On the fine spectrum of the second order difference operator over the sequence spaces ? p and bvp,(1<p<?)

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Mathematical and Computer Modelling

journal homepage:www.elsevier.com/locate/mcm

On the fine spectrum of the second order difference operator over the sequence spaces ℓ

^p

^{and b} v

^p

, ( ¹ < ^p < ∞)

Vatan Karakaya

^a,∗

, Manaf Dzh. Manafov

^b

, Necip Şimşek

^c

aDepartment of Mathematical Engineering, Yildiz Technical University, Esenler, İstanbul, Turkey

bAdiyaman University, Faculty of Arts and Sciences, Department of Mathematics, Adıyaman, Turkey

cİstanbul Commerce University, Faculty of Arts and Sciences, Department of Mathematics, İstanbul, Turkey

a r t i c l e i n f o

Article history:

Received 26 May 2011 Accepted 17 August 2011

Keywords:

Spectrum of an operator The sequence spacesℓpand bvp

Symmetric tri-band matrix

a b s t r a c t

In general, the behaviors of the symmetric tri-band matrices on the Hilbert spaces are well known. But the symmetric tri-band matrices have different behavior on the Banach spaces.

The main purpose of this work is to determine the fine spectra of the operator U(^s,^r,^s) defined by symmetric tri-band matrix over the sequence spacesℓpand bvp.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

As it is well known, the matrices play an important role in operator theory. The spectrum of an operator generalizes the notion of eigenvalues for matrices. In the calculation of the spectrum of an operator over a Banach space, we mostly deal with three disjoint parts of the spectrum, which are the point spectrum, the continuous spectrum and the residual spectrum.

The determination of these three parts of the spectrum of an operator is called the fine spectra.

Over the years and different names the spectrum and fine spectra of linear operators defined by some particular limitation matrices over some sequence spaces have been studied.

In the existing literature, there are many papers concerning the spectrum and the fine spectra of an operator over different sequence spaces. For example, Gonzàlez [1], and Akhmedov and Başar [2] computed the fine spectra of the Cesáro operator over the sequence spaces

ℓ

pand c₀, respectively. Also Reade [3] and Okutoyi [4] examined the spectrum of the Cesáro operator over the spaces c₀and b

v

, respectively. Later on Akhmedov and Başar [5,6] also determined the spectrum of the Cesáro operator and the fine spectrum of the difference operator∆over the sequence space b

v

p

(

¹

<

^p

< ∞)

. Also, in [7], Wenger studied the fine spectra of Hölder summability operators over the space c and Rhoades [8] extended this result to the weighed mean method.

Recently, Karakaya and Altun [9,10] have computed, respectively, the fine spectra of the upper triangular double-band matrices and the lacunary matrices as an operator over the sequence spaces c₀and c, which are defined below.

Further information on the spectrum and fine spectra of different operators over some sequence spaces can be found in the list of Refs. [11–14].

The main purpose of our work is to determine the fine spectra of the operator for which the corresponding matrix is the symmetrical tri-band matrix U

(

^s

,

^r

,

^s

)

over the sequence spaces

ℓ

pand b

v

p

(

¹

<

^p

< ∞)

^.

∗Corresponding author.

E-mail addresses:vkkaya@yahoo.com,vkkaya@yildiz.edu.tr(V. Karakaya),mmanafov@adiyaman.edu.tr(M.Dzh. Manafov),necsimsek@yahoo.com (N. Şimşek).

0895-7177/$ – see front matter©2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.mcm.2011.08.021

2. Preliminaries and notations

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 , i.e., R

(

^T

) = {

^y

∈

Y

:

y

=

Tx

;

x

∈

X

} .

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 . Let X

̸= { θ}

be a complex normed space and T

:

_D

(

^T

) →

X be a linear operator with domainD

(

^T

) ⊂

X . By T , we associate the operator

T_λ

=

_T

− λ

^I

;

where

λ

is a complex number and I is the identity operator onD

(

^T

)

^{. If T}λhas an inverse, which is linear, we denote it by T_λ⁻¹, that is

T_λ⁻¹

= (

^T

− λ

^I

)

⁻¹

,

and it is called the resolvent operator of T . Many properties of T_λand T_λ⁻¹depend on

λ

, and the spectral theory is concerned with those properties. For instance, we shall interest in the set of all

λ

in the complex plane such that T_λ⁻¹exists. Boundedness of T_λ⁻¹is another property that will be essential. We shall also ask for what all

λ

in the domain of T_λ⁻¹ is dense in X . For the investigation of T , T_λ and T_λ⁻¹, we need some basic concepts in spectral theory which are given as follows (see [15, pp. 370–371]).

Let X

̸= { θ}

be a complex normed space and T

:

_D

(

^T

) →

X be a linear operator with domainD

(

^T

) ⊂

X . A regular value

λ

of T is a complex number such that

(R1) T_λ⁻¹exists, (R2) T_λ⁻¹is bounded,

(R3) T_λ⁻¹is defined on a set which is dense in X .

The resolvent set

ρ(

^T

)

of T is the set of all regular values

λ

of T . Its complement

σ (

^T

) =

C

\ ρ(

^T

)

in the complex plane C is called the spectrum of T . Furthermore, the spectrum

σ (

^T

)

is partitioned into three disjoint sets as follows. The point spectrum

σ

p

(

^T

)

is the set such that T_λ⁻¹does not exist. A

λ ∈ σ

p

(

^T

)

is called an eigenvalue of T . The continuous spectrum

σ

c

(

^T

)

is the set such that T_λ⁻¹exists and satisfies (R3) but not (R2). The residual spectrum

σ

r

(

^T

)

is the set such that T_λ⁻¹exists but not satisfy (R3).

We shall write

ℓ

^{∞, c and c}0for the spaces of all bounded, convergent and null sequences, respectively. The sequence spaces

ℓ

pand b

v

pare defined by

ℓ

p

=



x

∈ ω : −

k

|

x_k

|

^p

< ∞



b

v

p

=



x

∈ ω : −

k

|

x_k

−

x_k+1

|

^p

< ∞

 .

Let

µ

^and

γ

be two sequence spaces and A

= (

^ank

)

be an infinite matrix of real or complex numbers a_nk, where n, k

∈

_N

= {

0

,

¹

,

²

, . . .}

. Then, we say that A defines a matrix mapping from

µ

^into

γ

, and we denote it by writing A

: µ → γ

^, if for every sequence x

= (

^xk

) ∈ µ

the sequence Ax

= { (

^Ax

)

n

}

, the A-transform of x, is in

γ

^{, where}

(

^Ax

)

n

= −

k

a_nkx_k

(

ⁿ

∈

_N

).

^(2.1)

By

(µ : γ )

, we denote the class of all matrices A such that A

: µ → γ

^{. Thus, A}

∈ (µ : γ )

if and only if the series on the right side of(2.1)converges for each n

∈

N and every x

∈ µ

, and we have Ax

= { (

^Ax

)

n

}

_n∈N

∈ γ

^{for all x}

∈ µ

^.

The symmetrical tri-band matrix used in our work is of the following form:

U

(

^s

,

^r

,

^s

) =













r s 0 0

· · ·

s r s 0

· · ·

0 s r s

· · ·

0 0 s r

· · ·

· · · · ...











 .

Now let us give some lemmas which we need in the sequel.

Lemma 2.1. Define the sets D_∞and D_qby

D∞

=



x

= (

^xk

) ∈ w :

^sup

k∈N











∞

−

j=k

x_j











< ∞



and D_q

=



x

= (

^xk

) ∈ w : −

k











∞

−

j=k

x_j











q

< ∞



, (

¹

<

^q

< ∞).

Then, the sets D∞and D_qare the Banach spaces with the norms

‖

a

‖

_D∞

=

sup

k∈N











∞

−

j=k

a_j











and

‖

a

‖

_Dq

=



−

k











∞

−

j=k

a_j











q



1/q

.

Additionally,

(i) D∞is isometrically isomorphic to b

v

^∗1, [16, Theorem 3.3]

(ii) D_qis isometrically isomorphic to b

v

p^∗, [17, Theorem 2.3].

The basis of the space b

v

pis also constructed and is given by the following lemma.

Lemma 2.2 ([18, Theorem 3.1]). Define the sequence b^(k)

=



b⁽_n^k)



n∈_Nof the elements of the space b

v

pfor every fixed k

∈

_{N by} b⁽_n^k)

=



0

, (

ⁿ

<

^k

)

1

, (

ⁿ

≥

k

)



for all n

∈

_N

.

Then the sequence



b⁽_n^k)



n∈N

is a basis for the space b

v

pand any x

∈

b

v

phas a unique representation of the form

x

= −

k

λ

kb^(k)

,

where

λ

k

=

x_k

−

x_k−1for all k

∈

_N.

Lemma 2.3 ([19, p. 253, Theorem 34.16]). The matrix A

= (

^ank

)

gives rise to a bounded linear operator T

∈

B

(ℓ

1

)

^from

ℓ

1to itself if and only if the supremum of

ℓ

1norms of the columns of A is bounded.

Lemma 2.4 ([19, p. 245, Theorem 34.3]). The matrix A

= (

^ank

)

gives rise to a bounded linear operator T

∈

B

(ℓ

^∞

)

^from

ℓ

^∞to itself if and only if the supremum of

ℓ

1norms of the rows of A is bounded.

Lemma 2.5 ([19, p. 254, Theorem 34.18]). Let 1

<

^p

< ∞

^{and A}

∈ (

^l∞

,

^l∞

) ∩ (

^l1

,

^l1

)

^{. Then A}

∈ (

^lp

,

^lp

)

^. Corollary 2.6. Let

µ ∈ 

^lp

,

^b

v

p

 (

¹

<

^p

< ∞)

^{. U}

(

^s

,

^r

,

^s

) : µ → µ

is a bounded linear operator and

‖

U

(

^s

,

^r

,

^s

)‖

(µ,µ)

=

2

|

s

| + |

r

|

.

3. The spectrum of the operator U

(

^s

,

^r

,

^s

)

on the sequence space

ℓ

p,

(

¹

<

^p

< ∞)

In this section, the fine spectrum of the second order difference operator U

(

^s

,

^r

,

^s

)

over the sequence space

ℓ

p,

(

¹

<

^p

<

∞ )

have been examined. We begin with a theorem concerning the bounded linearity of the operator U

(

^s

,

^r

,

^s

)

acting on the sequence space

ℓ

p,

(

¹

<

^p

< ∞).

Theorem 3.1. U

(

^s

,

^r

,

^s

) : ℓ

p

→ ℓ

pis a bounded linear operator satisfying the inequalities

|

_r

|

^p

+

₂

|

_s

|

^p



¹_p

≤ ‖

_U

(

^s

,

^r

,

^s

)‖

_ℓp

≤

₂

|

_s

| + |

_r

| .

Proof. The linearity of U

(

^s

,

^r

,

^s

)

is trivial and so it is omitted. Let us take e⁽¹⁾

= (

⁰

,

¹

,

⁰

, . . .)

ⁱⁿ

ℓ

p. Then U

(

^s

,

^r

,

^s

)

^e(1)

= (

^s

,

^r

,

^s

,

⁰

,

⁰

, . . .)

and observe that





U

(

^s

,

^r

,

^s

)

^e(1)





_ℓ

p

= |

_r

|

^p

+

2

|

s

|

^p



¹_p

≤ ‖

U

(

^s

,

^r

,

^s

)‖

ℓp





e⁽¹⁾





_ℓ

p

which gives the fact that

|

_r

|

^p

+

2

|

s

|

^p



¹_p

≤ ‖

U

(

^s

,

^r

,

^s

)‖

_ℓp (3.1)

for any p

>

1. Now take any x

= (

^xk

) ∈ ℓ

psuch that

‖

_x

‖ =

1. Then, using Minkowski’s inequality and taking x−₁

=

_{0, we} have

‖

U

(

^s

,

^r

,

^s

)

^x

‖

_ℓ

p

=



_∞

−

k=0

|

sx_k−1

+

rx_k

+

sx_k+1

|

^p



¹_p

≤



_∞

−

k=0

|

sx_k−1

|

^p



¹_p

+



_∞

−

k=0

|

rx_k

|

^p



¹_p

+



_∞

−

k=0

|

sx_k+1

|

^p



¹_p

= (|

^r

| +

2

|

s

| ) ‖

^x

‖

_ℓp

,

which gives

‖

U

(

^s

,

^r

,

^s

)‖

ℓp

≤

2

|

s

| + |

r

| .

^(3.2)

Combining the inequalities(3.1)and(3.2)we complete the proof. _ Theorem 3.2.

σ 

^U

(

^s

,

^r

,

^s

), ℓ

p

 = [

_r

−

2s

,

^r

+

2s

]

.

Proof. First, we prove that

(

^U

(

^s

,

^r

,

^s

) − λ

^I

)

⁻¹exists and in B

(ℓ

p

)

^for

λ ̸∈ {λ ∈

^C

: λ =

^r

+

2s

.

^cos

θ, θ ∈ [

⁰

,

²

π]}

and next that the operator

(

^U

(

^s

,

^r

,

^s

) − λ

^I

)

is not invertible for

λ ∈ {λ ∈

^C

: λ =

^r

+

2s

.

^cos

θ, θ ∈ [

⁰

,

²

π]} .

Let

λ ̸∈ σ 

^U

(

^s

,

^r

,

^s

), ℓ

p



. Let

α

1and

α

2be the roots of the polynomial P

(

^x

) =

^sx2

+ (

^r

− λ)

^x

+

s

,

^with

| α

2

| >

¹

> |α

1

| .

Solving the system of equations

(

^r

− λ)

^x1

+

sx₂

=

y₁ sx₁

+ (

^r

− λ)

^x2

+

sx₃

=

y₂ sx₂

+ (

^r

− λ)

^x3

+

sx₄

=

y₃

· · ·

(3.3)

for x

= (

^xk

)

in terms of y

= (

^yk

)

gives the matrix of

(

^U

(

^s

,

^r

,

^s

) − λ

^I

)

⁻¹. This is a non-homogeneous linear recurrence relation. Using the fact that x

,

^y

∈ ℓ

p, for(3.3)we can reach to a solution with generating functions (see [20]). This solution can be given by

x_k

=

¹

s

(α

²1

−

1

)

∞

−

n=0

t_kny_n

,

^(3.4)

where t_kn

=

 α

1^k+1−n

− α

^k1⁺³⁻ⁿ

;

if k

≥

n

, α

1^n+1−k

− α

ⁿ1^+3−k

;

if k

<

ⁿ

.

Thus, we obtain that



 (

^U

(

^s

,

^r

,

^s

) − λ

^I

)

⁻¹





_(ℓ

1,ℓ1)

=

sup

k

∞

−

n=k

|

x_k

| ≤

sup

k

| α

1

| + | α

1

|

^2k+3



k

−

n=0

| α

1

|

ⁿ

< ∞,

i.e.

(

^U

(

^s

,

^r

,

^s

) − λ

^I

)

⁻¹

∈ (ℓ

1

, ℓ

1

)

. Similarly



 (

^U

(

^s

,

^r

,

^s

) − λ

^I

)

⁻¹





_(ℓ

∞,ℓ∞)

< ∞.

ByLemma 2.5, we have

(

^U

(

^s

,

^r

,

^s

) − λ

^I

)

⁻¹

∈ (

^lp

,

^lp

)

. This shows that

σ 

^U

(

^s

,

^r

,

^s

), ℓ

p

 ⊆ { λ ∈

^C

: λ =

^r

+

2s

.

^cos

θ, θ ∈ [

0

,

²

π]}

^.

Let

λ ∈ σ 

^U

(

^s

,

^r

,

^s

), ℓ

p



and

λ ̸=

^{r. Then}

(

^U

(

^s

,

^r

,

^s

) − λ

^I

)

⁻¹ exists but y

= (

¹

,

⁰

,

⁰

, . . .) ∈

^lpand x

= (

^xk

)

^not in l_p, hence

| α

2

| >

¹

> |α

1

|

is not satisfied, i.e.

(

^U

(

^s

,

^r

,

^s

) − λ

^I

)

⁻¹ is not in B

(ℓ

p

)

^{. If}

λ =

^{r, then U}

(

^s

,

^r

,

^s

) − λ

^I

=

U

(

^s

,

⁰

,

^s

)

^{. Since U}

(

^s

,

⁰

,

^s

)

^x

= θ

^{implies x}

̸= θ = (

⁰

,

⁰

,

⁰

, . . .)

^{, U}

(

^s

,

^r

,

^s

) : ℓ

p

→ ℓ

pis not invertible. This shows that

{ λ ∈

^C

: λ =

^r

+

2s

.

^cos

θ, θ ∈ [

⁰

,

²

π]} ⊆ σ 

^U

(

^s

,

^r

,

^s

), ℓ

p



. On the other hand,

α

1

.α

2

=

1,

| α

2

| >

¹

> |α

1

|

is not satisfied means, the roots can be only of the form

α

1

=

¹

α

2

=

e^iθ

for some

θ ∈ [

⁰

,

²

π)

^{. Thenλ−}_s^r

= α

1

+ α

2

=

e^iθ

+

e^−iθ

=

2 cos

θ

^{. Hence}

λ =

^r

+

2s

.

^cos

θ

, which means

λ

can be only on the line segment [r

−

2s

,

^r

+

2s]. This completes the proof. 

We should remark that the index p has different meanings in the notation of the spaces

ℓ

p,

ℓ

^∗p

≃ ℓ

qwith p⁻¹

+

q⁻¹

=

1 and the point spectra

σ

p



U

(

^s

,

^r

,

^s

), ℓ

p



,

σ

p



U^∗

(

^s

,

^r

,

^s

), ℓ

q



which occur in the following theorems.

Theorem 3.3.

σ

p



U

(

^s

,

^r

,

^s

), ℓ

p

 =

_∅.

Proof. Let

λ

be an eigenvalue of the operator U

(

^s

,

^r

,

^s

)

. An eigenvector x

= (

^x1

,

^x2

, . . .) ∈ ℓ

p corresponding to this eigenvalue satisfies the linear system of equations

rx₁

+

sx₂

= λ

^x1

sx₁

+

rx₂

+

sx₃

= λ

^x2

sx₂

+

rx₃

+

sx₄

= λ

^x3

· · ·

(3.5)

If x₁

=

0, then x_k

=

0 for all k

∈

_{N. Hence x1}

̸=

0. Then the system of equations turn into the linear homogeneous recurrence relation

x_k+2

+

qx_k+1

+

x_k

=

0

,

^{for k}

≥

1

,

where q

=

^r−λ

s . The characteristic polynomial of the recurrence relation is x²

+

qx

+

1

=

0

.

There are two cases here.

Case 1.

|

q

| =

2.

Then characteristic polynomial has one root:

α = 

_{1,if q= −2}

−1,^{if q}=2. Hence, the solution of the recurrence relation is of the form

x_n

=



nx₁

;

if q

= −

2

, (−

¹

)

^n+1nx1

;

if q

=

2

.

This means

(

^xn

) ̸∈ ℓ

p. So, we conclude that there is no eigenvalue in this case.

Case 2.

|

q

| ̸=

2.

Then characteristic polynomial has two distinct roots

| α

1

| ̸=

_{1 and}

| α

2

| ̸=

_{1 with}

α

1

.α

2

=

1. Let

| α

2

| >

¹

> |α

1

|

_{. The} solution of the recurrence relation is of the form

x_n

=

A

(α

2

)

ⁿ

+

B

(α

1

)

ⁿ

.

Using the fact that qx₁

+

x₂

=

0, we get A

=

_α ¹

2−α1x₁, B

=

_α ¹

1−α2x₁. So we have x_n

= (α

2

)

ⁿ

− (α

1

)

ⁿ

α

2

− α

1

x₁

.

Again we have

(

^xn

) ̸∈ ℓ

p. Hence there is no eigenvalue also in this case.  Theorem 3.4.

σ

p



U^∗

(

^s

,

^r

,

^s

), ℓ

^∗p

 =

_∅.

Proof. Since U^∗

(

^s

,

^r

,

^s

) =

^Ut

(

^s

,

^r

,

^s

) =

^U

(

^s

,

^r

,

^s

)

^fromTheorem 3.3, the proof is obtained easily.  Corollary 3.5.

σ

r



U

(

^s

,

^r

,

^s

), ℓ

p

 =

_∅.

Theorem 3.6.

σ

c



U

(

^s

,

^r

,

^s

), ℓ

p

 =

_[r

−

2s

,

^r

+

2s].

Proof. Since

σ

p



U

(

^s

,

^r

,

^s

), ℓ

p

 = σ

r



U

(

^s

,

^r

,

^s

), ℓ

p

 =

_∅,

σ 

^U

(

^s

,

^r

,

^s

), ℓ

p



is the disjoint union of the parts

σ

p



U

(

^s

,

^r

,

^s

), ℓ

p



,

σ

r



U

(

^s

,

^r

,

^s

), ℓ

p



and

σ

r



U

(

^s

,

^r

,

^s

), ℓ

p



, we have

σ

c



U

(

^s

,

^r

,

^s

), ℓ

p

 =

_[r

−

2s

,

^r

+

2s]. 

4. The spectrum of the operator U

(

^s

,

^r

,

^s

)

on the sequence space b

v

p,

(

¹

<

^p

< ∞)

In this section, the fine spectrum of the second order difference operator U

(

^s

,

^r

,

^s

)

over the sequence space b

v

p,

(

¹

<

p

< ∞)

have been examined. We begin with a theorem concerning the bounded linearity of the operator U

(

^s

,

^r

,

^s

)

^acting on the sequence space b

v

p,

(

¹

<

^p

< ∞)

^.

Theorem 4.1. U

(

^s

,

^r

,

^s

) ∈

^B

(

^b

v

p

)

^.

Proof. The linearity of the operator U

(

^s

,

^r

,

^s

)

is trivial and so it is omitted. Let us take any x

= (

^xk

) ∈

^b

v

p. Then, using Minkowski’s inequality and taking the negative indices x−k

=

0, we have

‖

_U

(

^s

,

^r

,

^s

)

^x

‖

_bv

p

=



_∞

−

k=0

|

_sxk−1

+

rx_k

+

sx_k+1

− (

^sxk−2

+

rx_k−1

+

sx_k

)|

^p



¹_p

≤ (

²

|

s

| + |

r

| ) ‖

^x

‖

_bv

p

.

^

Theorem 4.2.

σ (

^U

(

^s

,

^r

,

^s

),

^b

v

p

) =

^[r

−

2s

,

^r

+

2s].

Proof. First, we prove that

(

^U

(

^s

,

^r

,

^s

) − λ

^I

)

⁻¹exists and is in B

(

^b

v

p

)

^for

λ ̸∈

^[r

−

2s

,

^r

+

2s] and next that the operator

(

^U

(

^s

,

^r

,

^s

) − λ

^I

)

is not invertible for

λ ∈

^[r

−

2s

,

^r

+

2s].

Let

λ ̸∈

^[r

−

2s

,

^r

+

2s]. Let y

= (

^yk

) ∈

^b

v

p. This implies that

(

^yk

−

y_k−1

) ∈ ℓ

p. Solving the equation

(

^U

(

^s

,

^r

,

^s

)−λ

^I

)

^x

=

y, we find the matrix in the proof ofTheorem 3.2. Then we obtain that

x_k

−

x_k−1

= (

^U

(

^s

,

^r

,

^s

) − λ

^I

)

⁻¹

(

^yk

−

y_k−1

).

Since

(

^U

(

^s

,

^r

,

^s

) − λ

^I

)

⁻¹

∈ (ℓ

p

, ℓ

p

)

^byTheorem 3.2,

(

^xk

) ∈

^b

v

p. This shows that

(

^U

(

^s

,

^r

,

^s

),

^b

v

p

) ⊆

^[r

−

2s

,

^r

+

2s].

Now, let

λ ∈

^[r

−

2s

,

^r

+

2s] and

λ ̸=

^{r. Then}

(

^U

(

^s

,

^r

,

^s

)−λ

^I

)

⁻¹exists. Using(3.4), it can be shown that it does not belong to B

(ℓ

p

)

^{. If}

λ =

r, then similar arguments as in the proof ofTheorem 3.2show that the operator U

(

^s

,

⁰

,

^s

) :

^b

v

p

→

_b

v

pis not invertible. This shows that [r

−

2s

,

^r

+

2s]

⊆ σ 

^U

(

^s

,

^r

,

^s

),

^b

v

p



. This completes the proof. 

Since the spectrum and fine spectrum of the matrix U

(

^s

,

^r

,

^s

)

as an operator on the sequence space b

v

pare similar to that of the space

ℓ

pin Section2, to avoid the repetition of the similar statements we give the results in the following theorem without proof.

Theorem 4.3. (i)

σ

p



U

(

^s

,

^r

,

^s

),

^b

v

p

 =

_∅, (ii)

σ

p



U^∗

(

^s

,

^r

,

^s

),

^b

v

p^∗

 =

_∅, (iii)

σ

r



U

(

^s

,

^r

,

^s

),

^b

v

p

 =

_∅, (iv)

σ

c



U

(

^s

,

^r

,

^s

),

^b

v

p

 =

_[r

−

2s

,

^r

+

2s].

References

[1] M. Gonzàlez, The fine spectrum of the Cesàro operator inℓp(¹<^p< ∞), Arch. Math. 44 (1985) 355–358.

[2] A.M. Akhmedov, F. Başar, On the fine spectrum of the Cesàro operator in c0, Math. J. Ibaraki Univ. 36 (2004) 25–32.

[3] J.B. Reade, On the spectrum of the Cesàro operator, Bull. Lond. Math. Soc. 17 (1985) 263–267.

[4] J.T. Okutoyi, On the spectrum of C1as an operator on bv, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 41 (1992) 197–207.

[5] A.M. Akhmedov, F. Başar, On spectrum of the Cesàro operator, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 19 (2004) 3–8.

[6] A.M. Akhmedov, F. Başar, The fine spectra of the difference operator∆over the sequence space bvp,(1≤p< ∞), Acta Math. Sin. (Engl. Ser.) 23 (2007) 1757–1768.

[7] R.B. Wenger, The fine spectra of Hölder summability operators, Indian J. Pure Appl. Math. 6 (1975) 695–712.

[8] B.E. Rhoades, The fine spectra for weighted mean operators, Pacific J. Math. 104 (1) (1983) 219–230.

[9] V. Karakaya, M. Altun, Fine spectra of upper triangular double-band matrices, J. Comput. Appl. Math. 234 (2010) 1387–1394.

[10] M. Altun, V. Karakaya, Fine spectra of lacunary matrices, J. Commun. Math. Anal. 7 (1) (2009) 1–10.

[11] H. Bilgiç, H. Furkan, On the fine spectrum of the generalized diffrerence operator B(^r,^s)over the sequence spacesℓpand bvp, Nonlinear Anal. 68 (2008) 499–506.

[12] H. Bilgiç, H. Furkan, On the fine spectrum of the operator B(^r,^s,^t)over the sequence spacesℓ1and bv, Math. Comput. Modelling 45 (2007) 883–891.

[13] C. Coşkun, The spectra and fine spectra for p-Cesàro operators, Turkish J. Math. 21 (1997) 207–212.

[14] H. Furkan, H. Bilgic, K. Kayaduman, On the fine spectrum of the generalized difference operator B(^r,^s)over the sequence spacesℓ1and bv^{, Hokkaido} Math. J. 35 (2006) 897–908.

[15] E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons Inc, New York, 1978.

[16] M. Imamınezhed, M.R. Miri, The dual space of the sequence space bvp, (¹≤p< ∞), Acta Math. Univ. Comenian (N.S.) 79 (1) (2010) 143–149.

[17] B. Altay, F. Başar, On the fine spectrum of the generalized difference operator B(^r,^s)over the sequence spaces c0and c, Int. J. Math. Math. Sci. 18 (2005) 3005–3013.

[18] F. Başar, B. Altay, On the space of sequence of p-bounded variation and related matrix mappings, Ukrainian Math. J. 55 (1) (2003) 135–147.

[19] B. Choudhary, S. Nanda, Functional Analysis with Applications, John Wiley & Sons Inc, New York, Chishester, Brisbane, Toronto, Singapore, 1989.

[20] R.L. Graham, D.E. Knuth, O. Patashnik, Concrete mathematics, Addison–Wesley Publishing Company, 1989.