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volume 7, issue 3, article 101, 2006.

Received 16 December, 2005;

accepted 07 January, 2006.

Communicated by:A.G. Babenko

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Journal of Inequalities in Pure and Applied Mathematics

THE DUAL SPACES OF THE SETS OF DIFFERENCE SEQUENCES OF ORDERm

Ç.A. BEKTA ¸S AND M. ET

Department of Mathematics Firat University

Elazig, 23119, TURKEY.

EMail:cbektas@firat.edu.tr EMail:mikailet@yahoo.com

c

2000Victoria University ISSN (electronic): 1443-5756 247-04

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The Dual Spaces of the Sets of Difference Sequences of Order

m

Ç.A. Bekta¸s and M. Et

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J. Ineq. Pure and Appl. Math. 7(3) Art. 101, 2006

http://jipam.vu.edu.au

Abstract

The idea of difference sequence spaces was introduced by Kızmaz [5] and the concept was generalized by Et and Çolak [3]. Letp= (p_k)be a bounded sequence of positive real numbers andv= (vk)be any fixed sequence of non- zero complex numbers. Ifx = (x_k)is any sequence of complex numbers we write∆^m_vxfor the sequence of them-th order differences ofxand∆^m_v(X) = {x= (xk) : ∆^m_vx∈X}for any setXof sequences. In this paper we determine theα-,β - andγ- duals of the sets∆^m_v(X)which are defined by Et et al. [2]

forX=`∞(p),c(p)andc₀(p).This study generalizes results of Malkowsky [9]

in special cases.

2000 Mathematics Subject Classification:40C05, 46A45.

Key words: Difference sequences,α−,β−andγ−duals.

1. Introduction, Notations and Known Results

Throughout this paperωdenotes the space of all scalar sequences and any subspace of ω is called a sequence space. Let `∞, c and c₀ be the linear space of bounded, convergent and null sequences with complex terms, respectively, normed by

kxk_∞ = sup

k

|x_k|,

where k ∈ N = {1,2,3, . . .}, the set of positive integers. Furthermore, let p= (pk)be bounded sequences of positive real numbers and

`∞(p) =

x∈ω : sup

k

|x_k|^p^k <∞

, c(p) =

n

x∈ω: lim

k→∞ |x_k−l|^p^k = 0, for some l∈C o

, c₀(p) = n

x∈ω: lim

k→∞ |x_k|^p^k = 0o (for details see [6], [7], [11]).

Letxandybe complex sequences, andE andF be subsets ofω. We write M(E, F) = \

x∈E

x⁻¹∗F ={a∈ω:ax∈F for allx∈E} [12].

In particular, the sets

E^α =M(E, l₁), E^β =M(E, cs) and E^γ =M(E, bs)

are called theα−,β−andγ−duals ofE, wherel₁,csandbsare the sets of all convergent, absolutely convergent and bounded series, respectively. IfE ⊂F,

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thenF^η ⊂E^ηforη=α, β, γ. It is clear thatE^α ⊂(E^α)^α=E^αα. IfE =E^αα, then E is anα -space. In particular, an α-space is called a Köthe space or a perfect sequence space.

Throughout this paper X will be used to denote any one of the sequence spaces`∞,candc₀.

Kızmaz [5] introduced the notion of difference sequence spaces as follows:

X(∆) ={x= (x_k) : (∆x_k)∈X}.

Later on the notion was generalized by Et and Çolak in [3], namely, X(∆^m) = {x= (x_k) : (∆^mx_k)∈X}.

Subsequently difference sequence spaces have been studied by Malkowsky and Parashar [8], Mursaleen [10], Çolak [1] and many others.

Letv = (v_k) be any fixed sequence of non-zero complex numbers. Et and Esi [4] generalized the above sequence spaces to the following ones

∆^m_v (X) = {x= (x_k) : (∆^m_v x_k)∈X},

where ∆⁰_vx = (v_kx_k), ∆^m_v x = (∆^m−1_v x_k −∆^m−1_v x_k+1) such that ∆^m_v x_k = Pm

i=0(−1)^{i m}_i

v_k+ix_k+i. Recently Et et al. [2] generalized the sequence spaces

∆^m_v (X)to the sequence spaces

∆^m_v (X(p)) ={x= (x_k) :(∆^m_v x_k)∈X(p)}

and showed that these spaces are complete paranormed spaces paranormed by g(x) =

m

X

i=1

|x_iv_i|+ sup

k

|∆^m_v x_k|^p^k^/M,

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whereH = sup_kp_kandM = max (1, H).

Let us define the operatorD: ∆^m_v X(p)→∆^m_v X(p)by Dx= (0,0, . . . , x_m+1, x_m+2, . . .),

wherex = (x₁, x₂, x₃, . . .). It is trivial that Dis a bounded linear operator on

∆^m_v X(p). Furthermore the set D[∆^m_v X(p)] =D∆^m_v X(p)

={x= (x_k) :x∈∆^m_v X(p), x₁ =x₂ =· · ·=x_m = 0}

is a subspace of ∆^m_v X(p). D∆^m_v X(p)andX(p)are equivalent as topological spaces, since

(1.1) ∆^m_v :D∆^m_v X(p)→X(p)

defined by ∆^m_v x = y = (∆^m_v x_k)is a linear homeomorphism. Let[X(p)]⁰ and [D∆^m_v X(p)]⁰denote the continuous duals ofX(p)andD∆^m_v X(p), respectively.

It can be shown that

T : [D∆^m_v X(p)]⁰ →[X(p)]⁰, f_∆ →f_∆◦(∆^m_v )⁻¹ =f, is a linear isometry. So[D∆^m_v X(p)]⁰is equivalent to[X(p)]⁰.

Lemma 1.1 ([5]). Let(t_n)be a sequence of positive numbers increasing mono- tonically to infinity, then

i) Ifsup_n|Pn

i=1t_ia_i|<∞, thensup_n t_nP∞

k=n+1a_k <∞, ii) IfP

kt_ka_kis convergent, thenlimn→∞t_nP∞

k=n+1a_k= 0.

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2. Main Results

In this section we determine theα−,β−andγ−duals of∆^m_v X(p).

Theorem 2.1. For every strictly positive sequencep= (pk), we have (i) [∆^m_v l∞(p)]^α =D₁^α(p),

(ii) [∆^m_v l_∞(p)]^αα =D^αα₁ (p) where

D^α₁(p) =

∞

\

N=2

(

a = (ak) :

∞

X

k=1

|ak| |vk|⁻¹

k−m

X

j=1

k−j−1 m−1

N^1/p^j <∞ )

and

D^αα₁ (p) =

∞

[

N=2







a= (a_k) : sup

k≥m+1

|a_k| |v_k|

"_k−m X

j=1

k−j−1 m−1

N^1/p^j

#⁻¹

<∞





 .

Proof.

(i) Leta∈D^α₁(p) andx∈∆^m_v l∞(p). We chooseN >max(1,sup_n|∆^m_v a_n|^pⁿ).

Since

k−m

X

j=1

k−j−1 m−1

N^1/p^j >

m

X

j=1

k−j−1 m−j

N^1/p^j ≥1

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for arbitraryN >1 (k = 2m,2m+ 1, . . .)and|∆^m−j_v x_j| ≤ M (1≤j ≤ m)for some constantM,a∈D^α₁(p)implies

∞

X

k=1

|a_k| |v_k|⁻¹

m

X

j=1

k−j −1 m−j

∆^m−j_v x_j <∞.

Then

∞

X

k=1

|a_kx_k|=

∞

X

k=1

|a_k| |v_k|⁻¹

k−m

X

j=1

(−1)^m

k−j−1 m−1

∆^m_v x_j

+

m

X

j=1

(−1)^m−j

k−j−1 m−j

∆^m−j_v xj

!

≤

∞

X

k=1

|a_k| |v_k|⁻¹

k−m

X

j=1

k−j −1 m−1

N^1/p^j

+

∞

X

k=1

|a_k| |v_k|⁻¹

m

X

j=1

k−j−1 m−j

∆^m−j_v x_j

<∞.

Conversely leta /∈D₁^α(p). Then we have

∞

X

k=1

|a_k||v_k|⁻¹

k−m

X

j=1

k−j−1 m−1

N^1/p^j =∞

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for some integerN >1. We define the sequencexby x_k =v_k⁻¹

k−m

X

j=1

k−j−1 m−1

N^1/p^j (k=m+ 1, m+ 2, . . .).

Then it is easy to see that x ∈ ∆^m_v l_∞(p) and P

k

|a_kx_k| = ∞. Hence a /∈[∆^m_v l∞(p)]^α. This completes the proof of (i).

(ii) Let a ∈ D₁^αα(p) and x ∈ [∆^m_v l∞(p)]^α = D₁^α(p), by part (i). Then for someN >1, we have

∞

X

k=m+1

|a_kx_k|=

∞

X

k=m+1

|a_k| |v_k|

"_k−m X

j=1

k−j−1 m−1

N^1/p^j

#⁻¹

× |x_k| |v_k|⁻¹

"_k−m X

j=1

k−j−1 m−1

N^1/p^j

#

≤ sup

k≥m+1



|a_k| |v_k|

"_k−m X

j=1

k−j −1 m−1

N^1/p^j

#⁻¹



×

∞

X

k=m+1

|x_k| |v_k|⁻¹

k−m

X

j=1

k−j −1 m−1

N^1/p^j

<∞.

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Conversely leta /∈D₁^αα(p).Then for all integersN >1, we have sup

k≥m+1

|a_k| |v_k|

"_k−m X

j=1

k−j−1 m−1

N^1/p^j

#⁻¹

=∞.

We recall that

k−m

X

j=1

k−j−1 m−1

y_j = 0 (k < m+ 1) for arbitraryy_j.

Hence there is a strictly increasing sequence (k(s)) of integers k(s) ≥ m+ 1such that

|a_k(s)| |v_k(s)|





k(s)−m

X

j=1

k(s)−j −1 m−1

s^1/p^j





−1

> s^m+1 (s=m+ 1, m+ 2, . . .).

We define the sequencexby x_k =







|a_k(s)|⁻¹, (k =k(s))

0, (k 6=k(s)) (k=m+ 1, m+ 2, . . .) Then for all integersN > m+ 1, we have

∞

X

k=1

|x_k| |v_k|⁻¹

k−m

X

j=1

k−j−1 m−1

N^1/p^j <

∞

X

s=m+1

s^−(m+1) <∞.

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Hencex ∈ [∆^m_v l∞(p)]^α andP∞

k=1|a_kx_k| = P∞

N=11 = ∞. Hencea /∈ [∆^m_v l∞(p)]^αα. The proof is completed.

Theorem 2.2. For every strictly positive sequencep= (p_k), we have (i) [∆^m_v c₀(p)]^α =M₀^α(p),

(ii) [∆^m_v c₀(p)]^αα =M₀^αα(p) where

M₀^α(p) =

∞

[

N=2

(

a∈ω:

∞

X

k=1

|a_k| |v_k|⁻¹

k−m

X

j=1

k−j −1 m−1

N^−1/p^j <∞ )

and

M₀^αα(p) =

∞

\

N=2







a∈ω : sup

k≥m+1

|a_k| |v_k|

"_k−m X

j=1

k−j−1 m−1

N^−1/p^j

#⁻¹

<∞





 .

Proof.

(i) Let a ∈ M₀^α(p) andx ∈ ∆^m_v c₀(p). Then there is an integerk₀ such that sup

k>k0

|∆^m_v x_k|^p^k ≤N⁻¹, whereN is the number inM₀^α(p). We put M = max

1≤k≤k₀|∆^m_v x_k|^p^k, n = min

1≤k≤k₀p_k, L= (M + 1)N

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and define the sequencey byy_k = x_k·L^−1/n (k = 1,2, . . .). Then it is easy to see thatsup_k|∆^m_v y_k|^p^k ≤N⁻¹.

Since

k−m

X

j=1

k−j−1 m−1

N^−1/p^j >

m

X

j=1

k−j−1 m−j

N^−1/p^j

for arbitraryN >1 (k = 2m,2m+ 1, . . .),a∈M₀^α(p) implies

∞

X

k=1

|a_k| |v_k|

m

X

j=1

k−j −1 m−j

|∆^m−j_v y_j|<∞.

Then

∞

X

k=1

|a_kx_k|=L^1/n

∞

X

k=1

|a_ky_k|

≤L^1/n

∞

X

k=1

|a_k| |v_k|⁻¹

k−m

X

j=1

k−j−1 m−1

N^−1/p^j

+L^1/n

∞

X

k=1

|a_k| |v_k|⁻¹

m

X

j=1

k−j −1 m−j

|∆^m−j_v y_j|

<∞.

So we havea∈[∆^m_v c0(p)]^α. ThereforeM₀^α(p)⊂[∆^m_v c0(p)]^α.

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Conversely, leta /∈ M₀^α(p). Then we can determine a strictly increasing sequence(k(s))of integers such thatk(1) = 1and

M(s) =

k(s+1)−1

X

k=k(s)

|a_k| |v_k|⁻¹

k−m

X

j=1

k−j−1 m−1

(s+1)^−1/p^j >1 (s= 1,2, . . .).

We define the sequencexby

x_k=v_k⁻¹





s−1

X

l=1

k(l+1)−1

X

j=k(l)

k−j−1 m−1

(l+ 1)^−1/p^j

+

k−m

X

j=k(s)

k−j−1 m−1

(s+ 1)^−1/p^j





(k(s)≤k≤k(s+ 1)−1;s= 1,2, . . .).

Then it is easy to see that

|∆^m_v x_k|^p^k = 1

s+ 1 (k(s)≤k ≤k(s+ 1)−1;s= 1,2, . . .) hencex∈∆^m_v c₀(p), andP∞

k=1|a_kx_k| ≥P∞

s=1 =∞,i.e.a /∈[∆^m_v c₀(p)]^α. (ii) Omitted.

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Theorem 2.3. For every strictly positive sequencep= (p_k), we have [∆^m_v c(p)]^α =M^α(p)

=M₀^α(p)∩ (

a∈ω:

∞

X

k=1

|a_k| |v_k|⁻¹

k−m

X

j=1

k−j−1 m−1

<∞ )

.

Proof. Let a ∈ M^α(p) and x ∈ ∆^m_v c(p). Then there is a complex number l such that|∆^m_v x_k−l|^p^k →0 (k→ ∞). We definey = (y_k) by

y_k=x_k+v⁻¹_k l(−1)^m+1

k−m

X

j=1

k−j −1 m−1

(k = 1,2, . . .).

Theny∈∆^m_v c₀(p) and

∞

X

k=1

|a_kx_k| ≤

∞

X

k=1

|a_k| |v_k|⁻¹

k−m

X

j=1

k−j−1 m−1

|∆^m_v y_j|

+

∞

X

k=1

|a_k| |v_k|⁻¹

m

X

j=1

k−j−1 m−j

|∆^m−j_v y_j|

+|l|

∞

X

k=1

|ak| |vk|⁻¹

k−m

X

j=1

k−j−1 m−1

<∞

by Theorem2.2(i) and sincea∈M^α(p).

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Now leta ∈ [∆^m_v c(p)]^α ⊂ [∆^m_v c₀(p)]^α =M₀^α(p)by Theorem2.2(i). Since the sequencexdefined by

x_k = (−1)^mv⁻¹_k

k−m

X

j=1

k−j−1 m−1

(k = 1,2, . . .)

is in∆^m_v c(p), we have

∞

X

k=1

|ak|

k−m

X

j=1

k−j−1 m−1

<∞.

Theorem 2.4. For every strictly positive sequencep= (pk), we have (i) [D∆^m_v `∞(p)]^β =M_∞^β(p),

(ii) [D∆^m_v `∞(p)]^γ =M_∞^γ(p) where

M_∞^β(p) = \

N >1

(

a∈ω :

∞

X

k=1

a_kv⁻¹_k

k−m

X

j=1

k−j−1 m−1

N^1/p^jconverges and

∞

X

k=1

|b_k|

k−m+1

X

j=1

k−j−1 m−2

N^1/p^j <∞ )

,

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M_∞^γ(p) = \

N >1

(

a∈ω : sup

n

|

n

X

k=1

a_kv_k⁻¹

k−m

X

j=1

k−j−1 m−1

N^1/p^j|<∞,

∞

X

k=1

|b_k|

k−m+1

X

j=1

k−j−1 m−2

N^1/p^j <∞ )

andb_k=P∞

j=k+1v_j⁻¹a_j (k = 1,2, . . .).

Proof.

(i) Ifx∈D∆^m_v `_∞(p)then there exists a uniquey= (y_k)∈`_∞(p) such that x_k =v_k⁻¹

k−m

X

j=1

(−1)^m

k−j−1 m−1

y_j

for sufficiently large k, for instance k > m by (1.1). Then there is an integer N >max{1,sup_k|∆^m_v x_k|^p^k}. Let a ∈ M_∞^β(p), and suppose that

−1

= 1. Then we may write

n

X

k=1

a_kx_k =

n

X

k=1

a_k v_k⁻¹

k−m

X

j=1

(−1)^m

k−j−1 m−1

y_j

!

= (−1)^m

n−m

X

k=1

b_k+m−1

k

X

j=1

k+m−j−2 m−2

y_j

−b_n

n−m

X

j=1

(−1)^m

n−j −1 m−1

y_j.

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Since

∞

X

k=1

|bk+m−1|

k

X

j=1

k+m−j −2 m−2

N^1/p^j <∞, the series

∞

X

k=1

b_k+m−1

k

X

j=1

k+m−j−2 m−2

y_j

is absolutely convergent. Moreover by Lemma1.1(ii), the convergence of

∞

X

k=1

akv_k⁻¹

k−m

X

j=1

k−j−1 m−1

N^1/p^j

implies

n→∞lim b_n

n−m

X

j=1

n−j−1 m−1

N^1/p^j = 0.

HenceP∞

k=1a_kx_k is convergent for allx∈D∆^m_v `∞(p), soa∈[D∆^m_v `∞(p)]^β. Conversely let a ∈ [D∆^m_v `∞(p)]^β. Then P∞

k=1a_kx_k is convergent for eachx∈D∆^m_v `∞(p). If we take the sequencex= (x_k) defined by

x_k=







0, k ≤m

v_k⁻¹Pk−m j=1

k−j−1 m−1

N^1/p^j, k > m

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then we have

∞

X

k=1

a_kv⁻¹_k

k−m

X

j=1

k−j −1 m−1

N^1/p^j =

∞

X

k=1

a_kx_k<∞.

Thus the seriesP∞

k=1a_kv_k⁻¹Pk−m j=1

k−j−1 m−1

N^1/p^j is convergent. This implies that

n→∞lim bn n−m

X

j=1

n−j−1 m−1

N^1/p^j = 0 by Lemma1.1(ii).

Now leta∈[D∆^m_v `_∞(p)]^β−M_∞^β(p). ThenP∞

k=1|b_k|Pk−m+1 j=1

k−j−1 m−2

N^1/p^j is divergent, that is,

∞

X

k=1

|b_k|

k−m+1

X

j=1

k−j−1 m−2

N^1/p^j =∞.

We define the sequencex= (xk) by x_k =







0, k ≤m

v⁻¹_k Pk−1

i=1 sgnb_iPi−m+1 j=1

i−j−1 m−2

N^1/p^j, k > m

wherea_k > 0for all k ora_k < 0 for all k. It is trivial that x = (x_k) ∈ D∆^m_v `∞(p). Then we may write forn > m

n

X

k=1

a_kx_k=−

m

X

k=1

bk−1∆_vxk−1−

n−m

X

k=1

bk+m−1∆_vxk+m−1−b_nx_nv_n.

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Since(b_nx_nv_n)∈c₀, now lettingn→ ∞we get

∞

X

k=1

a_kx_k =−

∞

X

k=1

bk+m−1∆_vxk+m−1

=

∞

X

k=1

|bk+m−1|

k

X

j=1

k+m−j−2 m−2

N^1/p^j =∞.

This is a contradiction toa∈[D∆^m_v `∞(p)]^β. Hencea∈M_∞^β(p).

(ii) Can be proved by the same way as above, using Lemma1.1(i).

Lemma 2.5. [D∆^m_v `∞(p)]^η = [D∆^m_v c(p)]^η forη =β orγ.

The proof is obvious and is thus omitted.

Theorem 2.6. Letc⁺₀ denote the set of all positive null sequences.

(a) We put

M₃^β(p) = (

a∈ω:

∞

X

k=1

a_kv_k⁻¹

k−m

X

j=1

k−j−1 m−1

N^1/p^j u_j converges and

∞

X

k=1

|b_k|

k−m+1

X

j=1

k−j−1 m−2

N^1/p^j u_j <∞,∀u∈c⁺₀ )

.

Then[D∆^m_v c0(p)]^β =M₃^β(p).

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(b) We put

M₄^γ(p) = (

a∈ω : sup

n

|

n

X

k=1

a_kv_k⁻¹

k−m

X

j=1

k−j−1 m−1

N^1/p^j u_j|<∞,

∞

X

k=1

|b_k|

k−m+1

X

j=1

k−j−1 m−2

N^1/p^j u_j <∞,∀u∈c⁺₀ )

.

Then[D∆^m_v c₀(p)]^γ =M₄^γ(p).

Proof. (a) and (b) can be proved in the same manner as Theorem 2.4, using Lemma1.1(i) and (ii).

Lemma 2.7.

i) [∆^m_v `∞(p)]^η = [D∆^m_v `∞(p)]^η, ii) [∆^m_v c(p)]^η = [D∆^m_v c(p)]^η, iii) [∆^m_v c0(p)]^η = [D∆^m_v c0(p)]^η forη=βorγ.

The proof is omitted.

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References

[1] R. ÇOLAK, Lacunary strong convergence of difference sequences with respect to a modulus function, Filomat, 17 (2003), 9–14.

[2] M. ET, H. ALTINOK AND Y. ALTIN, On some generalized sequence spaces, Appl. Math. Comp., 154 (2004), 167–173.

[3] M. ETANDR. ÇOLAK, On some generalized difference sequence spaces, Soochow J. Math., 21(4) (1995), 377-386.

[4] M. ET AND A. ESI, On Köthe-Toeplitz duals of generalized difference sequence spaces, Bull. Malays. Math. Sci. Soc., (2) 23 (2000), 25–32.

[5] H. KIZMAZ, On certain sequence spaces, Canad. Math. Bull., 24 (1981), 169–176.

[6] C.G. LASCARIDESANDI.J. MADDOX, Matrix transformations between some classes of sequences, Proc. Cambridge Philos. Soc., 68 (1970), 99–

104.

[7] I.J. MADDOX, Continuous and Köthe-Toeplitz duals of certain sequence spaces, Proc. Cambridge Philos. Soc., 65 (1967), 471–475.

[8] E. MALKOWSKY AND S.D. PARASHAR, Matrix transformations in spaces of bounded and convergent difference sequences of orderm, Anal- ysis, 17 (1997), 87–97.

[9] E. MALKOWSKY, Absolute and ordinary Köthe-Toeplitz duals of some sets of sequences and matrix transformations, Publ. Inst. Math., (Beograd) (N.S.) 46(60) (1989), 97–103.

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[10] MURSALEEN, Generalized spaces of difference sequences, J. Math.

Anal. Appl., 203 (1996), 738–745.

[11] S. SIMONS, The sequence spaces`(p_v) andm(p_v), Proc. London Math.

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[12] A. WILANSKY, Summability through Functional Analysis, North- Holland Mathematics Studies, 85 (1984).