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Journal of Mathematics
Volume 2013 (2013), Article ID 430965, 7 pages
The Spectrum of the Operator over the Sequence Spaces and
1Mathematical Sciences Division, Institute of Advanced Study in Science and Technology, Paschim Boragaon, Garchuk, Guwahati, Assam 781035, India
2Department of Mathematics, Cachar College, Silchar, Assam 788001, India
Received 7 November 2012; Revised 15 February 2013; Accepted 1 March 2013
Academic Editor: Inés Couso
Copyright © 2013 Binod Chandra Tripathy and Avinoy Paul. 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 examined the spectra of the operator on the sequence spaces and .
Spectral theory is an important branch of mathematics due to its application in other branches of science. It has been proved to be a standard tool of mathematical sciences because of its usefulness and application-oriented scope in different fields. In numerical analysis, the spectral values may determine whether a discretization of a differential equation will get the right answer or how fast a conjugate gradient iteration will converge. In aeronautics, the spectral values may determine whether the flow over a wing is laminar or turbulent. In electrical engineering, it may determine the frequency response of an amplifier or the reliability of a power system. In quantum mechanics, it may determine atomic energy levels and thus the frequency of a laser or the spectral signature of a star. In structural mechanics, it may determine whether an automobile is too noisy or whether a building will collapse in an earthquake. In ecology, the spectral values may determine whether a food web will settle into a steady equilibrium. In probability theory, they may determine the rate of convergence of a Markov process.
In summability theory, different classes of matrices have been investigated. Characterization of matrix classes is found in Rath and Tripathy , Tripathy , Tripathy and Sen , and many others. There are particular types of summability methods like Nörlund, Riesz, Euler, and Abel. Matrix methods have been studied from different aspects recently by Altin et al. , Tripathy and Baruah , and others. Still there is a lot to be explored on spectra of some matrix operators transforming one class of sequences into another class of sequences. The spectra of the difference operator have also been investigated on some classes of sequences. Altay and Başar [6–8] studied the spectra of difference operator and generalized difference operator on , and . Recently, the fine spectrum of over the sequence spaces , and has been studied by Furkan et al. [9, 10]. A detailed account of the development and initial works on spectra of some matrix classes are found in the monograph of Başar .
Throughout the paper , and denote the space of all bounded, convergent, and null sequences with complex terms, respectively, normed by . The zero sequence is denoted by . Kızmaz  defined the difference sequence spaces , and as follows: where
The previous spaces are Banach spaces, normed by .
Different classes of sequence spaces using the difference operator have been introduced and investigated in the recent past by Tripathy et al. [13, 14], Tripathy and Mahanta , Tripathy and Sarma , and many others. The idea of Kizmaz  was applied to introduce a new type of generalized difference operator on sequence spaces by Tripathy and Esi .
Let be fixed; then Tripathy and Esi  have introduced the following type of difference sequence spaces: where
Let be fixed integers; then Esi et al.  have introduced the following type of difference sequence spaces: where
Taking , we have the sequence spaces , , and studied by Tripathy and Esi . Taking , we have the sequence spaces , , and studied by Et and Çolak . Taking and , we have the sequence spaces , , and studied by Kızmaz .
2. Preliminaries and Definition
Let be a linear space. By , 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 denote the domain of and is a complex normed linear space. For we associate a complex number with the operator denoted by defined on the same domain , where is the identity operator. The inverse , denoted by , is known as the resolvent operator of .
A regular value of is a complex number of such that (R1) exists,(R2) is bounde, (R3) is defined on a set which is dense in .
The resolvent set of is the set of all such regular values of , denoted by . Its complement is given by in the complex plane which is called the spectrum of , denoted by . Thus the spectrum consists of those values of , for which is not invertible.
2.1. Classification of Spectrum
The spectrum is partitioned into three disjoint sets as follows.(i)The point (discrete) spectrum is the set such that does not exist. Further is called the eigenvalue of .(ii)The continuous spectrum is the set such that exists and satisfies (R3) but not (R2); that is, is unbounded.(iii)The residual spectrum is the set such that exists (and may be bounded or not) but does not satisfy (R3); that is, the domain of is not dense in .
This is to note that in finite dimensional case, continuous spectrum coincides with the residual spectrum and are equal to the empty set and the spectrum consists of only the point spectrum.
Let and be two sequence spaces and let be an infinite matrix of real or complex numbers , where . Then, we say that defines a matrix mapping from into , denoted by , if for every sequence the sequence is in where , provided the right hand side converges for every and .
Our main focus in this paper is on the matrix , where
We assume here and hereafter that and are complex parameters which do not simultaneously vanish.
Lemma 1. The matrix gives rise to a bounded linear operator from c to itself if and only if(1)the rows of A in and their norms are bounded,(2)the columns of are in ,(3)the sequence of row sums of is in .
The operator norm of T is the supremum of the norms of the rows.
Lemma 2. The matrix gives rise to a bounded linear operator from to itself if and only if (1)the rows of A in and their norms are bounded,(2)the columns of are in .
The operator norm of T is the supremum of the norms of the rows.
From the previous two lemmas we have the following results.
Lemma 3. is a bounded linear operator with
Lemma 4. is a bounded linear operator with
In this paper, our purpose is to determine the fine spectrum of the operator over the sequence spaces and .
3. The Fine Spectrum of the Operator over the Sequence Space
Theorem 5. Let s be a complex number such that and define the set by Then
Proof. At first we have to prove that exists and is in for and secondly we have to show that is not invertible for .
Without loss of generality we may assume that . Let ; then it is easy to see that and so is triangle and has an inverse.
On solving the previous system of equations we get
In general this sequence is obtained recursively by
It is easy to verify that
On taking we have
If one assumes , then we will get the same sequence as in the case of .
If , then
If we put , then , for and , for .
Thus, on simple calculation we get if and only if . Therefore implies , as .
Next, we assume that . Since , therefore . Now we have to show that .
We have that implies .
Since for any , we must have which leads us to conclude that and hence in this case also implies that as .
Since and , we have .
Thus, and hence .
Next we have to show that .
Let . If , then is represented by the matrix
Since does not have a dense range, so it is not invertible.
Again if , then for and , for .
Now and hence ; therefore, we have .
Next we assume that and .
Since , we have that is a triangle. Further , so we must have . This implies and hence . This shows that .
This completes the proof.
Theorem 6. One has .
Proof. Suppose for in . Then by solving the system of linear equations we have
If is the first nonzero entry of the sequence , then from the previous system of linear (22) and (23) we have and we obtain that and from the next of either (22) or (23) we get =0 which is a contradiction. This completes the proof.
If is a bounded linear operator with the matrix , then it is known that the adjoint operator is defined by the transpose of the matrix . It should be noted that the dual space of is isometrically isomorphic to the Banach space of absolutely summable sequences normed by .
Theorem 7. One has , where
Proof. Suppose for in ; that is, considering the following system of linear equations we have
If , then we may choose and so is an eigenvector corresponding to and hence .
Next we assume that ; then from the previous system of equations we have
If is a number such that , then we may choose and .
We can easily verify that and ; using these results and combining the fact with relation (26) we observe that
Similarly we can show that , for .
The same result may be obtained in the case , that is, for the case .
Now since ; this implies that and hence .
Conversely, let ; therefore ; that is, . We have to show that ; that is, to show for this it is sufficient to show that is divergent.
Now consider .
Here, Thus we have .
Now we examine three cases.
Case 1 (). In this case and
Now, if , then we get which is not in since . Otherwise
Case 2 (). In this case and using the formula
We have and so
Case 3 (). In this case and so we have and and so . Our aim is to show . On the contrary we assume that . This implies that .
Now, if we consider the series in place of , one will get results parallel to all the results obtained previous just by replacing in place of and in place of .
Since , we must have from (35) and similarly we get . This implies , a contradiction and so we must have .
In Cases 1 and 2, by the d’Alembert ratio test we get that is divergent and similarly we get that is also divergent and hence is divergent, since the sum of two absolutely divergent series is divergent. In Case 3 leads to contradiction. Thus in all the previous cases and hence . This completes the proof of theorem.
Lemma 8. T has a dense range if and only if is one to one, where denotes the adjoint operator of T.
Theorem 9. , where is defined as in Theorem 7.
Proof. One has ; then is not one-to-one for all . Therefore by Lemma 8, have a dense range for all and hence .
Theorem 10. Consider , where
Proof. The proof immediately follows from the fact that the set of spectra is the disjoint union of the point spectrum, residual spectrum, and continuous spectrum; that is,
4. The Fine Spectrum of the More Generalized Operator on the Sequence Space
Theorem 11. Consider , where is defined as in Theorem 5.
Proof. This is obtained in a similar way used in the proof of Theorem 5.
Theorem 12. One has .
Proof. This is obtained in a similar way that is used in the proof of Theorem 6.
If is a bounded matrix operator with the matrix , then acting on has a matrix representation of the form , where is the limit of the sequence of row sums of minus the sum of the columns of and is the column vector whose th entry is the limit of the th column of for each . For , the matrix is of the form .
Theorem 13. Consider .
Proof. Suppose for in ; that is, consider the following system of linear equations:
Then, we obtain that
If , then . So, is an eigenvalue with the corresponding eigenvector .
If , then and so using arguments similar to those in the proof of Theorem 7 one can see by (39) that . This completes the proof.
Theorem 14. Consider .
Theorem 15. Consider , where is defined as in Theorem 10.
5. Particular Case
The spectrum of the operator over the sequence space and may be derived as the set ; one can directly produce the same result from the present paper since .
The authors thank the referee for the comments and suggestions that improved the presentation of the paper.
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