## On Orlicz Difference Sequence Spaces. Hemen Dutta

SDU Journal of Science (E-Journal), 2010, 5 (1): 119-136 ___________________________________________________________________ On Orlicz Difference Seq...
Author: Alberta Watts
SDU Journal of Science (E-Journal), 2010, 5 (1): 119-136 ___________________________________________________________________

On Orlicz Difference Sequence Spaces Hemen Dutta Gauhati University, Department of Mathematics, Kokrajhar Campus, Assam, INDIA e-mail: [email protected] Received:13 November 2008, Accepted: 9 February 2010 Abstract: The main aim of this article is to generalize the famous Orlicz sequence space by using difference operators and a sequence of non-zero scalars and investigate some topological structure relevant to this generalized space. Key words: Difference sequence space, multiplier sequence space, Orlicz function, AK-BK space, topological isomorphism and Köthe-Toeplitz dual.

Orlicz Fark Dizi Uzayları Üzerine Özet: Bu makalenin amacı, sıfırdan farklı skalerlerden oluşan bir diziyi ve fark operatörlerini kullanarak Orlicz dizi uzaylarını genelleştirmek ve bu yeni tanımladığımız uzayın topolojik yapısını incelemektir. Anahtar kelimeler: Fark dizi uzayı, çok indisli dizi uzayı, Orlicz fonksiyonu, AK-BK uzayı, toplojik izomorfizm, Köthe-Toeplitz duali. 2000 Mathematics Subject Classification: 40A05, 40C05, 46A45.

1. Introduction Throughout this paper w, l ∞ , ℓ1, c and c° denote the spaces of all, bounded, absolutely summable, convergent and null sequences x = ( xk ) with complex terms respectively. The notion of difference sequence space was introduced by Kizmaz , who studied the difference sequence spaces l ∞ (∆ ) , c(∆ ) and c0 ( ∆ ) , where

Z ( ∆ ) = { x = ( xk ) ∈ w : ( ∆xk ) ∈ Z } ,

where ∆x = ( ∆xk ) = ( xk − xk +1 ) and ∆ 0 xk = xk for all k, for Z= l ∞ , c and c0 . An Orlicz function M :[0, ∞) → [0, ∞) is a function, which is continuous, non-decreasing and convex with M (0) = 0 , M ( x) > 0 , for x > 0 and M ( x) → ∞ , as x → ∞ . An Orlicz function M can always be represented in the following integral form: x

M(x) = ∫ p (t ) dt , 0

where p, known as kernel of M, is right differentiable for t ≥0, p(0) = 0, p(t) > 0 for t > 0, p is non-decreasing, and p (t ) → ∞ as t → ∞ .

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Consider the kernel p(t) associated with the Orlicz function M(t), and let q(s) = sup{t: p(t) ≤ s } Then q possesses the same properties as the function p. Suppose now x

Φ ( x) = ∫ q ( s ) ds 0

Then Φ is an Orlicz function. The functions M and Φ are called mutually complementary Orlicz functions. Now we state the following well known results which can be found in . Let M and F are mutually complementary Orlicz functions. Then we have (Young’s inequality) (1) (i) For x, y ≥ 0, xy ≤ M(x) + Φ ( y ) We also have (2) (ii) For x≥ 0, xp(x) = M(x) + Φ ( p ( x) ) (iii) M(λx) < λM(x) (3)

for all x ≥ 0 and λ with 0< λ0 there exist Rk>0 and xk>0 such that M(kx) ≤ RkM(x)

for all x ∈ (0, xk].

Moreover an Orlicz function M is said to satisfy the ∆2-condition if and only if M (2 x) 0  . k =1    ρ  For more details about Orlicz functions and sequence spaces associated with Orlicz functions one may refer to [2-5].

Let Λ = (λk) be a sequence of non-zero scalars. Then for a sequence space E, the multiplier sequence space E(Λ), associated with the multiplier sequence Λ is defined as E(Λ) =

{(xk ) ∈ w : (λk xk ) ∈ E}.

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The scope for the studies on sequence spaces was extended by using the notion of associated multiplier sequences. Goes and Goes  defined the differentiated sequence space dE and integrated sequence space ∫ E for a given sequence space E, using the

multiplier sequences (k-1) and (k) respectively. A multiplier sequence can be used to accelerate the convergence of the sequences in some spaces. In some sense, it can be viewed as a catalyst, which is used to accelerate the process of chemical reaction. Sometimes the associated multiplier sequence delays the rate of convergence of a sequence. Thus it also covers a larger class of sequences for study. In the present article we shall consider a general multiplier sequence Λ = (λk) of non-zero scalars.

The notion of duals of sequence spaces was introduced by Köthe and Toeplitz . Later on it was studied by Kizmaz , Kamthan  and many others. Let E and F be two sequence spaces. Then the F dual of E is defined as

EF = {(xk)∈ w : (xkyk)∈ F for all(yk)∈ E }. For F = ℓ1, the dual is termed as Köthe-Toeplitz or α-dual of E and denoted by Eα. More precisely, we have the following definition of Köthe Toeplitz dual of E:   E α = a = (ak ) : ∑ ak xk < ∞, for all x ∈ E  . k   It is known that if X Ì Y , then Yα ⊂ Xα. If EFF=E, where EFF= (EF)F, then E is said to be F-reflexive or F-perfect. In particular, if Eαα = E, then E is also said to be a Köthe space. Let Λ = (λk) be a sequence of non-zero scalars. Then we define the following spaces.

Definition 1.1. Let M be any Orlicz function. Then we define ∞   l% M ( ∆, Λ ) =  x ∈ w : δ ∆Λ ( M , x ) = ∑ M ( ∆λk xk ) < ∞  , k =1   where ∆λk xk = λk xk − λk +1 xk +1 for all k ≥ 1. ~ We can write l% M ∆ 0 , Λ = l M (Λ ) and if λk= 1 for all k ≥ 1, then we write ~ l% M ∆ 0 , Λ = l M .

(

)

(

)

Similarly we can define l% M ( ∇, Λ ) , where ∇λk xk = λk xk − λk −1 xk −1 for all k ≥ 1. Definition 1.2. Let M and Φ be mutually complementary functions. Then we define ∞   l M ( ∆, Λ ) =  x ∈ w : ∑ (∆λk xk ) yk converges for all y ∈ l% Φ  . k =1   We call this sequence space as Orlicz difference sequence space associated with the multiplier sequence Λ = (λk).

We can write l M ( ∆ 0 , Λ ) = l M (Λ ) and if λk= 1 for all k ≥ 1, then we write

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(

)

l M ∆0 , Λ = l M . Similarly we can define l M ( ∇, Λ ) where ∇λk xk = λk xk − λk −1 xk −1 for all k ≥ 1. One can easily observe in the special case M(x) = xp with 0 0, there exists a positive integer n0 such that

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xi − x j

M

inf 0 : ρ M = λ x +   ≤ 1 , ∑ 1 1 (M ) k =1   ρ   ∆

(11)

(ii) l M ( ∇, Λ ) is a normed linear space under the norm . ( M ) defined by ∇

x

Proof. (i) Clearly x

∆ (M )

∞  ∇λk xk    > = inf 0 : ρ M   ≤ 1 . ∑ (M ) k =1   ρ  

=0 if x=θ. Next suppose x

∆ (M )

(12)

=0. Then from (11) we have

λ1 x1 =0 and so λ1 x1 = 0 .

(13)

∞  ∆λk xk    Again inf  ρ > 0 : ∑ M   ≤ 1 =0. This implies that for a given ε > 0 , there k =1   ρ   exists some ρ ε (0 < ρ ε < ε ) such that

 ∆λk xk  sup M   ≤ 1. ρ k ε    ∆λk xk  This implies that M   ≤ 1 for all k≥ 1. Thus  ρε   ∆λk xk   ∆λk xk  M ≤M  ≤1 ε ρ ε     for all k≥ 1. ∆λni xni → ∞ . It follows that Suppose ∆λni xni ≠ 0 , for some i. Let ε → 0 , then

ε

 ∆λn xn i i M  ε 

  → ∞ as ε → 0 for some ni ∈ N . This is a contradiction. Therefore   ∆λk xk =0

(14)

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for all k≥ 1. Thus, by (13) and (14), it follows that λ k x k =0 for all k≥ 1. Hence x = θ , since (λk) is a sequence of non-zero scalars. Let x = (xk) and y = (yk) be any two elements of l M ( ∆, Λ ) . Then there exist ρ1 , ρ 2 >0 such that  ∆λk xk   ∆λk yk  sup M   ≤ 1 and sup M   ≤ 1. k k  ρ1   ρ2  Let ρ = ρ1 + ρ 2 . Then by convexity of M, we have

 ∆λk ( xk + yk ) sup M   ρ k 

  ∆λk xk   ∆λk yk  ρ1 ρ2 + ≤ sup M  sup M    ≤1.  ρ1 + ρ 2 k  ρ1  ρ1 + ρ 2 k  ρ2  

Hence we have

x+ y

  ∆λk ( xk + yk ) + + = ( ) inf λ x y  ρ > 0 : sup M  1 1 1 (M ) ρ k   ∆

   ≤ 1   

 ∆λk xk    ≤ λ1 x1 + inf  ρ1 > 0 : sup M   ≤ 1 + λ1 y1 k   ρ1     ∆λk yk   + inf  ρ 2 > 0 : sup M   ≤ 1 . k  ρ 2    This implies x + y

∆ (M )

≤ x

∆ (M )

+ x

∆ (M )

.

Finally, let ν be any scalar. Then

νx

∆ (M )

 ∆νλk xk    = νλ1 x1 + inf  ρ > 0 : sup M   ≤ 1 ρ   k    ∆λk xk    = ν λ1 x1 + inf r ν > 0 : sup M   ≤ 1 k  r    =ν x

where r =

∆ (M )

ρ . This completes the proof. ν

(ii) Proof is easy than part (i). ∆

Remark. It is obvious that the norms . ( M ) and . ( M ) are equivalent. Proposition 2.7. For x∈ l M ( ∇, Λ ) , we have

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 ∇λk xk  M ∑  || x ||(∆M−1 ) k =1  ∞

  ≤1.  

Proof. Proof is immediate from (12). ∇

Now we show that the norms . ( M ) and . M are equivalent. To prove this some other results are required. First we prove those results.

Proposition 2.8. Let x ∈ l M ( ∇, Λ ) with

(

δ Φ, { p ( ∇λn xn

x

)}) ≤1.

M

{ p ( ∇λ x )} ∈ l%

≤1. Then

n n

Φ

and

~ Proof. For any z ∈ lΦ , we may write if δ (Φ, z ) ≤ 1

∇ || x ||M (∇λi xi ) zi ≤  ∑ ∇ i =1 δ (Φ, z ) || x ||M ∞

Let now x∈ l M ( ∇, Λ ) with x

M

if δ (Φ, z ) > 1

.

(15)

≤1. Also x(n) = (x1,… xn, 0,0, …..) ∈ l M ( ∇, Λ ) for n ≥1.

We observe that

x

≥ M

∑ (∇λ x ) y

(n) i

i i

i =1

=

∇ M

( n) i i

i =1

~ for every y ∈ lΦ with δ (Φ, y ) ≤1 and thus x( n)

∑ (∇λ x

≤ x

∇ M

Since n

(

∑ Φ p ( ∇λi xi i =1

We find that

{ p ( ∇λ x )} ∈ ~l (n) i i

))

) yi ,

≤ 1.

( (

= ∑ Φ p ∇λi xi( n ) i =1

n ≥1

)) .

for each n ≥1. Let l ≥1 be an integer such that

Φ l

∑ Φ ( p ( ∇λ x ) ) >1. ∑ Φ ( p ( ∇λ x ∞

Then

i =1

i

i i

i =1

)) >1. Using (2), we have Φ ( p ( ∇ λ x ) ) < M ( ∇λ x ) + Φ ( p ( ∇ λ x ) ) = ∇ λ x p ( ∇λ x ) (l )

i

(l )

i

(l )

i

i

l i i

for all i, l ≥1. So by (15), we get

∑ Φ ( p ( ∇λ x ∞

i =1

i

(l ) i

)) < || x

(l )

(l )

i

i

i

l i i

( {(

||∇M δ Φ, p ∇λi xil

)}) .

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This implies that || x (l ) ||∇M >1, a contradiction. This contradiction implies that l

∑ Φ ( p ( ∇λ x ) ) ≤1

{

for all l ≥1. Hence p ( ∇λi xi

)} ∈ l%

i i

i =1

Φ

( {

)}) ≤1.

and δ Φ, p ( ∇λi xi

x∈ l% M ( ∇, Λ )

and

~ Proof. Let y= p ( ∇λi xi ) / sgn(∇λi xi ) . Then from Proposition 2.8, y ∈ lΦ

and

Proposition

2.9.

δ ∇Λ ( M , x) ≤ x

∇ M

x∈ l M ( ∇, Λ )

Let

with

x

∇ M

≤1.

Then

.

{

}

δ (Φ, y ) ≤1. By (2), we get ∞

∑ M ( ∇λ x ) ≤ ∑ M ( ∇ λ x ) + ∑ Φ ( p ( ∇ λ x ) ) i i

i =1

i i

i =1 ∞

= ∑ ∇λi xi p ( ∇λi xi i =1 ∞

=

∑ (∇λ x ) y i i

i =1

This implies that δ ∇Λ ( M , x) ≤ x

∇ M

i

i i

i =1

≤ x

) ∇ M

.

. ∞

 ∇λk xk  ≤1. ∇  M  

∑ M  || x ||

Proposition 2.10. For x∈ l M ( ∇, Λ ) , we have

k =1

Proof. Proof is immediate from Proposition 2.9. Theorem 2.11. For x ∈ l M ( ∇, Λ ) , || x ||∇( M ) ≤ || x ||∇M ≤2 || x ||∇( M ) . Proof. We have ∞  ∇λk xk    inf 0 : > M ρ =   ≤ 1 . ∑ (M ) k =1   ρ   Then using Proposition 2.10, we get || x ||∇( M ) ≤ || x ||∇M .

x

Let us suppose that x∈ l M ( ∇, Λ ) with x

∇ (M )

≤1. Then x∈ l% M ( ∇, Λ ) and δ ∇Λ ( M , x) ≤1.

Indeed, 1 || x ||∇( M )

∑ M ( ∇λi xi ) ≤ i =1

 ∇λi xi   ≤ 1 , ∇ M ( )  

∑ M  || x || i =1

by Proposition 2.7.

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 x % ( ∇, Λ ) with δ  M , x ∈ l M  || x ||∇( M ) || x ||∇( M )  arbitrary z∈ l% M ( ∇, Λ ) , Thus

  ≤1. We further observe that for an 

 ∞  || z ||∇M = sup  ∑ (∇λi zi ) yi : δ ( Φ, y ) ≤ 1 ≤ 1+ δ ∇Λ ( M , z )  i =1  x using (1). Hence taking z = , we have || x ||∇( M )

x || x ||∇( M )

∞  x  ≤ 1+ ∑ M  ≤2  || x ||∇( M )  i =1   M

by Proposition 2.7. Thus || x ||∇M ≤ 2 || x ||∇( M ) . This completes the proof. Proposition 2.12. For any Orlicz function M, l M ( ∇, Λ ) = l /M ( ∇, Λ ) , where ∞   ∇λk xk  l /M ( ∇, Λ ) =  x ∈ w : ∑ M   0  . 

Proof. Proof follows from Proposition 2.10. In view of above Proposition we give the following definition. Definition 2.13. For any Orlicz function M, ∞    ∇λk xk  hM ( ∇, Λ ) =  x ∈ w : ∑ M   < ∞ , for each ρ > 0  . k =1    ρ 

Clearly hM ( ∇, Λ ) is a subspace of l M ( ∇, Λ ) . Henceforth we shall write ||.|| instead of . ( M ) provided it does not lead to any confusion. The topology of hM ( ∇, Λ ) is the one it ∇

inherits from ||.||. Proposition 2.14. Let M be an Orlicz function. Then ( hM ( ∇, Λ ) ,||.||) is an AK-BK space. Proof. First we show that hM ( ∇, Λ ) is an AK space. Let x∈ hM ( ∇, Λ ) . Then for each ε, 0< ε < 1, we can find an n0 such that  ∇λi xi   ≤1.  ε 

∑M 

i ≥ n0

Hence for n ≥ n0,    ∇λi xi    ∇λi xi   ||x-x(n)|| = inf  ρ > 0 : ∑ M   ≤ 1 ≤ inf  ρ > 0 : ∑ M   ≤ 1 < ε . i ≥ n +1 i≥n  ρ    ρ    

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Thus we can conclude that hM ( ∇, Λ ) is an AK space. Next to show hM ( ∇, Λ ) is an BK space it is enough to show hM ( ∇, Λ ) is a closed subspace of hM ( ∇, Λ ) . For this let {xn} be a sequence in hM ( ∇, Λ ) such that ||xn-x|| →0, where x∈ hM ( ∇, Λ ) . To complete the proof we need to show that x∈ hM ( ∇, Λ ) , i.e.,  ∇λi xi  0. To ρ>0 there corresponds an l such that ||xl-x|| ≤ convexity of M,

(

 2 ∇λi xil − 2 ∇λi xil − ∇λi xi  ∇λi xi  M  =∑M  ∑ 2ρ ρ i ≥1   i ≥1   2 ∇λi xil 1 ≤ ∑M  2 i ≥1  ρ   2 ∇λi xil 1 ≤ ∑M  2 i ≥1  ρ 

2

. Then using

)   

 1  2 ∇λi ( xil − xi )  + ∑M   2 i ≥1  ρ    1  2 ∇λi ( xil − xi )  + ∑M   2 i ≥1  || x l − x ||  

     < ∞  

by proposition 2.7. Thus x ∈ hM ( ∇, Λ ) and consequently hM ( ∇, Λ ) is a BK space.

Proposition 2.15. Let M be an Orlicz function. If M satisfies the ∆2-condition at 0, then l M ( ∇, Λ ) is an AK space. Proof. In fact we shall show that if M satisfies the ∆2-condition at 0, then l M ( ∇, Λ ) = hM ( ∇, Λ ) and the result follows. Therefore it is enough to show that l M ( ∇, Λ ) ⊂ hM ( ∇, Λ ) . Let x ∈ l M ( ∇, Λ ) , then ρ > 0,

 ∇λi xi   0. If ρ ≤ l, then ∑ M   < ∞ . Let now l < ρ and put k= . l i ≥1  l  Since M satisfies ∆2-condition at 0, there exist R ≡ Rk>0 and r ≡ rk > 0 with M(kx)≤RM(x) for all x ∈ (0, r]. By (16) there exists a positive integer n1 such that  ∇λi xi  1  r  M  < rp   ρ  2 2

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for all i ≥ n1. We claim that ∇λ j x j

∇λi xi

ρ

≤ r for all i ≥ n1. Otherwise, we can find j > n1 with

> r, and thus

ρ

∇λ j x j  ∇λ j x j  1 r  ≥ ∫ ρ p(t ) dt > rp  M  ρ  r /2 2 2   Is a contradiction. Hence our claim is true. Then we can find that  ∇λi xi   ∇λi xi  M  ≤∑M  , ∑ i ≥ n1  l  i ≥ n1  ρ  and hence  ∇λi xi  M  0. This completes our proof.

Proposition 2.16. Let M1 and M2 be two Orlicz functions. If M1 and M2 are equivalent then l M1 ( ∇, Λ ) = l M 2 ( ∇, Λ ) and the identity map

(

I: l M1 ( ∇, Λ ) , . M ∇

1

) → (l

( ∇, Λ ) , . M ∇

M2

2

)

is a topological isomorphism. Proof. Let M1 and M2 are equivalent and so satisfy (4). Suppose x ∈ l M 2 ( ∇, Λ ) , then  ∇λi xi   < ∞ i =1  ρ  ∇λi xi for some ρ > 0. Hence for some l ≥1, ≤ x0 for all i ≥1. Therefore, lρ ∞

∑M

2

 α ∇λi xi  ∞  ∇λi xi  M  ≤ ∑ M2   < ∞. ∑ 1 i =1  l ρ  i =1  ρ  ∞

Thus l M 2 ( ∇, Λ ) ⊂ l M1 ( ∇, Λ ) . Similarly l M1 ( ∇, Λ ) ⊂ l M 2 ( ∇, Λ ) . Let us abbreviate here . M and . M by . 1 and . 2 , respectively. For x ∈ l M 2 ( ∇, Λ ) , ∇

1

2

 ∇λi xi  M ≤ 1. ∑ 2  x  i =1 2   x  x  One can find µ >1 with  0  µp2  0  ≥1, where p2 is the kernel associated with M2. 2 2 Hence  ∇λi xi   x0  x  M2  ≤   µp2  0    x  2 2 2   ∞

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SDU Journal of Science (E-Journal), 2010, 5 (1): 119-136 ___________________________________________________________________

for all i ≥1. This implies that

∇λi xi

µ x

≤ x0 for all i ≥1. Therefore

2 ∞

∑M i =1

1

 α ∇λi xi   µ x2

  1 α  x  x  such that γβ  0  p1  0  ≥1. Thus αµ −1 x 1 ≤ x 2 ≤ βγ x 1 which establishes that I is a 2 2 topological isomorphism. Proposition 2.17. (i) l M ( Λ ) ⊂ l M ( ∇, Λ ) ,

(ii) l M ( Λ ) ⊂ l M ( ∆, Λ ) .

Proof. (i) Proof follows from the following inequality:  ∇λi xi  1 ∞  λi xi  1 ∞  λi −1 xi −1 ≤ M M    + ∑M  ∑ ∑ i =1  2 ρ  2 i =1  ρ  2 i =1  ρ ∞

 , 

(ii) Proof is similar to that of part (i). Proposition 2.18. Let M be an Orlicz function and p the corresponding kernel. If p(x) = 0 for all x in [0, x0] where x0 is some positive number, then l M ( ∇, Λ ) is

topologically isomorphic to l ∞ ( ∇, Λ ) and hM ( ∇, Λ ) is topologically isomorphic to c0 ( ∇, Λ ) .

Proof. Let p(x) = 0 for all x in [0, x0]. If y ∈ l ∞ ( ∇, Λ ) , then we can find a ρ > 0 such  ∇λi yi   < ∞, giving thus y ∈ l M ( ∇, Λ ) . On ρ i =1  ρ  ∞  ∇λi yi  the other hand let y ∈ l M ( ∇, Λ ) , then ∑ M   < ∞, for some ρ > 0 and so i =1  ρ  ∇λi yi 0. (It is easy to show that ||y||∞= sup ( ∇λi yi i

i

)

is a norm on

l ∞ ( ∇, Λ ) ). For every ε, 0< ε < α, we can determine yj with ∇λ j y j > α-ε and so ∞

 ∇λi yi x1   (α − ε ) x1  . ≥M α α    

∑M  i =1

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 ∇λi yi x1   ≥1, and so ||y||∞≤ x1||y||, for otherwise α i =1   ∞ ∞  ∇λi yi   ∇λi yi x0  M  >1 is a contradiction by Proposition 2.7. Again, ∑ M   =0 ∑ α i =1 i =1  || y ||    1 and it follows that || y ||≤ || y ||∞ . Thus the identity map x0

Since M is continuous, we find

∑M 

I: ( l M ( ∇, Λ ) , . ) → ( l ∞ ( ∇, Λ ) , . )

is a topological isomorphism. For the last part, let y ∈ hM ( ∇, Λ ) , then for any ε > 0, ∇λi yi ≤ εx1, for all sufficiently large i, where x1 is some positive number with p(x1) > 0. Hence y ∈ c0 ( ∇, Λ ) . Next let y ∈ c0 ( ∇, Λ ) . Then for any ρ>0,

∇λi yi

ρ

1 < x0 for all sufficiently large i. Thus 2

 ∇λi yi  M  0 and so y ∈ hM ( ∇, Λ ) . Hence hM ( ∇, Λ ) = c0 ( ∇, Λ ) and we  ρ  are done.

Corollary 2.19. Let M be an Orlicz function and p the corresponding kernel. If p(x) = 0 for all x in [0, x0] where x0 is some positive number, then l M ( ∇, Λ ) is topologically

isomorphic to l ∞ and hM ( ∇, Λ ) is topologically isomorphic to c0 . Proof. Let us define the mapping for Z = l ∞ , c0

T: Z ( ∇, Λ ) → Z

by Tx = ( ∇λk xk ) , for every x∈ Z ( ∇, Λ ) . Then clearly T is a linear homeomorphism. Hence the proof follows from Proposition 2.18. Lemma 2.20. Let M be an Orlicz function. Then x ∈ l M ( ∆, Λ ) implies ( k −1λk xk ) ∈ l ∞ . Proof. Let x ∈ l M ( ∆, Λ ) . Then, one can easily prove that ( ∆λk xk ) ∈ l ∞ which gives the

result ( k −1λk xk ) ∈ l ∞ .

Proposition 2.21. Let M be an Orlicz function and p be the corresponding kernel of M. If p(x) = 0 for all x in [0, x0], where x0 is some positive number, then (i) Köthe-Toeplitz dual of l M ( ∆, Λ ) is D1, where ∞   D1= (ak ) : ∑ k λk−1ak < ∞  , k =1  

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(ii) Köthe-Toeplitz dual of D1 is D2, where

{

}

D2= (bk ) : sup k −1 λk bk < ∞ . k

Proof. (i) Let a ∈ D1 and x ∈ l M ( ∆, Λ ) . Then ∞

∑ a k xk = k =1

∑ k λk−1ak k −1 λk xk ≤ sup k −1 λk xk

∑k λ

k

k =1

−1 k k

k =1

α

a < ∞.

α

Hence a ∈  l M ( ∆, Λ )  . Thus, the inclusion D1 ⊂  l M ( ∆, Λ )  holds. α

Conversely suppose that a ∈  l M ( ∆, Λ )  . Then

∑a x k =1

k

k

< ∞ for every x ∈ l M ( ∆, Λ ) .

So we can take xk = λk−1k for all k ≥1, because then (xk) ∈ l ∞ ( ∆, Λ ) and hence

(xk) ∈ l M ( ∆, Λ ) as shown in Proposition 2.18. Now

k =1

k =1

α

∑ k λk−1ak = ∑ ak xk < ∞ and thus a∈ D1. Hence, the inclusion l M ( ∆, Λ ) ⊂ D1

holds. (ii) Proof follows by similar arguments used in the prove of case (i). Proposition 2.22. Let M be an Orlicz function and p be the corresponding kernel of M. If p(x) = 0 for all x in [0, x0], where x0 is some positive number, then Köthe-Toeplitz dual of hM ( ∆, Λ ) is D1, where D1 is defined as in Proposition 2.21. Proof. Let a ∈ D1 and x ∈ hM ( ∆, Λ ) . Then ∞

∑ a k xk = k =1

∑ k λk−1ak k −1 λk xk ≤ sup k −1 λk xk k

k =1

∑k λ

−1 k k

k =1

α

a < ∞.

α

Hence a ∈  hM ( ∆, Λ )  , that is the inclusion D1 ⊂  hM ( ∆, Λ )  holds. α

Conversely suppose that a ∈  hM ( ∆, Λ )  and a ∉ D1 . Then there exists a strictly increasing sequence (ni) of positive integers such that n1 < n2 i.

Define (xk) by , 0 xk =  −1 k λk sgn ak / i ,

1 ≤ k ≤ n1 ni < k ≤ ni +1

Then (xk) ∈ c0 ( ∆, Λ ) and so by Proposition 2.18, (xk) ∈ hM ( ∆, Λ ) . Then we have

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∑ a k xk = k =1

n2

∑ ak xk +…+

k = n1 +1 n2

=

k = n1 +1

α

ni +1

∑a

k = ni +1

k λk−1ak +…+

k

x k +…

1 ni+1 k λk−1ak +…> 1+1+…= ∞. ∑ i k = ni +1 α

This contradicts to a Î  hM ( ∆, Λ )  . Hence a ∈ D1, i.e. the inclusion  hM ( ∆, Λ )  ⊂ D1 also holds. This completes the proof. References  Kizmaz H., 1981. On certain sequence spaces, Canadian Mathematical Bulletin, 24 (2): 169-176.  Kamthan P.K., Gupta M., 1981. Sequence Spaces and Series, Marcel Dekker Inc., New York, USA, p. 368.  Lindenstrauss J., Tzafriri L., 1971. On Orlicz sequence spaces, Israel Journal of Mathematics, 10: 379-390.  Gribanov Y., 1957. On the theory of ℓM-spaces(Russian), Ucenyja Zapiski Kazansk un-ta, 117: 62-65.  Krasnoselskii M.A., Rutitsky Y.B., 1961. Convex functions and Orlicz spaces, Groningen, Netherlands, p. 249.  Goes G., Goes S., 1970. Sequences of bounded variation and sequences of Fourier coefficients, Mathematische Zeitschrift, 118 (2): 93-102.  Köthe G., Toeplitz O., 1934. Linear Raume mit unendlichvielen koordinaten and Ringe unendlicher Matrizen, Journal Für Die Reine und Angewandte Mathematik, 1934 (171): 193-226.  Kamthan P.K., 1976. Bases in a certain class of Frechet spaces, Tamkang Journal of Mathematics, 7 (1): 41-49.  Başar F., Altay B., 2003. On the space of sequences of p-bounded variation and related matrix mappings, Ukrainian Mathematical Journal, 55 (1): 136-147.  Altay B., Başar F., 2007. The fine spectrum and the matrix domain of the difference operator ∆ on the sequence space l p , ( 0 < p < 1) , Communications in Mathematical Analysis, 2 (2): 1-11.

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