .
Let be such that and . Since is triangular, so exists. From (6) we have , whenever , that is . When , the operator is represented by the matrix given by We observe that . Thus is injective but its range is not a dense set. Hence is not invertible. This shows that is not invertible for . This completes the proof.
Theorem 7. .
Proof. Let , for in . We have the following system of equations:
If is the first nonzero term of the sequence , , hence we have
This contradicts to the fact that .
Thus and hence is injective.
That is, .
Theorem 8. .
Proof. Suppose that , for . Now we have the following system of equations:
Solving the above systems of linear equations, we get We observe that if and only if .
This completes the proof.
Theorem 9. .
Proof. We prove that the range of is not dense in but has an inverse, bounded, or unbounded.
Theorem 8 implies that is not one-to-one and therefore by Lemma 3, is not dense in . We have also seen that exist and for those satisfying####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009;. Again for , is triangular and therefore has an inverse.
This completes the proof.
Theorem 10. .
Proof. It is well known that is the disjoint union of , , and since and = .
This completes the proof.
In this paper, the spectra of matrix operator on convergent series is established. In a similar way the spectra of other matrix operators can also be determined over the convergent series.
The authors thank the reviewer for his comments on the paper; those improved the presentation of the paper. The work of the authors is financially supported by the Council of Scientific and Industrial Research, India vide Grant no. 25(0182)/10/EMR-II.
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