Algebraic duals of some sets of difference sequences defined by Orlicz functions

R E S E A R C H

Open Access

Algebraic duals of some sets of difference

sequences defined by Orlicz functions

Hemen Dutta

_{and Iqbal H Jebril}

*_{Correspondence:}

[email protected] 2_{Department of Mathematics,}

Taibah University, Medina, Saudi Arabia

Full list of author information is available at the end of the article

Abstract

In this paper, we first give a description of some sets of sequences generated by difference operators and defined by Orlicz functions. Then their algebraic duals such as the

α-,

β

γ

MSC: 46A45; 47N40; 65J99; 46A20

Keywords: difference sequences; Orlicz function; algebraic duals

1 Introduction

Letwdenote the space of all scalar sequences, and any subspace ofwis called a sequence space. Let∞,candcbe the linear spaces of bounded, convergent and null sequences

x= (xk) with complex terms, respectively, normed byx∞=supk|xk|, wherek∈N= {, , . . .}, the set of positive integers.

Lindenstrauss and Tzafriri [] used the Orlicz function and introduced the sequence spaceMas follows:

M=

(xk)∈w:

∞

k=

M

|xk|

ρ

<∞for someρ> 

.

They proved thatMis a Banach space normed by

(xk)=inf

ρ>  :

∞

k=

M

|xk|

ρ

≤

.

Throughout this sectionXwill denote one of the sequence spaces∞,candc.

The notion of difference sequence spaces was introduced by Kizmaz []. For some other works on difference sequences, Orlicz functions and related literature, we refer to [–]. Letv= (vk) be any fixed sequence of non-zero complex numbers. Et and Esi [] generalized

the above sequence spaces to the following sequence spaces:

Xm_v =x= (xk) :

m_vxk ∈X

forX=∞,candc.

In this paper, for an Orlicz functionM, we can have the following spaces in the line of the spaces studied by Mursaleenet al. []:

XM,m_v =x= (xk) :

m_vxk ∈X(M)

.

In fact, we get the following spaces:

∞M,m_v =

x= (xk) :sup k

M

|m_vxk|

ρ

<∞for someρ> 

,

cM,m_v =

x= (xk) : lim

k→∞M

_|

m_vxk–L|

ρ

=  for someLandρ> 

,

c

M,m_v =

x= (xk) : lim

k→∞M

|m_vxk|

ρ

=  for someρ> 

,

wherem

vxk=mv–xk–vm–xk+,mvxk=

m

i=(–)i

_m

i vk+ixk+ifor allk∈N.

Bektaşet al.[] introduced the difference operator(vm)and defined it as follows:

(_vm)xk= m

i=

(–)i

m i

vk–ixk–i for allk∈N.

Using this difference operator, we can construct the following sequence space:

XM,(_vm) =x= (xk) :

(_vm)xk ∈X(M)

.

The operator

(m):w→w

is defined by

()xk= k

j=

xj (k= , , . . .), (m)=()o(m–) (m≥)

and

(m)o(m)=(m)o(m)= id, the identity onw(see [, ]).

Now, for subsequent use, we slightly generalize the above definition as follows. We define

_v(m):w→w

by

_v()xk= k

j=

vjxj (k= , , . . .), (vm)=v(m)o(vm–) (m≥)

and

Now, forx∈X(M,m

v), let us define

x= m

i=

|vixi|+inf

ρ>  :sup k

M

_| m_vxk|

ρ

≤

.

It can be shown that (X(M,m

v), · ) is aBK-space.

Again forx∈X(M,(vm)), let us define

x=inf

ρ>  :sup k

M

_| (vm)xk|

ρ

≤

.

It can be shown that (X(M,v(m)), · ) is aBK-space.

It is trivial that (m_vxk)∈X(M) if and only if (v(m)xk)∈X(M). Also the norms · and · are equivalent.

Let us define the operator

D:XM,_vm →XM,m_v

byDx= (, , . . . ,xm+,xm+, . . .), wherex= (x,x, . . . ,xm, . . .). It is trivial thatDis a bounded

linear operator onX(M,m_v),X=∞,candc. Furthermore, the set

DXM,m_v =DXM,m_v =x= (xk) :x∈X

M,m_v ,x=x=· · ·=xm= 

is a subspace ofX(M,m_v) and normed by

x=inf

ρ>  :sup k

M

_| m_vxk|

ρ

≤

inDXM,m_v .

DX(M,m_v) andX(M) are equivalent as topological spaces since

m_v :DXM,m_v →X(M),

defined by

m_vx=y=m_vxk , (.)

is a linear homeomorphism. Moreover, obviously

(_vm):XM,(_vm) →X(M), (_vm)x=y=(_vm)xk ,

_v(m):X(M)→XM,_v(m) , _v(m)x=y=_v(m)xk

are isometric isomorphisms forX=_∞,candc.

Hence ∞(M,m_v), c(M,_vm) and c(M,mv) are isometrically isomorphic to ∞(M),

Moreover,X(M,i

v)⊂X(M,mv) fori= , , . . . ,m– , which follows from the following

inequality and convexity ofM:

M

_| m_vxk|

ρ

≤

M

_| m_v–xk|

ρ

+

M

_|

m_v–xk+|

ρ

.

Investigation of spaces is often combined with that of duals. The algebraic dual space is defined for all vector spaces. When defined for a topological vector space, there is a sub-space of this dual sub-space, corresponding to continuous linear functionals, which constitutes a continuous dual space. For any finite-dimensional normed vector space or topological vector space, such as Euclideann-space, the continuous dual and the algebraic dual coin-cide. This is, however, false for any infinite-dimensional normed space. For some related literature on duality relevant to this paper, we refer to [, , , ]. Our results of this pa-per will generalize few existing results as well as generate some new results in the literature of algebraic duality within the field of functional analysis.

2 Computation of algebraic duals

In this section we compute theα-,β-,γ- andN-duals of the spaces∞(M,m_v),c(M,m_v) andc(M,mv).

Definition .[] LetXbe a sequence space and define

Xα₌

a= (ak) :

∞

k=

|akxk|<∞,∀x∈X

,

Xβ=

a= (ak) :

∞

k=

akxkis convergent,∀x∈X

,

Xγ=

a= (ak) :sup n

n

k=

akxk

<∞,∀x∈X

,

XN=

a= (ak) :lim

k akxk= ,∀x∈X

,

thenXα_,Xβ_,Xγ _andXN_{are called theα-,β-,γ- andN-(or null) duals ofX, respectively.}

It is known that ifX⊂Y, thenYη_⊂Xη_{forη=α,β,γ andN.}

Lemma .[] Let m be a positive integer.Then there exist positive constants Cand C

such that

Ckm≤

m+k k

≤Ckm, k= , , , . . . .

Lemma . x∈_∞(M,m

v)impliessupkM(

|k–m–

v xk|

ρ ) <∞for someρ> .

Proof Letx∈∞(M,m_v), then

sup k

M

|m_v–xk–mv–xk+|

ρ

Then there existsU>  such that

M

_| m–

v xk–mv–xk+|

ρ

<U for allk∈N.

Takingη=kρ, for an arbitrary fixed positive integerk, by the subadditivity of modulus, the monotonicity and convexity ofM:

M

|m_v–x–mv–xk+|

η

≤ 

k

k

l=

M

|m_v–xl–mv–xl+|

ρ

<U.

Then the above inequality, the inequality

|m_v–xk+|

(k+ )ρ ≤ 

k+  _|

m_v–x|

ρ +k

|m_v–x–mv–xk+|

kρ

and the convexity ofMimply

M

_|

m_v–xk+|

(k+ )ρ

≤ 

k+ 

M

_| m_v–x|

ρ

+kM

_|

m_v–x–mv–xk+|

kρ

≤max

M

_| m–

v x|

ρ

,U

<∞.

Hence we have the desired result.

Hence we have the following lemma.

Lemma .

(i) x∈_∞(M,m

v)impliessupkM(

|k–mvkxk|

ρ ) <∞for someρ> ,

(ii) x∈∞(M,_vm)impliessup_kk–m_|v

kxk|<∞. Theorem . Let M be an Orlicz function.Then

(i) [c(M,vm)]α= [c(M,mv)]α= [∞(M,mv)]α=D,

(ii) Dα  =D,

where

D=

a= (ak) :

∞

k=

kmv–_k ak<∞

,

D=

b= (bk) :sup k

k–m|vkbk|<∞

.

Proof (i) Leta∈D, then

∞

k=km|v–k ak|<∞. Now, for anyx∈∞(M,mv), we have sup_kk–m_|v

kxk|<∞. Then we have

∞

k=

|akxk| ≤sup k

k–m|vkxk|

∞

k=

kmakv–k <∞.

Conversely, suppose thata∈[X(M,m

v)]α forX=cand∞. Then

∞

k=|akxk|<∞for

eachx∈X(M,m_v). So we can take

xk=kmv–k , k≥.

Then

∞

k=

kmv–_k ak=

∞

k=

|akxk|<∞.

This implies thata∈D.

Again suppose thata∈[c(M,mv)]αanda∈/D. Then there exists a strictly increasing

sequence (ni) of positive integersniwithn<n<· · · such that ni+

k=ni+

kmv–_k ak>i.

Definex∈c(M,mv) by

xk=

, ≤k≤n,

km_v

ksgnak/i, ni<k≤ni+.

Then we have

∞

k=

|akxk|= n

k=n+

|akxk|+· · ·+ ni+

k=ni+

|akxk|+· · ·

=

n

k=n+

kmv–_k ak+· · ·+



i

ni+

k=ni+

kmv–_k ak+· · ·

>  +  +· · ·=∞.

This contradictsa∈[c(M,mv)]α. Hencea∈D. This completes the proof of (i).

(ii) The proof is similar to that of part (i).

If we takevk=  for allk∈Nin Theorem ., then we obtain the following corollary. Corollary . Let M be an Orlicz function.Then

(i) [c(M,m)]α= [c(M,m)]α= [∞(M,m)]α=E,

(ii) Eα  =E,

where

E=

a= (ak) :

∞

k=

km|ak|<∞

,

E=

b= (bk) :sup k

k–m|bk|<∞

Theorem . Let M be an Orlicz function.Then[c(M,m

v)]N= [∞(M,mv)]N=F,where

F=

a= (ak) :lim

k k

m_v– k ak= 

.

Proof The proof is immediate using Lemma .(ii). The following lemma will be used in the next theorem.

Lemma .[] Let(pn)be a sequence of positive numbers increasing monotonically to

infinity.

(i) Ifsup_n|n_v=pvav|<∞,thensupn|pn

∞

k=n+ak|<∞.

(ii) If_kpkakis convergent,thenlimnpn

∞

k=n+ak= .

Theorem . Let M be an Orlicz function and c+_denote the set of all positive null se-quences.Then

(i) [D_∞(M,m

v)]β= [Dc(M,mv)]β=G,

(ii) [Dc(M,mv)]β=G,

(iii) [D_∞(M,m

v)]γ = [Dc(M,mv)]γ=H,

(iv) [Dc(M,mv)]γ =H,

where

G=

a= (ak) :

∞

k=

akv–k k–m

j=

k–j– 

m– 

is convergent, ∞ k= ∞

j=k+

v–_j aj

k–m+

j=

k–j– 

m– 

<∞

,

G=

a= (ak) :

∞

k=

akv–k k–m

j=

k–j– 

m– 

ujconverges and

∞ k= ∞

j=k+

v–_j aj

k–m+

j=

k–j– 

m– 

uj<∞,∀u∈c+

,

H=

a= (ak) :sup n n k=

akv–k k–m

j=

k–j– 

m– 

<∞,

∞ k= ∞

j=k+

v–_j aj

k–m+

j=

k–j– 

m– 

<∞

,

H=

a= (ak) :sup n n k=

akv–k k–m

j=

k–j– 

m– 

uj

<∞,

∞ k= ∞

j=k+

v–_j aj

k–m+

j=

k–j– 

m– 

uj<∞,∀u∈c+

.

For eachx∈D_∞(M,m

v), there exists one and only oney= (yk)∈∞(M) such that

xk=v–k k–m

j=

k–j– 

m– 

yj, y–m=y–m=· · ·=y= 

for sufficiently largekby (.). Leta∈G. Then, using the same technique as applied in

[, p.], we can show thata∈[D∞(M,m_v)]β_.

Leta∈[D_∞(M,m

v)]β. Again, using the same technique as applied in [, pp.-],

we can show thata∈G.

This completes the proof.

3 Conclusion

Although we conclude this paper here, the following further suggestion remains open:

What is theN-dual of the spacec(M,mv)?

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

HD carried out the initial description of the spaces. IHJ proposed the structure of the paper. HD and IHJ both formulated and proved the results. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, Gauhati University, Guwahati, 781014, India.}2_{Department of Mathematics, Taibah}

University, Medina, Saudi Arabia.

Received: 29 May 2014 Accepted: 8 July 2014 Published:19 Aug 2014

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