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El-Shabrawy Journal of Inequalities and Applications 2014, 2014:241
http://www.journalofinequalitiesandapplications.com/content/2014/1/241
RESEARCH
Open Access
Spectra and fine spectra of certain lower
triangular double-band matrices as operators
on c
Saad R El-Shabrawy*
*
Correspondence:
[email protected]
Mathematics Department, Faculty
of Science, Damietta University,
New Damietta, Egypt
Abstract
In this paper we determine the fine spectrum of the generalized dierence operator
a,b defined by a lower triangular double-band matrix over the sequence space c0 .
The class of the operator a,b contains as special cases many operators that have
been studied recently in the literature. Illustrative examples showing the advantage of
the present results are also given.
MSC: Primary 47A10; secondary 47B37
Keywords: spectrum of an operator; infinite lower triangular matrices; sequence
spaces
1 Introduction
Several authors have studied the spectrum and fine spectrum of linear operators defined
by lower and upper triangular matrices over some sequence spaces [–].
Let X be a Banach space. By R(T), T ∗ , X ∗ , B(X), σ (T, X), σp (T, X), σr (T, X) and σc (T, X),
we denote the range of T, the adjoint operator of T, the space of all continuous linear
functionals on X, the space of all bounded linear operators on X into itself, the spectrum of
T on X, the point spectrum of T on X, the residual spectrum of T on X and the continuous
spectrum of T on X, respectively. We shall write c and c for the spaces of all convergent
and null sequences, respectively. Also by l we denote the space of all absolutely summable
sequences.
We assume here some familiarity with basic concepts of spectral theory and we refer to
Kreyszig [, pp.-] for basic definitions such as spectrum, point spectrum, residual
spectrum, and continuous spectrum of linear operators in normed spaces. Also, we refer
to Goldberg [, pp.-] for Goldberg?s classification of spectra.
Now, let (ak ) and (bk ) be two convergent sequences of nonzero real numbers with
lim ak = a
k→∞
and
lim bk = b =.
()
k→∞
We consider the operator a,b : c → c , which is defined as follows:
a,b x = a,b (xk ) = (ak xk + bk– xk– )∞
k=
with x– = b– =.
()
©2014 El-Shabrawy; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
El-Shabrawy Journal of Inequalities and Applications 2014, 2014:241
http://www.journalofinequalitiesandapplications.com/content/2014/1/241
Page 2 of 9
It is easy to verify that the operator a,b can be represented by a lower triangular doubleband matrix of the form
⎛
a
⎜b
⎜ 
a,b = ⎜
⎜
⎝
..
.

a
b
..
.


a
..
.
⎞
···
· · ·⎟
⎟
⎟
· · ·⎟ .
⎠
..
.
()
We begin by determining when a matrix A induces a bounded linear operator from c
to itself.
Lemma . (cf. [, ])
p.
The matrix A = (ank ) gives rise to a bounded linear operator
T ∈ B(c ) from c to itself if and only if
() the rows of A are in l and their l norms are bounded,
() the columns of A are in c .
The operator norm of T is the supremum of the l norms of the rows.
As a consequence of the above lemma, we have the following corollary for the bounded
linearity of the operator a,b on the space c .
Corollary . The operator a,b : c
a,b c = supk (|ak | + |bk– |).
c is a bounded linear operator with the norm
The rest of the paper is organized as follows. In Section , we analyze the spectrum of the
operator a,b on the sequence space c . In Section  we give some illustrative examples.
Finally, Section  concludes with remarks and some special cases.
2 Fine spectrum of the operator a,b on c0
In this section we examine the spectrum, the point spectrum, the residual spectrum and
the continuous spectrum of the operator a,b on the sequence space c .
Theorem .
Let D = {λ ∈ C : |λ – a| ≤ |b|} and E = {ak : k ∈ N, |ak – a| > |b|}. Then
σ (a,b , c ) = D ∪ E.
Proof First, we prove that (a,b – λI)– exists and is in B(c ) for λ ∈/ D ∪ E and then the
operator a,b – λI is not invertible for λ ∈ D ∪ E.
Let λ ∈/ D ∪ E. Then |λ – a| > |b| and λ = ak , for all k ∈ N. So, (a,b – λI) is triangle and
hence (a,b – λI)– exists. We can calculate that
⎛

(a –λ)
⎜
⎜
–b
⎜ (a –λ)(a  –λ)
(a,b – λI)– = (skj ) = ⎜
⎜
b b
⎜ (a –λ)(a–λ)(a
 –λ)
⎝
..
.
Now, for each k ∈ N, the series Sk =
that supk Sk is finite.



(a –λ)
–b
(a –λ)(a –λ)

..
.

(a –λ)
..
.
⎞
···
⎟
· · ·⎟
⎟
⎟.
· · ·⎟
⎟
⎠
..
.
j |skj | is convergent since it is finite. Next, we prove
El-Shabrawy Journal of Inequalities and Applications 2014, 2014:241
http://www.journalofinequalitiesandapplications.com/content/2014/1/241
Page 3 of 9
b
k
k
Since limk→∞ | a b–λ
| = | a–λ
| < , then there exist k ∈ N and q < such that | a b–λ
| < q ,
k
k
for all k ≥ k . Then, for each k > k ,
Sk =
|bk– |
|bk– ||bk– |

+
+
|ak – λ|
|ak– – λ| |ak– – λ||ak– – λ|
+ ··· +
|bk– ||bk– | · · · |bk |
|ak– – λ||ak– – λ| · · · |ak – λ|
|bk– ||bk– | · · · |bk + ||bk | · · · |b |
|ak– – λ||ak– – λ| · · · |ak + – λ||ak – λ| · · · |a – λ|

|bk – |
|bk – ||bk – |
k–k
k–k
k–k
≤
 + q + q + · · · + q  + q 
+ q 
|ak – λ|
|ak – – λ|
|ak – – λ||ak – – λ|
|bk – ||bk – | · · · |b |
k–k
.
+ · · · + q 
|ak – – λ||ak – – λ| · · · |a – λ|
+ ··· +
Therefore
Sk ≤
 k–k
 + q + q + · · · + q  mk ,
|ak – λ|
where
mk = +
|bk – ||bk – |
|bk – ||bk – | · · · |b |
|bk – |
+
+ ··· +
.
|ak – – λ| |ak – – λ||ak – – λ|
|ak – – λ||ak – – λ| · · · |a – λ|
Then mk > and so
Sk ≤
mk k–k  + q + q + · · · + q  .
|ak – λ|
But there exist k ∈ N and a real number q <
Sk ≤

|b|
such that

|ak –λ|
< q , for all k ≥ k . Then
q mk
,
 – q
for all k > max{k , k }. Thus supk Sk < ∞.
Also, it is easy to see that limk→∞ |skj | = , for all j ∈ N, since
sk+,j bk b <.
=
= lim
lim
k→∞ sk,j k→∞ ak+ – λ a – λ So, the sequence (sj , sj , sj , . . .) converges to zero, for each j ∈ N. This shows that the
columns of (a,b – λI)– are in c . Then, from Lemma ., (a,b – λI)– ∈ B(c ) and so,
λ ∈/ σ (a,b , c ). Thus σ (a,b , c ) ⊆ D ∪ E.
Conversely, suppose that λ ∈/ σ (a,b , c ). Then (a,b – λI)– ∈ B(c ). Since the (a,b –
λI)– transform of the unite sequence e = (,,, . . .) is in
c , we have limk→∞ | a bk–λ | =
k+
b
| a–λ
| ≤ and λ = ak , for all k ∈ N. Then {λ ∈ C : |λ – a| < |b|} σ (a,b , c ) and {ak : k ∈
N} ⊆ σ (a,b , c ). But σ (a,b , c ) is a compact set, and so it is closed. Then D = {λ ∈ C :
|λ – a| ≤ |b|} σ (a,b , c ) and E = {ak : k ∈ N, |ak – a| > |b|} σ (a,b , c ). This completes
the proof.
El-Shabrawy Journal of Inequalities and Applications 2014, 2014:241
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Theorem .
Page 4 of 9
σp (a,b , c ) = E ∪ K , where
∞
bi–
K = aj : j ∈ N, |aj – a| = |b|,
diverges to zero for some m ∈ N .
a – ai
i=m j
Proof Suppose a,b x = λx for any x ∈ c . Then we obtain
(a – λ)x =
and
for all k ∈ N.
bk xk + (ak+ – λ)xk+ = ,
If the sequence (ak ) is constant, then we can easily see that x = θ and so, σp (a,b , c ) = ∅
and the result follows immediately. Now, if the sequence (ak ) is not constant, then for all
λ ∈/ {ak : k ∈ N}, we have xk = for all
k ∈ N. So, λ ∈/ σp (a,b , c ). Also, we can easily prove
that a ∈/ σp (a,b , c ). Thus σp (a,b , c ) ⊆ {ak : k ∈ N}\{a}. Now, we will prove that
λ ∈ σp (a,b , c )
if and only if
λ ∈ E ∪ K.
If λ ∈ σp (a,b , c ), then λ = aj = a for some j ∈ N and there exists x ∈ c , x = θ such that
a,b x = aj x. Then
xk+ b ≤ .
lim
=
k→∞ xk a – aj Then λ = aj ∈ E or |aj – a| = |b|. In the case when |aj – a| = |b|, we have
xk =
Then
k
bk– bk– · · · bm–
bi–
xm– = xm–
,
(aj – ak )(aj – ak– ) · · · (aj – am )
a – ai
i=m j
∞
bi–
i=m aj –ai
k ≥ m.
diverges to , since x ∈ c . Therefore λ ∈ E ∪ K . Thus σp (a,b , c ) ⊆ E ∪ K .
Conversely, let λ ∈ E ∪ K . If λ ∈ E, then there exists i ∈ N such that λ = ai = a and so we
can take x = θ such that a,b x = ai x and
xk+ b =
<,
lim k→∞ xk a – ai that is, x ∈ c . Also, if λ ∈ K , then there exists j ∈ N such that λ = aj = a and |aj – a| = |b|,
∞ bi–
i=m aj –ai diverges to , for some m ∈ N. Then we can take x ∈ c , x = θ such that a,b x =
aj x. Thus E ∪ K ⊆ σp (a,b , c ). This completes the proof.
Theorem .
σp (∗a,b , c∗ ) = {λ ∈ C : |λ – a| < |b|}
E ∪ H, where
k
∞ λ – ai H = λ ∈ C : |λ – a| = |b|,
<∞ .
bi k= i=
El-Shabrawy Journal of Inequalities and Applications 2014, 2014:241
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Page 5 of 9
Proof Suppose that ∗a,b f = λf for f = (f , f , f , . . .) = θ in c∗ ∼
= l . Then, by solving the system of linear equations
a f + b f = λf ,
a f + b f = λf ,
a f + b f = λf ,
..
.
we obtain
λ – ak
fk ,
bk
fk+ =
k ∈ N.
Then we must take f = since otherwise we would have f = θ . It is clear that for all k ∈ N,
the vector f = (f , f , . . . , fk ,,, . . .) is an eigenvector of the operator
∗a,b corresponding to
n–
the eigenvalue λ = ak , where f = and fn = λ–a
f , for all n =,,, . . . , k
. Then {ak : k ∈
bn– n–
∞
∗
∗
N} ⊆ σp (a,b , c ). Also, if λ = ak for all k ∈ N, then fk = , for all k ≥ and
k= |fk | < ∞ if
fk+
λ–a
limk→∞ | f | = | b | <. Also, if |λ – a| = |b|, we can easily see that
k
k– λ – ai (λ – a )(λ – a ) · · · (λ – ak– ) |fk | = |f | = |f | b ,
b b · · · bk–
i
i=
and so
∞
k= |fk | < ∞
if
∞
k= |
k
i=
λ–ai
|
bi
for all k ≥ ,
< ∞. This implies that H ⊆ σp (∗a,b , c∗ ). Thus
λ ∈ C : |λ – a| < |b| ∪ E ∪ H ⊆ σp ∗a,b , c∗ .
The second inclusion can be proved analogously.
The following lemma is required in the proof of the next theorem.
Lemma .
Theorem .
[, p.]
T has a dense range if and only if T ∗ is one to one.
σr (a,b , c ) = σp (∗a,b , c∗ )\σp (a,b , c ).
Proof For λ ∈ σp (∗a,b , c∗ )\σp (a,b , c ), the operator a,b – λI is one to one and hence has
an inverse. But ∗a,b – λI is not one to one. Now, Lemma . yields the fact that the range
of the operator a,b – λI is not dense in c . This implies that λ ∈ σr (a,b , c ).
Theorem .
σr (a,b , c ) = {λ ∈ C : |λ – a| < |b|}
(H\K).
Proof The proof follows immediately from Theorems ., ., and ..
Theorem .
σc (a,b , c ) = {λ ∈ C : |λ – a| = |b|}\H.
Proof Since σ (a,b , c ) is the disjoint union of the parts σp (a,b , c ), σr (a,b , c ) and
σc (a,b , c ) we must have σc (a,b , c ) = {λ ∈ C : |λ – a| = |b|}\H.
Also, we have the following result.
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Con
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equ
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in
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k→∞ a
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l
so
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ra
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lk∈N.Thu
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ra
l
l
λ∈Cw
i
th|
λ–|
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o
r
a
l
lk
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i
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on
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P
age6o
f9
El-Shabrawy Journal of Inequalities and Applications 2014, 2014:241
http://www.journalofinequalitiesandapplications.com/content/2014/1/241
Page 7 of 9
σr (a,b , c ) = λ ∈ C : |λ – | <  ,
σc (a,b , c ) = λ ∈ C : |λ – | =  .
k+
k+
Example .
Let ak = k+
and bk = k+
for all k ∈ N. Then limk→∞ ak = a = and
limk→∞ bk = b = . Similarly, as in Example ., we can prove that E = K = H = ∅ and so
σ (a,b , c ) = λ ∈ C : |λ – | ≤  ,
σp (a,b , c ) = ∅,
σr (a,b , c ) = λ ∈ C : |λ – | <  ,
σc (a,b , c ) = λ ∈ C : |λ – | =  .
Example .
relations:
Consider the sequences (ak ) and (bk ) defined by the following recurrence
a =,
a =,
b =,
b =,
ak = for k ≥ ,
k+ 
bk =
for k ≥ .
k
Therefore, limk→∞ ak = a = and limk→∞ bk = b =. Then E = {}, K = ∅ and H = {λ ∈ C :
|λ – | = }, and so
σ (a,b , c ) = λ ∈ C : |λ – | ≤  ∪ {},
σp (a,b , c ) = {},
σr (a,b , c ) = λ ∈ C : |λ – | ≤  ,
σc (a,b , c ) = ∅.
Remark .
From the above examples, we note that the spectrum of the operator a,b
on the space c may include also a finite number of points outside a region enclosed by a
circle. Also, we may have σp (a,b , c ) = ∅.
Example .
Let the sequences (ak ) and (bk ) be taken such that ak = –bk =
k ∈ N. Then we can prove that E = K = H = ∅ and so we have
σ (a,b , c ) = λ ∈ C : λ –
σp (a,b , c ) = ∅,
σr (a,b , c ) = λ ∈ C : λ –
 
,
≤
 
 
<
,
 
 
.
σc (a,b , c ) = λ ∈ C : λ – =


(k+)
,
(k+) +(k+)
El-Shabrawy Journal of Inequalities and Applications 2014, 2014:241
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Page 8 of 9
4 Remarks and some special cases
In this section we are going to give some special cases of the operator a,b which has been
studied recently. More precisely, we show that special conditions on the sequences (ak )
and (bk ) characterize certain special cases of the operator a,b .
The dierence operator : If ak =  and bk = ? for all k ∈ N, then the operator a,b
reduces to the backward dierence operator (cf. []).
The generalized dierence operator B(r, s): If ak = r and bk = s =  for all k ∈ N, then
the operator a,b reduces to the operator B(r, s) (cf. []).
The generalized dierence operator v : If ak = –bk = vk for all k ∈ N, then the operator
a,b reduces to the operator v (cf. []).
The generalized dierence operator uv : If (ak ) is a sequence of positive real numbers
such that ak =  for all k ∈ N with limk→∞ ak = U =  and (bk ) is either constant or
strictly decreasing sequence of positive real numbers with limk→∞ bk = V =  and
supk ak < U + V , then the operator a,b reduces to the operator uv (cf. []).
Remark .
If (ak ) and (bk ) are convergent sequences of nonzero real numbers such that
lim ak = a >,
()
k→∞
lim bk = b;
k→∞
|b| = a,
()
and
sup ak ≤ a,
k
bk ≤ ak
for all k ∈ N, )
(
then we can prove that H = ∅ and so we have:
σ (a,b , c ) = λ ∈ C : |λ – a| ≤ a ∪ ak : k ∈ N, |ak – a| > a ,
σp (a,b , c ) = ak : k ∈ N, |ak – a| > a ,
σp ∗a,b , c∗ = λ ∈ C : |λ – a| < a ∪ ak : k ∈ N, |ak – a| > a ,
σr (a,b , c ) = λ ∈ C : |λ – a| < a ,
σc (a,b , c ) = λ ∈ C : |λ – a| = a .
It is immediate that our new results cover a wider class of linear operators which are
represented by infinite lower triangular double-band matrices on the sequence space c .
For this reason, our study is more general and more comprehensive than the previous
work. We note that our new results in this paper improve and generalize the results which
have been stated in [, ].
Competing interests
The author declares that he has no competing interests.
Acknowledgements
I wish to express my thanks to Prof. Ali M Akhmedov, Baku State University, Faculty of Mechanics & Mathematics, Baku,
Azerbaijan, for his kind help, careful reading, and making useful comments on the earlier version of this paper. Also,
I thank the editor and the anonymous referees for their careful reading and making some useful comments which
improved the presentation of the paper.
Received: 29 November 2013 Accepted: 27 May 2014 Published: 19 Jun 2014
El-Shabrawy Journal of Inequalities and Applications 2014, 2014:241
http://www.journalofinequalitiesandapplications.com/content/2014/1/241
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10.1186/1029-242X-2014-241
Cite this article as: El-Shabrawy: Spectra and fine spectra of certain lower triangular double-band matrices as
operators on c0 . Journal of Inequalities and Applications 2014, 2014:241
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