Hindawi Publishing Corporation
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 𝐺 (𝑢, V; Δ) over the Sequence Space ℓ1
P. Baliarsingh1 and S. Dutta2
1
Department of Mathematics, School of Applied Sciences, KIIT University,
Bhubaneswar 751024, India
2
Department of Mathematics, Utkal University, Vanivihar,
Bhubaneswar 751004, India
Correspondence should be addressed to P. Baliarsingh; pb.math10@gmail.com
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.
In the present work the generalized weighted mean difference operator 𝐺 (𝑢, 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 𝐺 (𝑢, V; Δ) is defined by
𝑘
(𝐺(𝑢, V; Δ)𝑥)𝑘 = ∑𝑖=0 𝑢𝑘 V𝑖 (𝑥𝑖 − 𝑥𝑖−1 ) with 𝑥𝑘 = 0 for all 𝑘 < 0. Furthermore, we compute the spectrum and the fine spectrum of the
operator 𝐺 (𝑢, 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 .
1. Introduction, Preliminaries, and Definitions
Let 𝑢 = (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
lim 𝑢𝑘 = 𝑢,
𝑘󳨀→∞
󵄨󵄨 V − V 󵄨󵄨
󵄨
𝑘+1 󵄨󵄨
lim 󵄨󵄨󵄨 𝑘
󵄨=1
𝑘󳨀→∞ 󵄨󵄨 V𝑘−1 − V𝑘 󵄨󵄨󵄨
(1)
(for strictly decreasing sequence) .
(2)
By ℓ1 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.
Let 𝑥 = (𝑥𝑘 ) be any sequence in ℓ1 , and we define the
weighted mean difference transform 𝐺(𝑢, V; Δ)𝑥 of 𝑥 by
𝑘
(𝐺 (𝑢, V; Δ) 𝑥)𝑘 = ∑𝑢𝑘 V𝑖 (𝑥𝑖 − 𝑥𝑖−1 )
𝑖=0
(𝑘 ∈ N0 ) ,
(3)
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
𝑔𝑛𝑘
𝑢𝑘 V𝑘 ,
{
{
{
= {𝑢𝑘 (V𝑘 − V𝑘+1 ) ,
{
{
{0,
(𝑘 = 𝑛) ,
(0 ≤ 𝑘 ≤ 𝑛 − 1) ,
(4)
(𝑘 > 0) .
Equivalently, in componentwise the triangle (𝑔𝑛𝑘 ) can be
represented by
2
International Journal of Analysis
𝑢0 V0
𝑢1 (V0 − V1 )
0
𝑢1 V1
0
0
0
0
(
𝑢2 V2
0
(𝑔𝑛𝑘 ) = (𝑢2 (V0 − V1 ) 𝑢2 (V1 − V2 )
𝑢3 (V0 − V1 ) 𝑢3 (V1 − V2 ) 𝑢3 (V2 − V3 ) 𝑢3 V3
..
..
..
..
.
.
.
.
(
The main objective of this paper is to determine the spectrum of the operator 𝐺(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 Δ ] over the sequence spaces ℓ1 and 𝑐0 , where (]𝑘 ) 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 Δ𝑟] (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.
Let 𝑋 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,
R (𝑇) = {𝑦 ∈ 𝑌 : 𝑦 = 𝑇𝑥; 𝑥 ∈ 𝑋} .
0 ⋅⋅⋅ )
).
0 ⋅⋅⋅
..
. d )
(5)
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]). Let 𝑋 ≠ {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.
The resolvent set 𝜌(𝑇, 𝑋) of 𝑇 is the set of all regular
values of 𝑇. Its complement 𝜎(𝑇, 𝑋) = C \ 𝜌(𝑇, 𝑋) in the
complex plane C is called the spectrum of 𝑇. Furthermore,
the spectrum 𝜌(𝑇, 𝑋) is partitioned into three disjoint sets as
follows.
(I) Point Spectrum 𝜎𝑝 (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 Spectrum 𝜎𝑐 (𝑇, 𝑋). It is the set of all 𝜆 ∈ C
such that (R1) holds and satisfies (R3) but does not
satisfy (R2).
(III) Residual Spectrum 𝜎𝑟 (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.
(6)
By 𝐵(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
the adjoint 𝑇∗ 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∗ ‖.
Let 𝑋 ≠ {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,
𝑇𝜆−1 = (𝑇 − 𝜆𝐼)−1
0 ⋅⋅⋅
0 ⋅⋅⋅
(7)
and is called the resolvent operator of 𝑇. Many properties
of 𝑇𝜆 and 𝑇𝜆−1 depend on 𝜆 and the spectral theory is
Goldberg’s Classification of the Operator 𝑇𝜆 (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 ) ≠ 𝑋
and
(1) 𝑇𝜆 is injective and 𝑇𝜆−1 is continuous;
(2) 𝑇𝜆 is injective and 𝑇𝜆−1 is discontinuous;
(3) 𝑇𝜆 is not injective.
International Journal of Analysis
3
Taking permutations (A), (B), (C) and (1), (2), (3), we get
nine different states. These are labelled by 𝐴 1 , 𝐴 2 , 𝐴 3 , 𝐵1 , 𝐵2 ,
𝐵3 , 𝐶1 , 𝐶2 , and 𝐶3 . If 𝜆 is a complex number such that 𝑇𝜆 ∈
𝐴 1 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 ∈ 𝐵2 𝜎(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 operator 𝑇 has a dense
range if and only if the adjoint 𝑇∗ is one to one.
Lemma 3 (see [17, page 60]). The adjoint operator 𝑇∗ is onto
if and and only if 𝑇 has a bounded inverse.
Let 𝑃, 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
𝐴 𝑛 (𝑥) = ∑𝑎𝑛𝑘 𝑥𝑘
(𝑛 ∈ N) .
𝑘
Lemma 4 (see [18, page 126]). The matrix 𝐴 = (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). If 𝑎𝑘 (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:
1
,
(𝑘 = 𝑛) ,
{
{
{
𝑎
(0)
𝑛
{
{
{
{ (−1)𝑛−𝑘
{
{
{
𝑐𝑛𝑘 = { ∏𝑛𝑗=𝑘 𝑎𝑗 (0)
{
{
{
{
(𝑘)
{
{
×𝐷𝑛−𝑘
(𝑎 [𝑚]) , (0 ≤ 𝑘 ≤ 𝑛 − 1) ,
{
{
{
(𝑘 > 𝑛) ,
{0,
(𝑛, 𝑘 ∈ N0 ) ,
(8)
The sequence 𝐴𝑥 = (𝐴 𝑛 (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.
where (𝐵(𝑎[𝑚])𝑥)𝑘 = 𝑎𝑘 (0)𝑥𝑘 + 𝑎𝑘−1 (1)𝑥𝑘−1 + ⋅ ⋅ ⋅ +
𝑎𝑘−𝑚 (𝑚)𝑥𝑘−𝑚 , (𝑘 ∈ N0 ) and
󵄨󵄨 𝑎 (1)
𝑎𝑘+1 (0)
0
⋅⋅⋅
0
0
⋅⋅⋅
0 󵄨󵄨󵄨
󵄨󵄨 𝑘
󵄨󵄨
󵄨󵄨
󵄨󵄨 𝑎𝑘 (2)
𝑎𝑘+1 (1)
𝑎𝑘+2 (0)
⋅⋅⋅
0
0
⋅⋅⋅
0 󵄨󵄨󵄨󵄨
󵄨󵄨
󵄨
󵄨󵄨󵄨 .
..
..
..
..
..
..
.. 󵄨󵄨󵄨󵄨
󵄨󵄨 ..
.
.
.
.
.
.
. 󵄨󵄨󵄨
󵄨󵄨
󵄨
󵄨󵄨
󵄨
⋅⋅⋅
𝑎𝑚−1 (1) 𝑎𝑚 (0) ⋅ ⋅ ⋅
0 󵄨󵄨󵄨󵄨 .
𝐷𝑛(𝑘) (𝑎 [𝑚]) = 󵄨󵄨󵄨󵄨𝑎𝑘 (𝑚) 𝑎𝑘+1 (𝑚 − 1) 𝑎𝑘+2 (𝑚 − 2)
󵄨󵄨
󵄨󵄨
󵄨󵄨 0
⋅⋅⋅
𝑎𝑚−1 (2) 𝑎𝑚 (1) ⋅ ⋅ ⋅
0 󵄨󵄨󵄨
𝑎𝑘+1 (𝑚) 𝑎𝑘+2 (𝑚 − 1)
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨 ..
..
..
..
..
..
.. 󵄨󵄨󵄨
󵄨󵄨 .
.
.
.
.
.
d
. 󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨 0
⋅⋅⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 𝑎𝑛−1 (1)󵄨󵄨󵄨
0
⋅⋅⋅
𝑎𝑛−𝑚 (𝑚)
2. The Spectrum of the Operator 𝐺(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 matrix 𝐺(𝑢, V; Δ) on the sequence space ℓ1 .
Theorem 6. The operator 𝐺(𝑢, V; Δ) : ℓ1 󳨀→ ℓ1 is a linear
operator and
(10)
(i) If (𝑢𝑘 ) is a strictly decreasing sequence and (V𝑘 ) is a
constant sequence of positive reals (say V𝑘 = V for all
𝑘), then
‖𝐺 (𝑢, V; Δ)‖(ℓ1 :ℓ1 ) = 𝑢0 V.
(12)
(ii) If (𝑢𝑘 ) and (V𝑘 ) are strictly decreasing sequences of
positive reals, then
‖𝐺 (𝑢, V; Δ)‖(ℓ1 :ℓ1 ) = 𝑢0 V0 .
(13)
(iii) If (𝑢𝑘 ) and (V𝑘 ) are constant sequences of positive
reals, then
𝑛−1
󵄨 󵄨
󵄨
󵄨 󵄨 󵄨
‖𝐺(𝑢, V; Δ)‖(ℓ1 :ℓ1 ) = sup 󵄨󵄨󵄨𝑢𝑛 󵄨󵄨󵄨 ( ∑ 󵄨󵄨󵄨V𝑘 − V𝑘+1 󵄨󵄨󵄨 + 󵄨󵄨󵄨V𝑛 󵄨󵄨󵄨) .
𝑛
(9)
‖𝐺 (𝑢, V; Δ)‖(ℓ1 :ℓ1 ) = 𝑢V.
𝑘=0
(11)
Proof. Proof of this theorem follows from Lemma 4 and in
particular cases.
(14)
(iv) If (𝑢𝑘 ) is a constant sequence and (V𝑘 ) is a strictly
decreasing sequence of positive reals, then
‖𝐺 (𝑢, V; Δ)‖(ℓ1 :ℓ1 ) = 𝑢V0 .
(15)
4
International Journal of Analysis
Theorem 7. The spectrum of 𝐺(𝑢, V; Δ) over the sequence space
ℓ1 is given by
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 −
𝑢𝑘 V𝑘+1 )/(𝜆 − 𝑢𝑘 V𝑘 )| < 1 for all 𝑘 ∈ N0 ; then (𝐺(𝑢, V; Δ) − 𝜆𝐼)
is a triangle and hence has an inverse, that is,
󵄨󵄨 𝜆 − 𝑢 V 󵄨󵄨
󵄨
𝑘 𝑘+1 󵄨󵄨
𝜎 (𝐺 (𝑢, V; Δ) , ℓ1 ) = {𝜆 ∈ C : 󵄨󵄨󵄨
󵄨 ≥ 1; (𝑘 ∈ N0 )} .
󵄨󵄨 𝜆 − 𝑢𝑘 V𝑘 󵄨󵄨󵄨
(16)
(𝐺 (𝑢, V; Δ) − 𝜆𝐼)−1
1
0
𝑢0 V0 − 𝜆
0
0
(
(
𝑢 (V − V1 )
1
(
− 11 0
0
0
(
(
𝑢
V
∏𝑗=0 (𝑢𝑗 V𝑗 − 𝜆)
1 1−𝜆
(
(
(
=(
𝑢 (V − V2 )
𝑢2 (V0 − V1 ) (𝜆 − 𝑢1 V2 )
1
(
− 22 1
0
(
2
𝑢
V
(
(𝑢
V
−
𝜆)
(𝑢
V
−
𝜆)
∏
∏
2 2−𝜆
𝑗
𝑗
𝑗
𝑗
𝑗=0
𝑗=1
(
(
(
( 𝑢3 (V0 − V1 ) (𝜆 − 𝑢1 V2 ) (𝜆 − 𝑢2 V3 ) 𝑢3 (V1 − V2 ) (𝜆 − 𝑢2 V3 )
𝑢 (V − V3 )
1
(−
− 33 2
3
3
∏𝑗=0 (𝑢𝑗 V𝑗 − 𝜆)
∏𝑗=1 (𝑢𝑗 V𝑗 − 𝜆)
∏𝑗=2 (𝑢𝑗 V𝑗 − 𝜆) 𝑢3 V3 − 𝜆
..
..
..
..
.
.
.
.
(
⋅⋅⋅
⋅⋅⋅
)
)
)
)
)
)
)
)
).
)
)
)
)
)
)
)
)
d
)
⋅⋅⋅
⋅⋅⋅
(17)
In general, by using Lemma 5, we observe that
󵄨󵄨𝑢 (V − V ) (𝑢 V − 𝜆)
0
󵄨󵄨 𝑘+1 𝑘
𝑘+1
𝑘+1 𝑘+1
󵄨󵄨
󵄨󵄨𝑢 (V − V ) 𝑢 (V − V ) (𝑢 V − 𝜆)
󵄨󵄨 𝑘+2 𝑘
𝑘+1
𝑘+2 𝑘+1
𝑘+2
𝑘+2 𝑘+2
󵄨󵄨
󵄨󵄨
(𝑘)
𝐷𝑛 (𝑢, V, 𝜆) = 󵄨󵄨󵄨󵄨𝑢𝑘+3 (V𝑘 − V𝑘+1 ) 𝑢𝑘+3 (V𝑘+1 − V𝑘+2 ) 𝑢𝑘+3 (V𝑘+2 − V𝑘+3 )
󵄨󵄨
..
..
..
󵄨󵄨󵄨
.
.
.
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨 𝑢𝑛 (V𝑘 − V𝑘+1 ) 𝑢𝑛 (V𝑘+1 − V𝑘+2 ) 𝑢𝑛 (V𝑘+2 − V𝑘+3 )
= (V𝑘 − V𝑘+1 ) (V𝑘+1 − V𝑘+2 ) ⋅ ⋅ ⋅ 𝑢𝑛 (V𝑛−1 − V𝑛 ) × (𝑢𝑘+1 −
× (𝑢𝑘+2 −
((𝐺 (𝑢, V; Δ) − 𝜆𝐼)−1 )𝑛,𝑘
𝑛−1 (𝜆 − 𝑢 V
(V𝑘 − V𝑘+1 ) 𝑢𝑛
𝑗 𝑗+1 )
.
∏
(𝑢𝑘 V𝑘 − 𝜆) (𝑢𝑛 V𝑛 − 𝜆) 𝑗=𝑘+1 (𝑢𝑗 V𝑗 − 𝜆)
By using Lemma 4, we obtain that
0
⋅⋅⋅
0
⋅⋅⋅
0
d
..
.
⋅ ⋅ ⋅ 𝑢𝑛 (V𝑛−1 − V𝑛 )
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨
(𝑢𝑘+1 V𝑘+1 − 𝜆)
)
(V𝑘+1 − V𝑘+2 )
(𝑢𝑘+2 V𝑘+2 − 𝜆)
(𝑢 V − 𝜆)
) × ⋅ ⋅ ⋅ × (𝑢𝑛−1 − 𝑛−1 𝑛−1
).
(V𝑘+2 − V𝑘+3 )
(V𝑛−1 − V𝑛 )
Therefore, for all 𝑛 = 𝑘, ((𝐺(𝑢, V; Δ) − 𝜆𝐼)−1 )𝑛,𝑘 = 1/(𝑢𝑛 V𝑛 − 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 󵄩󵄩󵄩
󵄩(ℓ1 :ℓ1 )
󵄩
𝑛 󵄨
󵄨󵄨
󵄨
= sup ∑ 󵄨󵄨󵄨((𝐺 (𝑢, V; Δ) − 𝜆𝐼)−1 )𝑛,𝑘 󵄨󵄨󵄨
󵄨
󵄨
𝑛
𝑘=0
󵄨󵄨
󵄨󵄨 𝑛−1 󵄨󵄨󵄨
(V𝑘 − V𝑘+1 ) 𝑢𝑛
1
󵄨
󵄨󵄨
≤ sup (󵄨󵄨󵄨
󵄨󵄨 + ∑ 󵄨󵄨󵄨󵄨
󵄨󵄨 𝑢𝑛 V𝑛 − 𝜆 󵄨󵄨 𝑘=0 󵄨󵄨 (𝑢𝑘 V𝑘 − 𝜆) (𝑢𝑛 V𝑛 − 𝜆)
𝑛
󵄨
(𝜆 − 𝑢𝑗 V𝑗+1 ) 󵄨󵄨󵄨󵄨
󵄨󵄨)
×∏
󵄨󵄨
𝑗=𝑘+1 (𝑢𝑗 V𝑗 − 𝜆) 󵄨󵄨
𝑛−1
(18)
International Journal of Analysis
5
󵄨
𝑛−1 󵄨󵄨
󵄨󵄨
󵄨󵄨
1
󵄨󵄨 (V𝑘 − V𝑘+1 ) 𝑢𝑛 󵄨󵄨󵄨
󵄨
󵄨󵄨
󵄨
󵄨󵄨
(1
+
= sup 󵄨󵄨󵄨
∑
󵄨󵄨
󵄨󵄨
𝑛 󵄨󵄨 𝑢𝑛 V𝑛 − 𝜆 󵄨󵄨
󵄨 (𝑢𝑘 V𝑘 − 𝜆) 󵄨󵄨󵄨
𝑘=0 󵄨
󵄨󵄨
𝑛−1 󵄨󵄨󵄨 (𝜆 − 𝑢 V
󵄨
𝑗 𝑗+1 ) 󵄨󵄨󵄨
󵄨󵄨)
× ∏ 󵄨󵄨󵄨󵄨
󵄨󵄨
󵄨
𝑗=𝑘+1 󵄨󵄨 (𝑢𝑗 V𝑗 − 𝜆) 󵄨󵄨
󵄨
𝑛−1 󵄨󵄨
󵄨󵄨 𝑢
󵄨󵄨
1
󵄨 (V𝑘 − V𝑘+1 ) 󵄨󵄨󵄨 𝑛−1 󵄨󵄨 󵄨󵄨
󵄨
󵄨󵄨
𝑛
󵄨󵄨 ∏ 󵄨𝑟𝜆 󵄨)
≤ sup 󵄨󵄨󵄨
󵄨󵄨 ( 󵄨󵄨 󵄨󵄨 + ∑ 󵄨󵄨󵄨󵄨
𝑛 󵄨󵄨 𝑢𝑛 V𝑛 − 𝜆 󵄨󵄨
󵄨󵄨𝑢𝑛 󵄨󵄨 𝑘=0 󵄨󵄨 (𝑢𝑘 V𝑘 − 𝜆) 󵄨󵄨󵄨 𝑗=𝑘+1 󵄨 󵄨
󵄨󵄨 𝑄 𝑢 󵄨󵄨
1
1 𝑛−1󵄨󵄨 󵄨󵄨𝑛−𝑘−1
󵄨
󵄨
≤ sup 󵄨󵄨󵄨 𝜆 𝑛 󵄨󵄨󵄨 ( 󵄨󵄨
+
)
󵄨
󵄨
󵄨󵄨𝑟𝜆 󵄨󵄨󵄨 ∑ 󵄨󵄨𝑟𝜆 󵄨󵄨
𝑛 󵄨󵄨 𝑢𝑛 V𝑛 − 𝜆 󵄨󵄨 󵄨󵄨𝑄𝜆 𝑢𝑛 󵄨󵄨󵄨
󵄨 󵄨 𝑘=0
󵄨 󵄨𝑛−1
󵄨󵄨 𝑄 𝑢 󵄨󵄨
1 − 󵄨󵄨󵄨𝑟𝜆 󵄨󵄨󵄨
1
1
󵄨
󵄨
+
(
= sup 󵄨󵄨󵄨 𝜆 𝑛 󵄨󵄨󵄨 [ 󵄨󵄨
󵄨 󵄨󵄨 󵄨󵄨
󵄨 󵄨 )] < ∞.
𝑛 󵄨󵄨 𝑢𝑛 V𝑛 − 𝜆 󵄨󵄨 󵄨󵄨𝑄𝜆 𝑢𝑛 󵄨󵄨󵄨
1 − 󵄨󵄨󵄨𝑟𝜆 󵄨󵄨󵄨
󵄨󵄨𝑟𝜆 󵄨󵄨
(20)
For simplicity, we write 𝑄𝜆 = sup𝑘 ((V𝑘 − V𝑘+1 )/(𝑢𝑘 V𝑘 − 𝜆)) and
𝑟𝜆 = sup𝑗 ((𝜆 − 𝑢𝑗 V𝑗+1 )/(𝑢𝑗 V𝑗 − 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−𝑢𝑘 V𝑘+1 )/(𝜆−𝑢𝑘 V𝑘 )| < 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,
󵄨󵄨󵄨 𝜆 − 𝑢𝑘 V𝑘+1 󵄨󵄨󵄨
󵄨󵄨 ≥ 1; (𝑘 ∈ N0 )} .
𝜎 (𝐺 (𝑢, V; Δ) , ℓ1 ) ⊆ {𝜆 ∈ C : 󵄨󵄨󵄨
󵄨
󵄨󵄨 𝜆 − 𝑢𝑘 V𝑘 󵄨󵄨
(21)
Conversely, consider 𝜆 ≠ 𝑢𝑘 V𝑘 and |(𝜆−𝑢𝑘 V𝑘+1 )/(𝜆−𝑢𝑘 V𝑘 )| ≥ 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 = 𝑢𝑘 V𝑘 , the matrix (𝐺(𝑢, V; Δ) −
𝑢𝑘 V𝑘 𝐼)(𝑘 ∈ N0 ) is not invertible. Thus,
󵄨󵄨 𝜆 − 𝑢 V 󵄨󵄨
󵄨
𝑘 𝑘+1 󵄨󵄨
{𝜆 ∈ C : 󵄨󵄨󵄨
󵄨 ≥ 1; (𝑘 ∈ N0 )} ⊆ 𝜎 (𝐺 (𝑢, V; Δ) , ℓ1 ) .
󵄨󵄨 𝜆 − 𝑢𝑘 V𝑘 󵄨󵄨󵄨
(22)
Since 𝑄 = 0, ‖(𝐺(𝑢, V; Δ) − 𝜆𝐼)−1 ‖(ℓ1 :ℓ1 ) = sup𝑛 |1/(𝑢𝑛 V − 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 operator 𝐺(𝑢, V; Δ) over
ℓ1 is given by
𝑢𝑘 V,
𝜎𝑝 (𝐺 (𝑢, V; Δ) , ℓ1 ) = {
0,
(𝑖𝑓 (V𝑘 ) 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒)
(𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒) .
(24)
Proof. Suppose 𝑥 ∈ ℓ1 and consider the system of linear
equations 𝐺(𝑢, V; Δ)𝑥 = 𝜆𝑥 for 𝑥 ≠ 0 = {0, 0, 0, . . .} in ℓ1 ,
𝑢0 V0 𝑥0 = 𝜆𝑥0
𝑢1 (V0 − V1 ) 𝑥0 + 𝑢1 V1 𝑥1 = 𝜆𝑥1
𝑢2 (V0 − V1 ) 𝑥0 + 𝑢2 (V1 − V2 ) 𝑥1 + 𝑢2 V2 𝑥2 = 𝜆𝑥2
..
.
𝑢𝑛 (V0 − V1 ) 𝑥0 + 𝑢𝑛 (V1 − V2 ) 𝑥1 + ⋅ ⋅ ⋅ + 𝑢𝑛 V𝑛 𝑥𝑛 = 𝜆𝑥𝑛
..
.
(25)
On solving the system of (25), we obtain that 𝜆 = 𝑢0 V0
whenever 𝑥0 ≠ 0 and
𝑥1 =
𝑢1 (V0 − V1 )
𝑥
𝑢0 V0 − 𝑢1 V1 0
Combining (21) and (22) we complete the proof for this
case.
𝑥2 =
𝑢2 (V0 − V1 ) (𝑢0 V0 − 𝑢1 V2 )
𝑥 and in general
(𝑢0 V0 − 𝑢1 V1 ) (𝑢0 V0 − 𝑢2 V2 ) 0
Case 2. If (𝑢𝑘 ) is a strictly decreasing sequence and (V𝑘 ) is a
constant sequence of positive reals, then
𝑥𝑛 =
𝑢𝑛 (V0 − V1 ) 𝑛−1 𝑢0 V0 − 𝑢𝑗 V𝑗+1
𝑥.
∏
𝑢0 V0 − 𝑢𝑛 V𝑛 𝑗=1 𝑢0 V0 − 𝑢𝑗 V𝑗 0
(𝐺 (𝑢, V; Δ) − 𝜆𝐼)−1
1
𝑢0 V − 𝜆
(
(
(
=(
(
(
(
(
(26)
Now, we have the following cases.
0
0
0
⋅⋅⋅
0
1
𝑢1 V − 𝜆
0
0
0
0
1
𝑢2 V − 𝜆
0
0
0
0
⋅ ⋅ ⋅)
)
)
⋅ ⋅ ⋅)
).
)
)
⋅⋅⋅
..
.
..
.
..
.
1
𝑢3 V − 𝜆
..
.
d)
(23)
Case 1. If (𝑢𝑘 ) and (V𝑘 ) are strictly decreasing sequences
of positive reals, then we need to show that |(𝑢0 V0 −
𝑢𝑗 V𝑗+1 )/(𝑢0 V0 − 𝑢𝑗 V𝑗 )| ≥ 1. Suppose, for the contrary, if
|(𝑢0 V0 − 𝑢𝑗 V𝑗+1 )/(𝑢0 V0 − 𝑢𝑗 V𝑗 )| < 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 ) = 0.
Case 2. If (𝑢𝑘 ) is a strictly decreasing and (V𝑘 ) is a constant sequence of positive reals, then for all 𝑘 ∈ N0 ,
𝜆 = 𝑢𝑘 V is an eigen value corresponding to the eigen
6
International Journal of Analysis
vector 𝑥 = {0, 0, . . . , 0, 1, 0, . . .} whose 𝑘th entry is 1. Thus,
𝜎𝑝 (𝐺(𝑢, V; Δ), ℓ1 ) = 𝑢𝑘 V.
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
(𝑖𝑓 (V𝑘 ) 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒)
𝑢 V,
{
{ 𝑘
󵄨󵄨 𝜆 − 𝑢 V 󵄨󵄨
𝜎𝑝 ((𝐺 (𝑢, V; Δ))∗ , ℓ∞ ) = {
𝑘 𝑘+1 󵄨󵄨
{{𝜆 ∈ C : 󵄨󵄨󵄨
󵄨󵄨 𝜆 − 𝑢 V 󵄨󵄨󵄨 > 1; (𝑘 ∈ N0 )} , (𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒) .
󵄨
{
𝑘 𝑘 󵄨
Proof. Consider (𝐺(𝑢, V; Δ))∗ 𝑓 = 𝜆𝑓 and 0 ≠ 𝑓 ∈ ℓ1 , and then
the system of equations
V𝑘 − V𝑘+1 (𝜆 − 𝑢𝑘−1 V𝑘−1 )
(
𝑓 − 𝑢𝑘 𝑓𝑘 ) .
𝜆 − 𝑢𝑘 V𝑘
(V𝑘−1 − V𝑘 ) 𝑘−1
(31)
Therefore, by using ratio text, it is observed that
𝑢1 V1 𝑓1 + (V1 − V2 ) 𝑢2 𝑓2 + (V1 − V2 ) 𝑢3 𝑓3 + ⋅ ⋅ ⋅ = 𝜆𝑓1
󵄨󵄨 (V − V ) (𝜆 − 𝑢 V ) 󵄨󵄨
󵄨󵄨 𝑓 󵄨󵄨
󵄨
󵄨
󵄨
𝑘+1
𝑘−1 𝑘−1 󵄨󵄨
󵄨󵄨 .
lim 󵄨󵄨󵄨 𝑘 󵄨󵄨󵄨 = lim 󵄨󵄨󵄨󵄨 𝑘
𝑘󳨀→∞ 󵄨󵄨 𝑓𝑘−1 󵄨󵄨
𝑘󳨀→∞ 󵄨󵄨 (V𝑘−1 − V𝑘 ) (𝜆 − 𝑢𝑘 V𝑘+1 ) 󵄨󵄨󵄨
Now, we have the following cases.
𝑢2 V2 𝑓2 + (V2 − V3 ) 𝑢3 𝑓3 + (V2 − V3 ) 𝑢4 𝑓4 + ⋅ ⋅ ⋅ = 𝜆𝑓2
..
.
(32)
Case 1. If (𝑢𝑘 ) and (V𝑘 ) are strictly decreasing sequences
of positive reals, then by using (2), we conclude that
lim𝑘󳨀→∞ |𝑓𝑘 /𝑓𝑘−1 | < 1 if and only if |(𝜆 − 𝑢𝑘−1 V𝑘−1 )/(𝜆 −
𝑢𝑘 V𝑘+1 )| < 1 for all 𝑘 ≥ 1. Therefore, 𝜎𝑝 ((𝐺(𝑢, V; Δ))∗ , ℓ∞ ) =
{𝜆 ∈ C : |(𝜆 − 𝑢𝑘 V𝑘+1 )/(𝜆 − 𝑢𝑘 V𝑘 )| > 1; (𝑘 ∈ N0 )}.
𝑢𝑘 V𝑘 𝑓𝑘 + (V𝑘 − V𝑘+1 ) 𝑢𝑘+1 𝑓𝑘+1 + (V𝑘 − V𝑘+1 ) 𝑢𝑘+1 𝑓𝑘+1
+ ⋅ ⋅ ⋅ = 𝜆𝑓𝑘
..
.
(28)
From the system of linear equations (28), we obtain that
𝑓𝑘−1 =
Combining (29) and (30), we have
𝑓𝑘 =
𝑢0 V0 𝑓0 + (V0 − V1 ) 𝑢1 𝑓1 + (V0 − V1 ) 𝑢2 𝑓2 + ⋅ ⋅ ⋅ = 𝜆𝑓0
(27)
V𝑘−1 − V𝑘
(𝑢 𝑓 + 𝑢𝑘+1 𝑓𝑘+1 + 𝑢𝑘+2 𝑓𝑘+2 + ⋅ ⋅ ⋅ ) ,
𝜆 − 𝑢𝑘−1 V𝑘−1 𝑘 𝑘
(29)
V − V𝑘+1
(𝑢 𝑓 + 𝑢𝑘+2 𝑓𝑘+2 + 𝑢𝑘+3 𝑓𝑘+3 + ⋅ ⋅ ⋅ )
𝑓𝑘 = 𝑘
𝜆 − 𝑢𝑘 V𝑘 𝑘+1 𝑘+1
(𝑘 ≥ 1) .
(30)
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 = 𝑢𝑘 V 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∞ ) = 𝑢𝑘 V.
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 operator 𝐺(𝑢, V; Δ)
over the sequence space ℓ1 is given by
0,
{
{
󵄨󵄨
󵄨󵄨
𝜎𝑟 (𝐺 (𝑢, V; Δ) , ℓ1 ) = {
{{𝜆 ∈ C : 󵄨󵄨󵄨 𝜆 − 𝑢𝑘 V𝑘+1 󵄨󵄨󵄨 > 1; (𝑘 ∈ N0 )} ,
󵄨󵄨 𝜆 − 𝑢 V 󵄨󵄨
󵄨
{
𝑘 𝑘 󵄨
Proof. Proof follows from Lemma 2 and Theorems 8 and 9.
(𝑖𝑓 (V𝑘 ) 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒)
(𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒) .
(33)
Theorem 11. The continuous spectrum of the operator
𝐺(𝑢, V; Δ) over the sequence space ℓ1 is given by
International Journal of Analysis
7
0,
(𝑖𝑓 (V𝑘 ) 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒)
{
{
󵄨󵄨 𝜆 − 𝑢 V 󵄨󵄨
𝜎𝑐 (𝐺 (𝑢, V; Δ) , ℓ1 ) = {
𝑘 𝑘+1 󵄨󵄨
{{𝜆 ∈ C : 󵄨󵄨󵄨
󵄨󵄨 𝜆 − 𝑢 V 󵄨󵄨󵄨 = 1; (𝑘 ∈ N0 )} , (𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒) .
󵄨
{
𝑘 𝑘 󵄨
Proof. The proof of this theorem follows from Theorems 7, 8,
and 10 and along with the fact that
[10]
𝜎 (𝐺 (𝑢, V; Δ) , ℓ1 )
= 𝜎𝑝 (𝐺 (𝑢, V; Δ) , ℓ1 ) ∪ 𝜎𝑟 (𝐺 (𝑢, V; Δ) , ℓ1 )
(35)
[11]
∪ 𝜎𝑐 (𝐺 (𝑢, V; Δ) , ℓ1 ) .
3. Conclusion
In this work, the authors have determined the spectrum of
the generalized weighted mean difference operator 𝐺(𝑢, V; Δ)
over the Banach space ℓ1 .
[12]
[13]
[14]
Conflict of Interests
The authors declare that there is no conflict of interests
regarding the publication of this paper.
[15]
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