Volume 2011, Article ID 143974,6pages doi:10.1155/2011/143974

*Research Article*

**The Semi-Difference Entire Sequence Space**

*cs*

## ∩

*d*

### 1

**N. Subramanian,**

**1**

_{K. Chandrasekhara Rao,}

_{K. Chandrasekhara Rao,}

**2**

**and K. Balasubramanian**

**1**

*1 _{Department of Mathematics, SASTRA University, Thanjavur 613 401, India}*

*2*

_{Srinivasa Ramanujan Centre, SASTRA University, Kumbakonam 612 001, India}Correspondence should be addressed to N. Subramanian,nsmaths@yahoo.com

Received 24 November 2010; Accepted 7 February 2011 Academic Editor: Ricardo Estrada

Copyrightq2011 N. Subramanian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

LetΓdenote the space of all entire sequences. LetΛdenote the space of all analytic sequences. In
this paper, we introduce a new class of sequence space, namely, the semi-diﬀerence entire sequence
space*cs*∩*d*1. It is shown that the intersection of all semi-diﬀerence entire sequence spaces*cs*∩*d*1

is*I*⊂*cs*∩*d*1andΓΔ⊂*I.*

**1. Introduction**

A complex sequence, whose*kth term isxk*, is denoted by{x*k}*or simply*x. Let w be the set*

of all sequences and*φ*be the set of all finite sequences. Let∞*, c, c0*be the classes of bounded,

convergent, and null sequence, respectively. A sequence *x* {x*k}* is said to be analytic if

sup* _{k}*|xk|1/k

*<*∞. The vector space of all analytic sequences will be denoted byΛ. A sequence

*x*is called entire sequence if limk→ ∞|xk|1/k 0. The vector space of all entire sequences will

be denoted byΓ.

Given a sequence*x*{x*k}, itsnth section is the sequencex*n{x1*, x*2*, . . . , xn,*0,0, . . .}.

Let *δ*n 0,0, . . . ,1,0,0, . . ., 1 in the *nth place and zeros elsewhere,* *s*k

0,0, . . . ,1,−1,0, . . ., 1 in the*nth place,* −1 in the *n* 1th place and zeros elsewhere. An

FK-space Fr´echet coordinate space is a Fr´echet space which is made up of numerical

sequences and has the property that the coordinate functionals*pkx* *xk* *k* 1,2,3, . . .

are continuous.

We recall the following definitionsone may refer to Wilansky1.

An FK-space is a locally convex Fr´echet space which is made up of sequences and has

the property that coordinate projections are continuous. A metric space*X, d*is said to have

AKor sectional convergenceif and only if*dx*n*, x* → *x*as*n* → ∞,see1. The space is

said to have ADorbe an AD-space if*φ*is dense in*X, whereφ*denotes the set of all finitely

If*X*is a sequence space, we define

i*X*the continuous dual of*X;*

ii*Xα*_{}_{{a} _{a}

*k*:

_{∞}

*k1*|a*kxk|<*∞, for each*x*∈*X};*

iii*Xβ*_{}_{{a} _{a}

*k*:

_{∞}

*k1akxk* is convergent, for each*x*∈*X};*

iv*Xγ* _{}_{{a} _{a}

*k*: sup*n*|

_{n}

*k1akxk|<*∞, for each*x*∈*X};*

vlet*X*be an FK-space⊃*φ. Then,Xf* _{}_{{f}_{}* _{δ}*n

_{}

_{:}

_{f}_{∈}

_{X}_{}.}

*Xα _{, X}β_{, X}γ*

_{are called the}

_{α}_{}

_{or K ¨othe-T ¨oeplitz}

_{}

_{dual of}

_{X,}_{β—}_{}

_{or generalized K ¨othe-T ¨oeplitz}

_{}

dual of*X,γ*dual of*X. Note thatXα*_{⊂}_{X}β_{⊂}_{X}γ_{. If}_{X}_{⊂}_{Y}_{, then}_{Y}μ_{⊂}_{X}μ_{, for}_{μ}_{}_{α,}_{β, or}_{γ.}

Let *p* *pk* be a sequence of positive real numbers with sup*kpk* *G* and *D*

max{1,2*G−1*}. Then, it is well known that for all*ak, bk* ∈ *C, the field of complex numbers,*

for all*k*∈*N,*

|a*k* *bk|pk* ≤*D*

|a*k|pk* _{|b}_{k|}pk_{.}_{}_{1.1}_{}

**Lemma 1.1**Wilansky1, Theorem 7.2.7*. LetXbe an FK-space*⊃*φ. Then,*

i*Xγ* _{⊂}_{X}f_{;}

ii*ifXhas AK,Xβ*_{}_{X}f_{;}

iii*ifXhas AD,Xβ*_{}_{X}γ_{.}

**2. Definitions and Preliminaries**

LetΔ:*w* → *w*be the diﬀerence operator defined byΔ*x* *xk*−*xk 1*∞*k1*. Let

Γ

*x*∈*w*: lim

*k*→ ∞

|xk|1/k_{}_{0}_{,}

Λ *x*∈*w*: sup

*k*

|x*k|*1/k_{<}_{∞}

*.*

2.1

Define the setsΓΔ {x∈*w*:Δ*x*∈Γ}andΛΔ {x∈*w*:Δ*x*∈Λ}.

The spacesΓΔandΛΔare the metric spaces with the metric

*dx, y*inf *ρ >*0 : sup

*k*

_{}

Δ*xk*−Δ*yk*1/k

≤1

*.* 2.2

Because of the historical roots of summability in convergence, conservative space and matrices play a special role in its theory. However, the results seem mainly to depend on a

weaker assumption, that the spaces be semi-conservative.See Wilansky1.

Snyder and Wilansky3introduced the concept of semi-conservative spaces. Snyder

4 studied the properties of semi-conservative spaces. Later on, in the year 1996 the semi

replete spaces were introduced by Rao and Srinivasalu5.

In a similar way, in this paper, we define semi-diﬀerence entire sequence space*cs*∩*d1,*

**3. Main Results**

**Proposition 3.1.** Γ⊂ΓΔ*and the inclusion is strict.*

*Proof. Letx*∈Γ. Then, we have

|x*k|*1/k_{−→}_{0,} _{as}_{k}_{−→ ∞,} _{}_{3.1}_{}

|Δ*xk|*1/k

2 ≤

1 2

|xk|1/k 1

2

|xk 1|1/k_{,}_{by}_{}_{1.1}_{}

−→0, as*k*−→ ∞ by3.1*.*

3.2

Let*x*∈Γ. Then, we have

|xk|1/k_{−→}_{0 as}_{k}_{−→ ∞.} _{}_{3.3}_{}

Then,*xk*∈ΓΔfollows from the inequality1.1and3.3.

Consider the sequence*e* 1,1, . . .. Then,*e* ∈ ΓΔbut*e /*∈ Γ. Hence, the inclusion

Γ⊂ΓΔis strict.

**Lemma 3.2.** *A*∈Γ*, cif and only if*

lim

*n*→ ∞*ank* *exists for eachk*∈*N,* 3.4

sup
*n,k*
*k*
*i0*
*ani*

*<*∞. 3.5

* Proposition 3.3. Define the setd*1 {a

*ak*∈

*w*: sup

*|*

_{n,k∈N}_{k}

*j0nijai*| *<* ∞}*. Then,*

ΓΔ*β*_{}_{cs}_{∩}_{d}

1*.*

*Proof. Consider the equation*

*n*
*k0*
*akxk*
*n*
*k0*
*ak*
⎛

⎝*k*

*j0*
*yj*

⎞

⎠_{}*n*

*k0*
⎛

⎝*n*

*jk*
*aj*

⎞

⎠_{yk}_{}_{Cy}

*n,* 3.6

where*C* *Cnk*is defined by

*Cnk*
⎧
⎪
⎨
⎪
⎩
*n*
*jk*

*aj,* if 0≤*k*≤*n,*

0, if*k > n;* *n, k*∈*N.*

Thus, we deduce from Lemma 3.2with 3.6 that*ax* *akxk* ∈ *cs* whenever*x*

*xk* ∈ΓΔif and only if*Cy*∈ *c*whenever*y* *yk* ∈Γ, that is*C*∈Γ*, c*. Thus,*ak*∈*cs*

and*ak*∈*d*1byLemma 3.2and3.5and3.6, respectively. This completes the proof.

**Proposition 3.4.** ΓΔ*has AK.*

*Proof. Letx*{x*k} ∈*ΓΔ. Then,|Δ*xk|*1/k∈Γ. Hence,

sup
*k≥n 1*

|Δ*xk|*1/k−→0, as*k*−→ ∞, _{}_{3.8}_{}

*dx, x*n inf *ρ >*0 : sup

*k≥n 1*

|Δ*xk|*1/k_{≤}_{1}

−→0, as*n*−→ ∞, by using3.8

⇒*x*n−→*x* as*n*−→ ∞,

⇒ΓΔhas AK.

3.9

This completes the proof.

**Proposition 3.5.** ΓΔ*is not solid.*

To proveProposition 3.5, consider*xk* 1∈ΓΔand*αk*{−1*k*}. Then*αkxk*∈*/*

ΓΔ. Hence,ΓΔis not solid.

**Proposition 3.6.** ΓΔ*μ* _{}_{cs}_{∩}_{d1}_{for}_{μ}_{}_{α, β, γ, f}_{.}

*Proof.*

*Step 1.* ΓΔhas AK byProposition 3.4. Hence, byLemma 1.1ii, we getΓΔ*β* ΓΔ*f*.

ButΓΔ*βcs*∩*d*1. Hence,

ΓΔ*f* _{}_{cs}_{∩}_{d}

1*.* 3.10

*Step 2. Since AK*⇒AD. Hence, byLemma 1.1iii, we getΓΔ*β* ΓΔ*γ*. Therefore,

ΓΔ*γ*_{}_{cs}_{∩}_{d1.}_{}_{3.11}_{}

*Step 3.* ΓΔ is not normal by Proposition 3.5. Hence by Proposition 2.7 of Kamthan and

Gupta6, we get

ΓΔ*α*

*/*

ΓΔ*γ*

*/*

*cs*∩*d*1*.* 3.12

From3.10and3.11, we have

ΓΔ*β*_{ ΓΔ}*γ* _{ ΓΔ}*f* _{}_{cs}_{∩}_{d}

**Lemma 3.7**Wilansky1, Theorem 8.6.1. *Y* ⊃*X*⇔*Yf* _{⊂}_{X}f_{where}_{X}_{is an AD-space and}_{Y}_{an}

*FK-space.*

**Proposition 3.8. Let**Y*be any FK-space* ⊃ *φ. Then,Y* ⊃ ΓΔ*if and only if the sequence* *δ*k *is*

*weakly converges incs*∩*d1.*

*Proof. The following implications establish the result.*

*Y* ⊃ΓΔ⇔*Yf* _{⊂}_{ΓΔ}*f*_{, since}_{ΓΔ}_{has AD by}_{Lemma 3.7}_{.}

⇔*Yf* _{⊂}_{cs}_{∩}_{d}

1, sinceΓΔ*f* *cs*∩*d*1.

⇔for each*f*∈*Y*, the topological dual of*Y*.

⇔*fδ*k∈*cs*∩*d*1.

⇔*δ*kis weakly converges in*cs*∩*d1.*

This completes the proof.

**4. Properties of Semi-Difference Entire Sequence Space**

*cs*

### ∩

*d*

1
*Definition 4.1. An FK-space*Δ*X*is called “semi-diﬀerence entire sequence space*cs*∩*d*1” if its

dualΔ*Xf* ⊂*cs*∩*d*1.

In other wordsΔ*X*is semi-diﬀerence entire sequence space*cs*∩*d1*if*fδ*k∈*cs*∩*d1*

for all*f*∈Δ*X*for each fixed*k.*

*Example 4.2.* ΓΔis semi-diﬀerence entire sequence space*cs*∩*d*1. Indeed, ifΓΔis the space

of all diﬀerence of entire sequences, then byLemma 4.3,ΓΔ*f* *cs*∩*d*1.

**Lemma 4.3**Wilansky1, Theorem 4.3.7*. Letzbe a sequence. Thenzβ _{, P}*

_{}

_{is an AK space with}*P* *Pk* :*k* 0,1,2, . . .*, whereP0x* sup* _{m}*|

*m*|xn|

_{k1}zkxk|, andPnx*. For anyksuch that*

*zk/*0, P*kmay be omitted. Ifz*∈*φ, P*0*may be omitted.*

**Proposition 4.4. Let**zbe a sequence.zβ_{is a semi-di}_{ﬀ}_{erence entire sequence space}_{cs}_{∩}_{d}

1*if and only*

*ifzis incs*∩*d1.*

*Proof. Suppose that* *zβ* _{is a semi-di}_{ﬀ}_{erence entire sequence space} _{cs}_{∩}_{d}

1. *zβ* has AK by

Lemma 4.3. Therefore*zββ* _{z}β_{}*f* _{by Lemma 1}_{1}_{. So}_{z}β_{is semi-di}_{ﬀ}_{erence entire sequence}

space*cs*∩*d*1if and only if*zββ* ⊂*cs*∩*d*1. But then*z*∈*zββ* ⊂*cs*∩*d*1. Hence,*z*is in*cs*∩*d*1.

Conversely, suppose that*z*is in*cs*∩*d1. Thenzβ*⊃ {cs∩*d1}β*and*zββ* ⊂ {cs∩*d1}ββ*

*cs*∩*d1. Butzβ*_{}*f* _{}_{z}ββ_{. Hence,}_{}_{z}β_{}*f* _{⊂}_{cs}_{∩}_{d1. Therefore}_{z}β_{is semi-di}_{ﬀ}_{erence entire sequence}

space*cs*∩*d1. This completes the proof.*

* Proposition 4.5. Every semi-diﬀerence entire sequence spacecs*∩

*d*1

*contains*Γ

*.*

*Proof. Let*Δ*X*be any semi-diﬀerence entire sequence space*cs*∩*d1. Hence,*Δ*Xf* ⊂*cs*∩*d1.*

Therefore*fδ*k ∈*cs*∩*d1* for all*f* ∈ Δ*X*. So,{δk}is weakly converges in*cs*∩*d1*with

respect toΔ*X. Hence,*Δ*X* ⊃ ΓΔbyProposition 3.8. ButΓΔ ⊃ Γ. Hence,Δ*X* ⊃ Γ. This

completes the proof.

*Proof. Let*Δ*X* ∞* _{n1}*Δ

*Xn. Then*Δ

*X*is an FK-space which contains

*φ. Also everyf*∈Δ

*X*

can be written as*f* *g*1 *g*2 *. . .* *gm*, where*gk* ∈ Δ*Xn*for some*n*and for 1 ≤ *k* ≤ *m.*

But then*fδk* _{g}

1*δk* *g*2*δk* · · · *gmδk*. SinceΔ*Xn* *n*1,2, . . .are semi-diﬀerence

entire sequence space*cs*∩*d1, it follows thatgiδk*_{}_{∈}_{cs}_{∩}_{d1}_{for all}_{i}_{} _{1,}_{2, . . . m. Therefore}

*fδk* ∈ *cs*∩*d1* for all*k* and for all*f. Hence,*Δ*X* is semi-diﬀerence entire sequence space

*cs*∩*d*1. This completes the proof.

* Proposition 4.7. The intersection of all semi-diﬀerence entire sequence spacecs*∩d1

*isI*⊂

*cs*∩

*d*1

*β*

*and*ΓΔ⊂*I.*

*Proof. Let* *I* be the intersection of all semi-diﬀerence entire sequence space *cs* ∩ *d1. By*

Proposition 4.4, we see that the intersection

*I* ⊂ ∩*zβ*:*z*∈*cs*∩*d*1

{cs∩*d*1}*β.* 4.1

ByProposition 4.6it follows that*I*is semi-diﬀerence entire sequence space*cs*∩*d*1. By

Proposition 4.5, consequently

Γ*M* ΓΔ⊂*I.* 4.2

From4.1and4.2, we get*I*⊂ {cs∩*d1}β*andΓΔ⊂*I. This completes the proof.*

* Corollary 4.8. The smallest semi-diﬀerence entire sequence spacecs* ∩

*d1*

*is*

*I*⊂

*cs*∩

*d1β*

*and*

ΓΔ⊂*I.*

**Acknowledgments**

The author wishes to thank the referees for their several remarks and valuable suggestions that improved the presentation of the paper and also thanks Professor Dr. Ricardo Estrada, Department of Mathematics, Louisiana State University, for his valuable moral support in connection with paper presentation.

**References**

1 *A. Wilansky, Summability Through Functional Analysis, vol. 85 of North-Holland Mathematics Studies,*
North-Holland Publishing, Amsterdam, The Netherlands, 1984.

2 H. I. Brown, “The summability field of a perfect − *method of summation,” Journal d’Analyse*

*Math´ematique, vol. 20, pp. 281–287, 1967.*

3 A. K. Snyder and A. Wilansky, “Inclusion theorems and semi-conservative*FKspaces,” The Rocky*

*Mountain Journal of Mathematics, vol. 2, no. 4, pp. 595–603, 1972.*

4 *A. K. Snyder, “Consistency theory in semi-conservative spaces,” Studia Mathematica, vol. 71, no. 1, pp.*
1–13, 1982.

5 *K. C. Rao and T. G. Srinivasalu, “The Hahn sequence space-II,” Journal of Faculty of Education, vol. 1,*
no. 2, pp. 43–45, 1996.

6 *P. K. Kamthan and M. Gupta, Sequence Spaces and Series, vol. 65 of Lecture Notes in Pure and Applied*

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