The Semi Difference Entire Sequence Space

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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,}

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,[email protected]

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 spacecs∩d1. It is shown that the intersection of all semi-diﬀerence entire sequence spacescs∩d1

isI⊂cs∩d1andΓΔ⊂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

sup_k|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, x → xasn → ∞,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;

iiXα_{a _a

k:

_∞

k1|akxk|<∞, for eachx∈X};

iiiXβ_{a _a

k:

_∞

k1akxk is convergent, for eachx∈X};

ivXγ _{a _a

k: supn|

_n

k1akxk|<∞, for eachx∈X};

vletXbe an FK-space⊃φ. Then,Xf _{f_δn_:_f_∈_X_}.

Xα_{, X}β_{, X}γ_{are called the}_α_{or K öthe-T öeplitz}_{dual of}_X,_β—_{or generalized K öthe-T öeplitz}

dual ofX,γdual ofX. Note thatXα_⊂_Xβ_⊂_Xγ_{. If}_X_⊂_Y_{, then}_Yμ_⊂_Xμ_{, for}_μ_α,_{β, or}_γ.

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, bk ∈ C, the field of complex numbers,

for allk∈N,

|ak bk|pk ≤D

|ak|pk _|b_k|pk_. _1.1

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

iXγ _⊂_Xf_;

iiifXhas AK,Xβ_Xf_;

iiiifXhas AD,Xβ_Xγ_.

2. Definitions and Preliminaries

LetΔ:w → wbe the diﬀerence operator defined byΔx xk−xk 1∞k1. Let

Γ

x∈w: lim

k→ ∞

|xk|1/k₀_,

Λ x∈w: sup

k

|xk|1/k_<_∞

.

2.1

Define the setsΓΔ {x∈w:Δx∈Γ}andΛΔ {x∈w:Δx∈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 spacecs∩d1,

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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_−→_0, _as_k_{−→ ∞,} _3.1

|Δxk|1/k

2 ≤

1 2

|xk|1/k 1

2

|xk 1|1/k_, _by_1.1

−→0, ask−→ ∞ by3.1.

3.2

Letx∈Γ. Then, we have

|xk|1/k_−→_{0 as}_k_{−→ ∞.} _3.3

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 eachk∈N, 3.4

sup n,k k i0 ani

<∞. 3.5

Proposition 3.3. Define the setd1 {a ak ∈ w : sup_n,k∈N|

_k

j0nijai| < ∞}. Then,

ΓΔβ_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

whereC Cnkis defined by

Cnk ⎧ ⎪ ⎨ ⎪ ⎩ n jk

aj, if 0≤k≤n,

0, ifk > n; n, k∈N.

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Thus, we deduce from Lemma 3.2with 3.6 thatax akxk ∈ cs wheneverx

xk ∈ΓΔif and only ifCy∈ cwhenevery yk ∈Γ, that isC∈Γ, c. Thus,ak∈cs

andak∈d1byLemma 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

|Δxk|1/k−→0, ask−→ ∞, _3.8

dx, xn inf ρ >0 : sup

k≥n 1

|Δxk|1/k_≤₁

−→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 _cs_∩_d1_for_μ_{α, β, γ, f}_.

Proof.

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

ButΓΔβcs∩d1. Hence,

ΓΔf _cs_∩_d

1. 3.10

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

ΓΔγ_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

ΓΔα

/

ΓΔγ

/

cs∩d1. 3.12

From3.10and3.11, we have

ΓΔβ_ΓΔγ _ΓΔf _cs_∩_d

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Lemma 3.7Wilansky1, Theorem 8.6.1. Y ⊃X⇔Yf _⊂_Xf _where_X_{is an AD-space and}_Y _an

FK-space.

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

weakly converges incs∩d1.

Proof. The following implications establish the result.

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

⇔Yf _⊂_cs_∩_d

1, sinceΓΔf cs∩d1.

⇔for eachf∈Y, the topological dual ofY.

⇔fδk∈cs∩d1.

⇔δkis weakly converges incs∩d1.

This completes the proof.

4. Properties of Semi-Difference Entire Sequence Space

cs

∩

d

1

Definition 4.1. An FK-spaceΔXis called “semi-diﬀerence entire sequence spacecs∩d1” if its

dualΔXf ⊂cs∩d1.

In other wordsΔXis semi-diﬀerence entire sequence spacecs∩d1iffδk∈cs∩d1

for allf∈ΔXfor each fixedk.

Example 4.2. ΓΔis semi-diﬀerence entire sequence spacecs∩d1. Indeed, ifΓΔis the space

of all diﬀerence of entire sequences, then byLemma 4.3,ΓΔf cs∩d1.

Lemma 4.3Wilansky1, Theorem 4.3.7. Letzbe a sequence. Thenzβ_{, P}_{is an AK space with}

P Pk :k 0,1,2, . . ., whereP0x sup_m|m_k1zkxk|, andPnx |xn|. For anyksuch that

zk/0, Pkmay be omitted. Ifz∈φ, P0may be omitted.

Proposition 4.4. Letzbe a sequence.zβ_{is a semi-di}_ﬀ_{erence entire sequence space}_cs_∩_d

1if 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. Thereforezββ _zβf _{by Lemma 1}₁_{. So}_zβ_{is semi-di}_ﬀ_{erence entire sequence}

spacecs∩d1if and only ifzββ ⊂cs∩d1. But thenz∈zββ ⊂cs∩d1. Hence,zis incs∩d1.

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

cs∩d1. Butzβf _zββ_{. Hence,}_zβf _⊂_cs_∩_{d1. Therefore}_zβ_{is semi-di}_ﬀ_{erence entire sequence}

spacecs∩d1. This completes the proof.

Proposition 4.5. Every semi-diﬀerence entire sequence spacecs∩d1containsΓ.

Proof. LetΔXbe any semi-diﬀerence entire sequence spacecs∩d1. Hence,ΔXf ⊂cs∩d1.

Thereforefδk ∈cs∩d1 for allf ∈ ΔX. So,{δk}is weakly converges incs∩d1with

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

completes the proof.

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Proof. LetΔX ∞_n1ΔXn. ThenΔXis an FK-space which containsφ. Also everyf ∈ΔX

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

But thenfδk _g

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

entire sequence spacecs∩d1, it follows thatgiδk_∈_cs_∩_d1 _{for all}_i _1,_{2, . . . m. Therefore}

fδk ∈ cs∩d1 for allk and for allf. Hence,ΔX is semi-diﬀerence entire sequence space

cs∩d1. This completes the proof.

Proposition 4.7. The intersection of all semi-diﬀerence entire sequence spacecs∩d1isI⊂cs∩d1β

andΓΔ⊂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∩d1

{cs∩d1}β. 4.1

ByProposition 4.6it follows thatIis semi-diﬀerence entire sequence spacecs∩d1. 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 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-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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