# On the Spectral Properties of the Weighted Mean Difference OperatorGu,v;Δ over the Sequence Space ℓ1

Authors: P. Baliarsingh; S. Dutta
Journal: International Journal of Analysis
 (2014)

---

## Abstract

In the present work  the generalized weighted mean difference operatorGu,v;Δ has been introduced by combining the generalized weighted mean and difference operator under certain special cases of sequences u=(uk) and v=(vk). For any two sequences u and v of either constant or strictly decreasing real numbers satisfying certain conditions the difference operator Gu,v;Δ is defined by (G(u,v;Δ)x)k=∑i=0k‍ukvi(xi-xi-1) with xk=0 for all k<0. Furthermore, we compute the spectrum and the fine spectrum of the operator Gu,v;Δ over the sequence space l1. In fact, we determine the spectrum, the point spectrum, the residual spectrum, and the continuous spectrum of this operator on the sequence space l1.


---

## Body

## 1. Introduction, Preliminaries, and Definitions

Letu=(uk) and v=(vk) be two bounded sequences of either constant or strictly decreasing positive real numbers such that uk≠0 and vk≠0 for all k, and
(1)lim⁡k→∞uk=u,(2)lim⁡k→∞|vk-vk+1vk-1-vk|=1(forstrictlydecreasingsequence).Byl1 and bvp, we denote the spaces of all absolutely summable and p-bounded variation series, respectively. Also, by l∞, c, and c0, 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 G(u,v;Δ) as follows.Letx=(xk) be any sequence in l1, and we define the weighted mean difference transform G(u,v;Δ)x of x by
(3)(G(u,v;Δ)x)k=∑i=0kukvi(xi-xi-1)(k∈N0),
where N0 denotes the set of nonnegative integers and we assume throughout that any term with negative subscript is zero. Instead of writing (3), the operator G(u,v;Δ) can be expressed as a lower triangular matrix (gnk), where
(4)gnk={ukvk,(k=n),uk(vk-vk+1),(0≤k≤n-1),0,(k>0).
Equivalently, in componentwise the triangle (gnk) can be represented by(5)(gnk)=(u0v00000⋯u1(v0-v1)u1v1000⋯u2(v0-v1)u2(v1-v2)u2v200⋯u3(v0-v1)u3(v1-v2)u3(v2-v3)u3v30⋯⋮⋮⋮⋮⋮⋱).The main objective of this paper is to determine the spectrum of the operatorG(u,v;Δ) over the basic sequence space l1. The operator G(u,v;Δ) has been studied by Polat et al. [1] in detail by introducing the difference sequence spaces l∞(u,v;Δ), c0(u,v;Δ), and c(u,v;Δ). 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 Δ over the sequence spaces lp for 0<p<1 and c, c0 has been studied by Altay and Başar [3, 4]. Kayaduman and Furkan [5] have determined the fine spectrum of the difference operator Δ over the sequence spaces l1 and bvp and on generalizing these results, Srivastava and Kumar [6, 7] have determined the fine spectrum of the operator Δν over the sequence spaces l1 and c0, where (νk) is a sequence of either constant or strictly deceasing sequence of reals satisfying certain conditions. Dutta and Baliarsingh [8–10] have computed the spectrum of the operator Δνr (r∈N0) and Δ2 over the sequence spaces l1, c0, and c0, respectively. The fine spectrum of the generalized difference operators B(r,s) and B(r,s,t) over the sequence spaces lp, bvp and l1, bv 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 [13–15] and many others.LetX and Y be Banach spaces and T:X→Y be a bounded linear operator. By R(T), we denote the range of T; that is,
(6)R(T)={y∈Y:y=Tx;x∈X}.ByB(X), we denote the set of all bounded linear operators on X into itself. If X is any Banach space and T∈B(X) then theadjointT* of T is a bounded linear operator on the dual X* of X defined by (T*ϕ)(x)=ϕ(Tx) for all ϕ∈X* and x∈X with ∥T∥=∥T*∥.LetX≠{0} be a normed linear space over the complex field and T:D(T)→X be a linear operator, where D(T) denotes the domain of T. With T, for a complex number λ, we associate an operator Tλ=(T-λI), where I is called identity operator on D(T) and if Tλ has an inverse, we denote it by Tλ-1, that is,
(7)Tλ-1=(T-λI)-1
and is called theresolvent operator of T. Many properties of Tλ and Tλ-1 depend on λ and the spectral theory is concerned with those properties. We are interested in the set of all λ’s in the complex plane such that Tλ-1 exists/Tλ-1 is bounded/domain of Tλ-1 is dense in X. Now, we state the following results which are essential for our investigation.Definition 1 (see [16, page 371]).
LetX≠{0} be normed linear space over the complex field and T:D(T)→X be a linear operator with domain D(T)⊆X. A regular value of T is a complex number λ such that (R1)
Tλ-1 exists;(R2)
Tλ-1 is bounded;(R3)
Tλ-1 is defined on a set which is dense in X.Theresolvent set ρ(T,X) of T is the set of allregular values of T. Its complement σ(T,X)=C∖ρ(T,X) in the complex plane C is called thespectrum of T. Furthermore, the spectrum ρ(T,X) is partitioned into three disjoint sets as follows.(I)
Point Spectrumσp(T,X). It is the set of all λ∈C such that (R1) does not hold. The elements of σp(T,X) are called eigen values of T.(II)
Continuous Spectrumσc(T,X). It is the set of all λ∈C such that (R1) holds and satisfies (R3) but does not satisfy (R2).(III)
Residual Spectrumσr(T,X). It is the set of all λ∈C such that (R1) holds but does not satisfy (R3). Condition (R2) may or may not hold.Goldberg’s Classification of the Operator
T
λ
(see [17], page 58–71). Let X be a Banach space and Tλ=(T-λI)∈B(X), where λ is a complex number. Again, let R(Tλ) and Tλ-1 denote the range and inverse of the operator Tλ, respectively. Then the following possibilities may occur: (A)
R(Tλ)=X;(B)
R(Tλ)≠R(Tλ)¯=X;(C)
R(Tλ)¯≠Xand(1)
Tλ is injective and Tλ-1 is continuous;(2)
Tλ is injective and Tλ-1 is discontinuous;(3)
Tλ is not injective.Taking permutations (A), (B), (C) and (1), (2), (3), we get nine different states. These are labelled byA1, A2, A3, B1, B2, B3, C1, C2, and C3. If λ is a complex number such that Tλ∈A1 or Tλ∈B1, then λ is in the resolvent set ρ(T,X) of T on X. The other classifications give rise to the fine spectrum of T. We use λ∈B2σ(T,X) which means the operator Tλ∈B2, that is, R(Tλ)≠R(Tλ)¯=X and Tλ is injective but Tλ-1 is discontinuous; similarly are the others.Lemma 2 (see [17, page 59]).
A linear operatorT has a dense range if and only if the adjoint T* is one to one.Lemma 3 (see [17, page 60]).
The adjoint operatorT* is onto if and and only if T has a bounded inverse.LetP, Q be two nonempty subsets of the space w of all real or complex sequences and let A=(ank) be an infinite matrix of complex numbers ank, where n,k∈N0. For every x=(xk)∈P, we write
(8)An(x)=∑kankxk(n∈N).
The sequence Ax=(An(x)), if it exists, is called the transformation of x by the matrix A. Infinite matrix A∈(P,Q) if and only if Ax∈Q whenever x∈P.Lemma 4 (see [18, page 126]).
The matrixA=(ank) gives rise to a bounded linear operator T∈B(l1) from l1 to itself if and only if the supremum l1 norms of the columns of A is bounded.Lemma 5 (see [19], Theorem 2).
Ifak(0)≠0 for all k∈N0, then the inverse of the difference operator B(a[m]) is given by a lower triangular Toeplitz matrix C=(cnk) as follows:(9)cnk={1an(0),(k=n),(-1)n-k∏j=knaj(0)×Dn-k(k)(a[m]),(0≤k≤n-1),0,(k>n),kkkkkkkkkkkkkkkkkkkkk(n,k∈N0),
where (B(a[m])x)k=ak(0)xk+ak-1(1)xk-1+⋯+ak-m(m)xk-m, (k∈N0) and(10)Dn(k)(a[m])=|ak(1)ak+1(0)0⋯00⋯0ak(2)ak+1(1)ak+2(0)⋯00⋯0⋮⋮⋮⋮⋮⋮⋮⋮ak(m)ak+1(m-1)ak+2(m-2)⋯am-1(1)am(0)⋯00ak+1(m)ak+2(m-1)⋯am-1(2)am(1)⋯0⋮⋮⋮⋮⋮⋮⋱⋮00⋯an-m(m)⋯⋯⋯an-1(1)|.


## 2. The Spectrum of the OperatorG(u,v;Δ) over the Sequence l1

In this section, we compute the spectrum, the point spectrum, the continuous spectrum, and the residual spectrum of the difference matrixG(u,v;Δ) on the sequence space l1.Theorem 6.
The operatorG(u,v;Δ):l1→l1 is a linear operator and
(11)∥G(u,v;Δ)∥(l1:l1)=sup⁡n|un|(∑k=0n-1‍|vk-vk+1|+|vn|).Proof.
Proof of this theorem follows from Lemma4 and in particular cases. (i)
If(uk) is a strictly decreasing sequence and (vk) is a constant sequence of positive reals (say vk=v for all k), then
(12)∥G(u,v;Δ)∥(l1:l1)=u0v.(ii)
If(uk) and (vk) are strictly decreasing sequences of positive reals, then
(13)∥G(u,v;Δ)∥(l1:l1)=u0v0.(iii)
If(uk) and (vk) are constant sequences of positive reals, then
(14)∥G(u,v;Δ)∥(l1:l1)=uv.(iv)
If(uk) is a constant sequence and (vk) is a strictly decreasing sequence of positive reals, then
(15)∥G(u,v;Δ)∥(l1:l1)=uv0.Theorem 7.
The spectrum ofG(u,v;Δ) over the sequence space l1 is given by
(16)σ(G(u,v;Δ),l1)={λ∈C:|λ-ukvk+1λ-ukvk|≥1;(k∈N0)}.Proof.
The proof is divided into four parts as follows.
Let(uk) and (vk) be two bounded sequences of positive reals satisfying (1) and (2) and λ∈C such that |(λ-ukvk+1)/(λ-ukvk)|<1 for all k∈N0; then (G(u,v;Δ)-λI) is a triangle and hence has an inverse, that is, (17)(G(u,v;Δ)-λI)-1=(1u0v0-λ000⋯-u1(v0-v1)∏j=01(ujvj-λ)1u1v1-λ00⋯u2(v0-v1)(λ-u1v2)∏j=02(ujvj-λ)-u2(v1-v2)∏j=12(ujvj-λ)1u2v2-λ0⋯-u3(v0-v1)(λ-u1v2)(λ-u2v3)∏j=03(ujvj-λ)u3(v1-v2)(λ-u2v3)∏j=13(ujvj-λ)-u3(v2-v3)∏j=23(ujvj-λ)1u3v3-λ⋯⋮⋮⋮⋮⋱).In general, by using Lemma 5, we observe that(18)Dn(k)(u,v,λ)=|uk+1(vk-vk+1)(uk+1vk+1-λ)0⋯0uk+2(vk-vk+1)uk+2(vk+1-vk+2)(uk+2vk+2-λ)⋯0uk+3(vk-vk+1)uk+3(vk+1-vk+2)uk+3(vk+2-vk+3)⋯0⋮⋮⋮⋱⋮un(vk-vk+1)un(vk+1-vk+2)un(vk+2-vk+3)⋯un(vn-1-vn)|=(vk-vk+1)(vk+1-vk+2)⋯un(vn-1-vn)×(uk+1-(uk+1vk+1-λ)(vk+1-vk+2))×(uk+2-(uk+2vk+2-λ)(vk+2-vk+3))×⋯×(un-1-(un-1vn-1-λ)(vn-1-vn)).Therefore, for all n=k, ((G(u,v;Δ)-λI)-1)n,k=1/(unvn-λ), for all k>n, ((G(u,v;Δ)-λI)-1)n,k=0, and for all 0≤k<n one can deduce that
(19)((G(u,v;Δ)-λI)-1)n,k=(vk-vk+1)un(ukvk-λ)(unvn-λ)∏j=k+1n-1(λ-ujvj+1)(ujvj-λ).
By using Lemma 4, we obtain that
(20)∥(G(u,v;Δ)-λI)-1∥(l1:l1)=sup⁡n∑k=0n‍|((G(u,v;Δ)-λI)-1)n,k|≤sup⁡n(|1unvn-λ|+∑k=0n-1‍|(vk-vk+1)un(ukvk-λ)(unvn-λ)jjjjjjjjjjjjjjjjjjjhhhhhh×∏j=k+1n-1‍(λ-ujvj+1)(ujvj-λ)|)=sup⁡n|1unvn-λ|(1+∑k=0n-1‍|(vk-vk+1)un(ukvk-λ)|hhhhhhhhhhhhgggg×∏j=k+1n-1‍|(λ-ujvj+1)(ujvj-λ)|)≤sup⁡n|ununvn-λ|(1|un|+∑k=0n-1‍|(vk-vk+1)(ukvk-λ)|∏j=k+1n-1‍|rλ|)≤sup⁡n|Qλununvn-λ|(1|Qλun|+1|rλ|∑k=0n-1‍|rλ|n-k-1)=sup⁡n|Qλununvn-λ|[1|Qλun|+1|rλ|(1-|rλ|n-11-|rλ|)]<∞.
For simplicity, we write Qλ=sup⁡k((vk-vk+1)/(ukvk-λ)) and rλ=sup⁡j((λ-ujvj+1)/(ujvj-λ)). Since (uk) and (vk) are two bounded sequences of positive reals, the quantity Qλ is finite. As per the assumption, λ∈C and |(λ-ukvk+1)/(λ-ukvk)|<1 for all k∈N0. Therefore, |rλ|<1. Now, we consider the four possible cases of the sequences (uk) and (vk).
Case  1. If (uk) and (vk) are strictly decreasing sequences of positive reals, then from the above relation we have (G(u,v;Δ)-λI)-1∈(l1:l1). Thus,
(21)σ(G(u,v;Δ),l1)⊆{λ∈C:|λ-ukvk+1λ-ukvk|≥1;(k∈N0)}.
Conversely, consider λ≠ukvk and |(λ-ukvk+1)/(λ-ukvk)|≥1 for all k∈N0, and clearly (G(u,v;Δ)-λI) is a triangle and hence (G(u,v;Δ)-λI)-1 exists, but ∥(G(u,v;Δ)-λI)-1∥(l1:l1) is not finite. Again, for λ=ukvk, the matrix (G(u,v;Δ)-ukvkI)(k∈N0) is not invertible. Thus,
(22){λ∈C:|λ-ukvk+1λ-ukvk|≥1;(k∈N0)}⊆σ(G(u,v;Δ),l1).
Combining (21) and (22) we complete the proof for this case.
Case  2. If (uk) is a strictly decreasing sequence and (vk) is a constant sequence of positive reals, then
(23)(G(u,v;Δ)-λI)-1=(1u0v-λ000⋯01u1v-λ00⋯001u2v-λ0⋯0001u3v-λ⋯⋮⋮⋮⋮⋱).
Since Q=0, ∥(G(u,v;Δ)-λI)-1∥(l1:l1)=sup⁡n|1/(unv-λ)|<∞ and the rest part of the proof is similar to that of Case 1.
Case  3. If (uk) and (vk) are constant sequences of positive reals, then the proof is similar to that of Case 2 and only the difference is ∥(G(u,v;Δ)-λI)-1∥(l1:l1)=|1/uv-λ|<∞.
Case  4. If (uk) is constant sequence and (vk) is strictly decreasing sequence of positive reals, then the proof is similar to that of Case 1.Theorem 8.
The point spectrum of the operatorG(u,v;Δ) over l1 is given by
(24)σp(G(u,v;Δ),l1)={ukv,(if(vk)isaconstantsequence)∅,(otherwise).Proof.
Supposex∈l1 and consider the system of linear equations G(u,v;Δ)x=λx for x≠0={0,0,0,…} in l1,
(25)u0v0x0=λx0u1(v0-v1)x0+u1v1x1=λx1u2(v0-v1)x0+u2(v1-v2)x1+u2v2x2=λx2⋮un(v0-v1)x0+un(v1-v2)x1+⋯+unvnxn=λxn⋮
On solving the system of (25), we obtain that λ=u0v0 whenever x0≠0 and
(26)x1=u1(v0-v1)u0v0-u1v1x0x2=u2(v0-v1)(u0v0-u1v2)(u0v0-u1v1)(u0v0-u2v2)x0andingeneralxn=un(v0-v1)u0v0-unvn∏j=1n-1u0v0-ujvj+1u0v0-ujvjx0.
Now, we have the following cases.
Case  1. If (uk) and (vk) are strictly decreasing sequences of positive reals, then we need to show that |(u0v0-ujvj+1)/(u0v0-ujvj)|≥1. Suppose, for the contrary, if |(u0v0-ujvj+1)/(u0v0-ujvj)|<1, then vj+1>vj which contradicts the fact that (vk) is strictly decreasing sequence. Therefore, x∉l1 and hence σp(G(u,v;Δ),l1)=∅.
Case  2. If (uk) is a strictly decreasing and (vk) is a constant sequence of positive reals, then for all k∈N0, λ=ukv is an eigen value corresponding to the eigen vector x={0,0,…,0,1,0,…} whose kth entry is 1. Thus, σp(G(u,v;Δ),l1)=ukv.
Case  3. If (uk) and (vk) are constant sequences of positive reals, then the proof is similar to that of Case 2 and σp(G(u,v;Δ),l1)=uv.
Case  4. If (uk) is a constant and (vk) 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(G(u,v;Δ))* over the sequence space l1*≊l∞ is given by(27)σp((G(u,v;Δ))*,l∞)={ukv,(if(vk)isaconstantsequence){λ∈C:|λ-ukvk+1λ-ukvk|>1;(k∈N0)},(otherwise).Proof.
Consider(G(u,v;Δ))*f=λf and 0≠f∈l1, and then the system of equations
(28)u0v0f0+(v0-v1)u1f1+(v0-v1)u2f2+⋯=λf0u1v1f1+(v1-v2)u2f2+(v1-v2)u3f3+⋯=λf1u2v2f2+(v2-v3)u3f3+(v2-v3)u4f4+⋯=λf2⋮ukvkfk+(vk-vk+1)uk+1fk+1+(vk-vk+1)uk+1fk+1+⋯=λfk⋮
From the system of linear equations (28), we obtain that
(29)fk-1=vk-1-vkλ-uk-1vk-1(ukfk+uk+1fk+1+uk+2fk+2+⋯),(30)fk=vk-vk+1λ-ukvk(uk+1fk+1+uk+2fk+2+uk+3fk+3+⋯)jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj(k≥1).
Combining (29) and (30), we have
(31)fk=vk-vk+1λ-ukvk((λ-uk-1vk-1)(vk-1-vk)fk-1-ukfk).
Therefore, by using ratio text, it is observed that
(32)lim⁡k→∞|fkfk-1|=lim⁡k→∞|(vk-vk+1)(λ-uk-1vk-1)(vk-1-vk)(λ-ukvk+1)|.
Now, we have the following cases.
Case  1. If (uk) and (vk) are strictly decreasing sequences of positive reals, then by using (2), we conclude that lim⁡k→∞|fk/fk-1|<1 if and only if |(λ-uk-1vk-1)/(λ-ukvk+1)|<1 for all k≥1. Therefore, σp((G(u,v;Δ))*,l∞)={λ∈C:|(λ-ukvk+1)/(λ-ukvk)|>1;(k∈N0)}.
Case  2. If (uk) is a strictly decreasing and (vk) is a constant sequence of positive reals, then from the system of (28), we observe that for all k∈N0, λ=ukv is an eigen value corresponding to the eigen vector f={0,0,…,0,1,0,…} whose kth entry is 1. Thus, σp((G(u,v;Δ))*,l∞)=ukv.
Case  3. If (uk) and (vk) are constant sequences of positive reals, then the proof is similar to that of Case 2.
Case  4. If (uk) is a constant and (vk) 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 operatorG(u,v;Δ) over the sequence space l1 is given by(33)σr(G(u,v;Δ),l1)={∅,(if(vk)isaconstantsequence){λ∈C:|λ-ukvk+1λ-ukvk|>1;(k∈N0)},(otherwise).Proof.
Proof follows from Lemma2 and Theorems 8 and 9.Theorem 11.
The continuous spectrum of the operatorG(u,v;Δ) over the sequence space l1 is given by(34)σc(G(u,v;Δ),l1)={∅,(if(vk)isaconstantsequence){λ∈C:|λ-ukvk+1λ-ukvk|=1;(k∈N0)},(otherwise).Proof.
The proof of this theorem follows from Theorems7, 8, and 10 and along with the fact that
(35)σ(G(u,v;Δ),l1)=σp(G(u,v;Δ),l1)∪σr(G(u,v;Δ),l1)∪σc(G(u,v;Δ),l1).


## 3. Conclusion

In this work, the authors have determined the spectrum of the generalized weighted mean difference operatorG(u,v;Δ) over the Banach space l1.



---

*Source: 786437-2014-06-16.xml*