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International Journal of Analysis

Volume 2014, Article ID 786437, 7 pages

http://dx.doi.org/10.1155/2014/786437
Research Article

On the Spectral Properties of the Weighted Mean Difference Operator over the Sequence Space

1Department of Mathematics, School of Applied Sciences, KIIT University, Bhubaneswar 751024, India

2Department of Mathematics, Utkal University, Vanivihar, Bhubaneswar 751004, India

Received 21 December 2013; Accepted 30 April 2014; Published 16 June 2014

Academic Editor: Baruch Cahlon

Copyright © 2014 P. Baliarsingh and S. Dutta. 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.

Abstract

In the present work the generalized weighted mean difference operator has been introduced by combining the generalized weighted mean and difference operator under certain special cases of sequences and . For any two sequences and of either constant or strictly decreasing real numbers satisfying certain conditions the difference operator is defined by with for all . Furthermore, we compute the spectrum and the fine spectrum of the operator over the sequence space . In fact, we determine the spectrum, the point spectrum, the residual spectrum, and the continuous spectrum of this operator on the sequence space .

1. Introduction, Preliminaries, and Definitions

Let and be two bounded sequences of either constant or strictly decreasing positive real numbers such that and for all , and

By and , we denote the spaces of all absolutely summable and p-bounded variation series, respectively. Also, by , , and , we denote the spaces of all bounded, convergent, and null sequences, respectively. The main perception of this paper is to introduce the weighted mean difference operator as follows.

Let be any sequence in , and we define the weighted mean difference transform of by where denotes the set of nonnegative integers and we assume throughout that any term with negative subscript is zero. Instead of writing (3), the operator can be expressed as a lower triangular matrix , where Equivalently, in componentwise the triangle can be represented by

The main objective of this paper is to determine the spectrum of the operator over the basic sequence space . The operator has been studied by Polat et al. [1] in detail by introducing the difference sequence spaces , , and . In the existing literature several researchers have been actively engaged in finding the spectrum and fine spectrum of different bounded linear operators over various sequence spaces. The spectrum of weighted mean operator has been studied by Rhoades [2], whereas that of the difference operator over the sequence spaces for and , has been studied by Altay and Başar [3, 4]. Kayaduman and Furkan [5] have determined the fine spectrum of the difference operator over the sequence spaces and and on generalizing these results, Srivastava and Kumar [6, 7] have determined the fine spectrum of the operator over the sequence spaces and , where is a sequence of either constant or strictly deceasing sequence of reals satisfying certain conditions. Dutta and Baliarsingh [810] have computed the spectrum of the operator ( ) and over the sequence spaces , , and , respectively. The fine spectrum of the generalized difference operators and over the sequence spaces , and , has been studied by Bilgiç and Furkan [11, 12], respectively. Recently, the spectrum of some particular limitation matrices in certain sequence spaces has been studied by [1315] and many others.

Let and be Banach spaces and be a bounded linear operator. By , we denote the range of ; that is,

By , we denote the set of all bounded linear operators on into itself. If is any Banach space and then the adjoint of is a bounded linear operator on the dual of defined by for all and with .

Let be a normed linear space over the complex field and be a linear operator, where denotes the domain of . With , for a complex number , we associate an operator , where is called identity operator on and if has an inverse, we denote it by , that is, and is called the resolvent operator of . Many properties of and depend on and the spectral theory is concerned with those properties. We are interested in the set of all ’s in the complex plane such that exists/ is bounded/domain of is dense in . Now, we state the following results which are essential for our investigation.

Definition 1 (see [16, page 371]). Let be normed linear space over the complex field and be a linear operator with domain . A regular value of is a complex number such that (R1) exists;(R2) is bounded;(R3) is defined on a set which is dense in .

The resolvent set of is the set of all regular values of . Its complement in the complex plane is called the spectrum of . Furthermore, the spectrum is partitioned into three disjoint sets as follows.(I)Point Spectrum . It is the set of all such that (R1) does not hold. The elements of are called eigen values of .(II)Continuous Spectrum . It is the set of all such that (R1) holds and satisfies (R3) but does not satisfy (R2).(III)Residual Spectrum . It is the set of all such that (R1) holds but does not satisfy (R3). Condition (R2) may or may not hold.

Goldberg’s Classification of the Operator (see [17], page 58–71). Let be a Banach space and , where is a complex number. Again, let and denote the range and inverse of the operator , respectively. Then the following possibilities may occur: (A) ;(B) ; (C)

and(1) is injective and is continuous;(2) is injective and is discontinuous;(3) is not injective.

Taking permutations (A), (B), (C) and (1), (2), (3), we get nine different states. These are labelled by , , , , , , , , and . If is a complex number such that or , then is in the resolvent set of on . The other classifications give rise to the fine spectrum of . We use which means the operator , that is, and is injective but is discontinuous; similarly are the others.

Lemma 2 (see [17, page 59]). A linear operator has a dense range if and only if the adjoint is one to one.

Lemma 3 (see [17, page 60]). The adjoint operator is onto if and and only if has a bounded inverse.

Let , be two nonempty subsets of the space of all real or complex sequences and let be an infinite matrix of complex numbers , where . For every , we write The sequence , if it exists, is called the transformation of by the matrix . Infinite matrix if and only if whenever .

Lemma 4 (see [18, page 126]). The matrix gives rise to a bounded linear operator from to itself if and only if the supremum norms of the columns of is bounded.

Lemma 5 (see [19], Theorem 2). If for all , then the inverse of the difference operator is given by a lower triangular Toeplitz matrix as follows: where , and

2. The Spectrum of the Operator over the Sequence

In this section, we compute the spectrum, the point spectrum, the continuous spectrum, and the residual spectrum of the difference matrix on the sequence space .

Theorem 6. The operator is a linear operator and

Proof. Proof of this theorem follows from Lemma 4 and in particular cases. (i)If is a strictly decreasing sequence and is a constant sequence of positive reals (say for all ), then (ii)If and are strictly decreasing sequences of positive reals, then (iii)If and are constant sequences of positive reals, then (iv)If is a constant sequence and is a strictly decreasing sequence of positive reals, then

Theorem 7. The spectrum of over the sequence space is given by

Proof. The proof is divided into four parts as follows.

Let and be two bounded sequences of positive reals satisfying (1) and (2) and such that for all ; then is a triangle and hence has an inverse, that is, In general, by using Lemma 5, we observe that Therefore, for all , , for all , , and for all one can deduce that By using Lemma 4, we obtain that For simplicity, we write and . Since and are two bounded sequences of positive reals, the quantity is finite. As per the assumption, and for all . Therefore, . Now, we consider the four possible cases of the sequences and .

Case  1. If and are strictly decreasing sequences of positive reals, then from the above relation we have . Thus, Conversely, consider and for all , and clearly is a triangle and hence exists, but is not finite. Again, for , the matrix is not invertible. Thus,

Combining (21) and (22) we complete the proof for this case.

Case  2. If is a strictly decreasing sequence and is a constant sequence of positive reals, then Since , and the rest part of the proof is similar to that of Case 1.

Case  3. If and are constant sequences of positive reals, then the proof is similar to that of Case 2 and only the difference is .

Case  4. If is constant sequence and is strictly decreasing sequence of positive reals, then the proof is similar to that of Case 1.

Theorem 8. The point spectrum of the operator over is given by

Proof. Suppose and consider the system of linear equations for in , On solving the system of (25), we obtain that whenever and Now, we have the following cases.

Case  1. If and are strictly decreasing sequences of positive reals, then we need to show that . Suppose, for the contrary, if , then which contradicts the fact that is strictly decreasing sequence. Therefore, and hence .

Case  2. If is a strictly decreasing and is a constant sequence of positive reals, then for all , is an eigen value corresponding to the eigen vector whose th entry is 1. Thus, .

Case  3. If and are constant sequences of positive reals, then the proof is similar to that of Case 2 and .

Case  4. If is a constant and is a strictly decreasing sequence of positive reals, then the proof is similar to that of Case 1.

Theorem 9. The point spectrum of the dual operator over the sequence space is given by

Proof. Consider and , and then the system of equations From the system of linear equations (28), we obtain that

Combining (29) and (30), we have Therefore, by using ratio text, it is observed that Now, we have the following cases.

Case  1. If and are strictly decreasing sequences of positive reals, then by using (2), we conclude that if and only if for all . Therefore, .

Case  2. If is a strictly decreasing and is a constant sequence of positive reals, then from the system of (28), we observe that for all , is an eigen value corresponding to the eigen vector whose th entry is 1. Thus, .

Case  3. If and are constant sequences of positive reals, then the proof is similar to that of Case 2.

Case  4. If is a constant and is a strictly decreasing sequence of positive reals, then the proof is similar to that of Case 1.

Theorem 10. The residual spectrum of the operator over the sequence space is given by

Proof. Proof follows from Lemma 2 and Theorems 8 and 9.

Theorem 11. The continuous spectrum of the operator over the sequence space is given by

Proof. The proof of this theorem follows from Theorems 7, 8, and 10 and along with the fact that

3. Conclusion

In this work, the authors have determined the spectrum of the generalized weighted mean difference operator over the Banach space .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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