The Semi Difference Entire Sequence Space

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

d

1

2

and K. Balasubramanian

1

1Department of Mathematics, SASTRA University, Thanjavur 613 401, India 2Srinivasa Ramanujan Centre, SASTRA University, Kumbakonam 612 001, India

Correspondence should be addressed to N. Subramanian,nsmaths@yahoo.com

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 spacecsd1. It is shown that the intersection of all semi-diﬀerence entire sequence spacescsd1

isIcsd1andΓΔ⊂I.

1. Introduction

A complex sequence, whosekth term isxk, is denoted by{xk}or simplyx. Let w be the set

of all sequences andφbe the set of all finite sequences. Let, c, c0be the classes of bounded,

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

supk|xk|1/k <∞. The vector space of all analytic sequences will be denoted byΛ. A sequence

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

be denoted byΓ.

Given a sequencex{xk}, itsnth section is the sequencexn{x1, x2, . . . , xn,0,0, . . .}.

Let δn 0,0, . . . ,1,0,0, . . ., 1 in the nth place and zeros elsewhere, sk

0,0, . . . ,1,−1,0, . . ., 1 in thenth 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 functionalspkx 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 spaceX, dis said to have

AKor sectional convergenceif and only ifdxn, xxasn → ∞,see1. The space is

said to have ADorbe an AD-space ifφis dense inX, whereφdenotes the set of all finitely

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IfXis a sequence space, we define

iXthe continuous dual ofX;

ii{a a

k:

k1|akxk|<∞, for eachxX};

iii{a a

k:

k1akxk is convergent, for eachxX};

iv {a a

k: supn|

n

k1akxk|<∞, for eachxX};

vletXbe an FK-space⊃φ. Then,Xf {fδn:fX}.

, Xβ, Xγare called theαor K ¨othe-T ¨oeplitzdual ofX,β—or generalized K ¨othe-T ¨oeplitz

dual ofX,γdual ofX. Note thatXαXβXγ. IfXY, thenYμXμ, forμα,β, orγ.

Let p pk be a sequence of positive real numbers with supkpk G and D

max{1,2G−1}. Then, it is well known that for allak, bkC, the field of complex numbers,

for allkN,

|ak bk|pkD

|ak|pk |bk|pk. 1.1

Lemma 1.1Wilansky1, Theorem 7.2.7. LetXbe an FK-spaceφ. Then,

i Xf;

iiifXhas AK,XβXf;

2. Definitions and Preliminaries

LetΔ:wwbe the diﬀerence operator defined byΔx xkxk 1k1. Let

Γ

xw: lim

k→ ∞

|xk|1/k0,

Λ xw: sup

k

|xk|1/k<

.

2.1

Define the setsΓΔ {x∈wx∈Γ}andΛΔ {x∈wx∈Λ}.

The spacesΓΔandΛΔare the metric spaces with the metric

dx, yinf ρ >0 : sup

k

Δxk−Δyk1/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 spacecsd1,

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3. Main Results

Proposition 3.1. Γ⊂ΓΔand the inclusion is strict.

Proof. Letx∈Γ. Then, we have

xk|1/k

2 ≤

1 2

|xk|1/k 1

2

|xk 1|1/k, by1.1

3.2

Letx∈Γ. Then, we have

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

Consider the sequencee 1,1, . . .. Then,e ∈ ΓΔbute /∈ Γ. Hence, the inclusion

Γ⊂ΓΔis strict.

Lemma 3.2. A∈Γ, cif and only if

lim

n→ ∞ank exists for eachkN, 3.4

sup n,k k i0 ani

<∞. 3.5

Proposition 3.3. Define the setd1 {a akw : supn,k∈N|

k

j0nijai| < ∞}. Then,

ΓΔβcsd

1.

Proof. Consider the equation

n k0 akxk n k0 ak

k

j0 yj

n

k0

n

jk aj

ykCy

n, 3.6

whereC Cnkis defined by

Cnk ⎧ ⎪ ⎨ ⎪ ⎩ n jk

aj, if 0≤kn,

0, ifk > n; n, kN.

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Thus, we deduce from Lemma 3.2with 3.6 thatax akxkcs wheneverx

xk ∈ΓΔif and only ifCycwhenevery yk ∈Γ, that isC∈Γ, c. Thus,akcs

andakd1byLemma 3.2and3.5and3.6, respectively. This completes the proof.

Proposition 3.4. ΓΔhas AK.

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

sup k≥n 1

dx, xn inf ρ >0 : sup

k≥n 1

xk|1/k1

−→0, asn−→ ∞, by using3.8

xn−→x asn−→ ∞,

⇒ΓΔhas AK.

3.9

This completes the proof.

Proposition 3.5. ΓΔis not solid.

To proveProposition 3.5, considerxk 1∈ΓΔandαk{−1k}. Thenαkxk/

ΓΔ. Hence,ΓΔis not solid.

Proposition 3.6. ΓΔμ csd1forμα, β, γ, f.

Proof.

Step 1. ΓΔhas AK byProposition 3.4. Hence, byLemma 1.1ii, we getΓΔβ ΓΔf.

ButΓΔβcsd1. Hence,

ΓΔf csd

1. 3.10

Step 2. Since AK⇒AD. Hence, byLemma 1.1iii, we getΓΔβ ΓΔγ. Therefore,

ΓΔγcsd1. 3.11

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

Gupta6, we get

ΓΔα

/

ΓΔγ

/

csd1. 3.12

From3.10and3.11, we have

ΓΔβ ΓΔγ ΓΔf csd

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Lemma 3.7Wilansky1, Theorem 8.6.1. YXYf Xf whereXis an AD-space andY an

FK-space.

Proposition 3.8. LetY be any FK-spaceφ. Then,Y ⊃ ΓΔif and only if the sequence δk is

weakly converges incsd1.

Proof. The following implications establish the result.

Y ⊃ΓΔ⇔Yf ΓΔf, sinceΓΔhas AD byLemma 3.7.

Yf csd

1, sinceΓΔf csd1.

⇔for eachfY, the topological dual ofY.

k∈csd1.

δkis weakly converges incsd1.

This completes the proof.

d

1

Definition 4.1. An FK-spaceΔXis called “semi-diﬀerence entire sequence spacecsd1” if its

dualΔXfcsd1.

In other wordsΔXis semi-diﬀerence entire sequence spacecsd1ifk∈csd1

for allf∈ΔXfor each fixedk.

Example 4.2. ΓΔis semi-diﬀerence entire sequence spacecsd1. Indeed, ifΓΔis the space

of all diﬀerence of entire sequences, then byLemma 4.3,ΓΔf csd1.

Lemma 4.3Wilansky1, Theorem 4.3.7. Letzbe a sequence. Thenzβ, Pis an AK space with

P Pk :k 0,1,2, . . ., whereP0x supm|mk1zkxk|, andPnx |xn|. For anyksuch that

zk/0, Pkmay be omitted. Ifzφ, P0may be omitted.

Proposition 4.4. Letzbe a sequence.zβis a semi-dierence entire sequence spacecsd

1if and only

ifzis incsd1.

Proof. Suppose that is a semi-dierence entire sequence space cs d

1. has AK by

Lemma 4.3. Thereforezββ zβf by Lemma 11. Sozβis semi-dierence entire sequence

spacecsd1if and only ifzββcsd1. But thenzzββcsd1. Hence,zis incsd1.

Conversely, suppose thatzis incsd1. Thenzβ⊃ {cs∩d1}βandzββ ⊂ {cs∩d1}ββ

csd1. Butzβf zββ. Hence,zβf csd1. Thereforezβis semi-dierence entire sequence

spacecsd1. This completes the proof.

Proposition 4.5. Every semi-diﬀerence entire sequence spacecsd1containsΓ.

Proof. LetΔXbe any semi-diﬀerence entire sequence spacecsd1. Hence,ΔXfcsd1.

Thereforek ∈csd1 for allf ∈ ΔX. So,{δk}is weakly converges incsd1with

respect toΔX. Hence,ΔX ⊃ ΓΔbyProposition 3.8. ButΓΔ ⊃ Γ. Hence,ΔX ⊃ Γ. This

completes the proof.

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Proof. LetΔXn1ΔXn. ThenΔXis an FK-space which containsφ. Also everyf ∈ΔX

can be written asf g1 g2 . . . gm, wheregk ∈ ΔXnfor somenand for 1 ≤ km.

But thenfδk g

1δk g2δk · · · gmδk. SinceΔXn n1,2, . . .are semi-diﬀerence

entire sequence spacecsd1, it follows thatgiδkcsd1 for alli 1,2, . . . m. Therefore

fδkcsd1 for allk and for allf. Hence,ΔX is semi-diﬀerence entire sequence space

csd1. This completes the proof.

Proposition 4.7. The intersection of all semi-diﬀerence entire sequence spacecs∩d1isIcsd1β

andΓΔ⊂I.

Proof. Let I be the intersection of all semi-diﬀerence entire sequence space csd1. By

Proposition 4.4, we see that the intersection

I ⊂ ∩:zcsd1

{cs∩d1}β. 4.1

ByProposition 4.6it follows thatIis semi-diﬀerence entire sequence spacecsd1. By

Proposition 4.5, consequently

ΓM ΓΔ⊂I. 4.2

From4.1and4.2, we getI⊂ {cs∩d1}βandΓΔ⊂I. This completes the proof.

Corollary 4.8. The smallest semi-diﬀerence entire sequence spacecsd1 is Icsd1β 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-conservativeFKspaces,” 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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