International Journal of Mathematics and Mathematical Sciences
Volume 2011, Issue 1 143974
Research Article
Open Access

The Semi-Difference Entire Sequence Space cs∩d1

N. Subramanian, 

Corresponding Author

N. Subramanian

Department of Mathematics, SASTRA University, Thanjavur 613 401, India sastra.edu

Search for more papers by this author
K. Chandrasekhara Rao, 

K. Chandrasekhara Rao

Srinivasa Ramanujan Centre, SASTRA University, Kumbakonam 612 001, India sastra.edu

Search for more papers by this author
K. Balasubramanian, 

K. Balasubramanian

Department of Mathematics, SASTRA University, Thanjavur 613 401, India sastra.edu

Search for more papers by this author
First published: 14 May 2011
https://doi.org/10.1155/2011/143974
Academic Editor: Ricardo Estrada

Abstract

Let Γ denote the space of all entire sequences. Let Λ denote 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∩d1. It is shown that the intersection of all semi-difference entire sequence spaces cs∩d1 is I ⊂ cs∩d1 and Γ(Δ) ⊂ I.

1. Introduction

A complex sequence, whose kth term is xk, is denoted by {xk} or simply x. Let w be the set of all sequences and ϕ be the set of all finite sequences. Let ℓ∞, c, c0 be the classes of bounded, convergent, and null sequence, respectively. A sequence x = {xk} is said to be analytic if . The vector space of all analytic sequences will be denoted by Λ. A sequence x is called entire sequence if . The vector space of all entire sequences will be denoted by Γ.

Given a sequence x = {xk}, its nth section is the sequence x(n) = {x1, x2, …, 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 + 1)th place and zeros elsewhere. An FK-space (Fréchet coordinate space) is a Fréchet space which is made up of numerical sequences and has the property that the coordinate functionals pk(x) = xk  (k = 1,2, 3, …) are continuous.

We recall the following definitions (one may refer to Wilansky [1]).

An FK-space is a locally convex Fréchet 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 AK (or sectional convergence) if and only if d(x(n), x) → x as n → ∞, (see [1]). The space is said to have AD (or) be an AD-space if ϕ is dense in X, where ϕ denotes the set of all finitely nonzero sequences. We note that AK implies AD (one may refer to Brown [2]).

If X is a sequence space, we define
  • (i)

    X′ = the continuous dual of X;

  • (ii)

    ;

  • (iii)

    ;

  • (iv)

    ;

  • (v)

    let X be an FK-space ⊃ϕ. Then, Xf = {f(δ(n)) : f ∈ X′}.

Xα, Xβ, Xγ are called the α (or Köthe-Töeplitz) dual of X, β—(or generalized Köthe-Töeplitz) dual of X, γ dual of X. Note that Xα ⊂ 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 sup kpk = G and D = max {1, 2G−1}. Then, it is well known that for all ak, bk ∈ C, the field of complex numbers, for all k ∈ N,
(1.1)

Lemma 1.1 (Wilansky [1, Theorem 7.2.7]). Let X be an FK-space ⊃ϕ. Then,

  • (i)

    Xγ ⊂ Xf;

  • (ii)

    if X has AK, Xβ = Xf;

  • (iii)

    if X has AD, Xβ = Xγ.

2. Definitions and Preliminaries

Let Δ : w → w be the difference operator defined by . Let
(2.1)
Define the sets Γ(Δ) = {x ∈ w : Δx ∈ Γ} and Λ(Δ) = {x ∈ w : Δx ∈ Λ}.
The spaces Γ(Δ) and Λ(Δ) are the metric spaces with the metric
(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 Wilansky [1]).

Snyder and Wilansky [3] introduced 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 Srinivasalu [5].

In a similar way, in this paper, we define semi-difference entire sequence space cs∩d1, and show that semi-difference entire sequence space cs∩d1 is I ⊂ cs∩d1 and Γ(Δ) ⊂ I.

3. Main Results

Proposition 3.1. Γ ⊂ Γ(Δ) and the inclusion is strict.

Proof. Let x ∈ Γ. Then, we have

(3.1)
(3.2)

Let x ∈ Γ. Then, we have

(3.3)

Then, (xk) ∈ Γ(Δ) follows from the inequality (1.1) and (3.3).

Consider the sequence e = (1,1, …). Then, e ∈ Γ(Δ) but e ∉ Γ. Hence, the inclusion Γ ⊂ Γ(Δ) is strict.

Lemma 3.2. A ∈ (Γ, c) if and only if

(3.4)
(3.5)

Proposition 3.3. Define the set . Then, [Γ(Δ)]β = cs∩d1.

Proof. Consider the equation

(3.6)
where C = (Cnk) is defined by
(3.7)

Thus, we deduce from Lemma 3.2 with (3.6) that ax = (akxk) ∈ cs whenever x = (xk) ∈ Γ(Δ) if and only if Cy ∈ c whenever y = (yk) ∈ Γ, that is C ∈ (Γ, c). Thus, (ak) ∈ cs and (ak) ∈ d1 by Lemma 3.2 and (3.5) and (3.6), respectively. This completes the proof.

Proposition 3.4. Γ(Δ) has AK.

Proof. Let x = {xk} ∈ Γ(Δ). Then, . Hence,

(3.8)
(3.9)
This completes the proof.

Proposition 3.5. Γ(Δ) is not solid.

To prove Proposition 3.5, consider (xk) = (1) ∈ Γ(Δ) and αk = {(−1)k}. Then (αkxk) ∉ Γ(Δ). Hence, Γ(Δ) is not solid.

Proposition 3.6. (Γ(Δ))μ = cs∩d1 for μ = α, β, γ, f.

Proof.

Step 1. Γ(Δ) has AK by Proposition 3.4. Hence, by Lemma 1.1(ii), we get (Γ(Δ))β = (Γ(Δ))f. But (Γ(Δ))β = cs∩d1. Hence,

(3.10)

Step 2. Since AK⇒AD. Hence, by Lemma 1.1(iii), we get (Γ(Δ))β = (Γ(Δ))γ. Therefore,

(3.11)

Step 3. Γ(Δ) is not normal by Proposition 3.5. Hence by Proposition 2.7 of Kamthan and Gupta [6], we get

(3.12)

From (3.10) and (3.11), we have

(3.13)

Lemma 3.7 (Wilansky [1, Theorem 8.6.1]). Y⊃X⇔Yf ⊂ Xf where X is an AD-space and Y an FK-space.

Proposition 3.8. Let Y be any FK-space ⊃ϕ. Then, Y⊃Γ(Δ) if and only if the sequence δ(k) is weakly converges in cs∩d1.

Proof. The following implications establish the result.

  • Y⊃Γ(Δ)⇔Yf ⊂ (Γ(Δ))f, since Γ(Δ) has AD by Lemma 3.7.

  • ⇔Yf ⊂ cs∩d1, since (Γ(Δ))f = cs∩d1.

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

  • ⇔f(δ(k)) ∈ cs∩d1.

  • ⇔δ(k) is weakly converges in cs∩d1.

This completes the proof.

4. Properties of Semi-Difference Entire Sequence Space cs∩d1

Definition 4.1. An FK-space ΔX is called “semi-difference entire sequence space cs∩d1” if its dual (ΔX)f ⊂ cs∩d1.

In other words ΔX is semi-difference entire sequence space cs∩d1 if f(δ(k)) ∈ cs∩d1 for all f ∈ (ΔX)′ for each fixed k.

Example 4.2. Γ(Δ) is semi-difference entire sequence space cs∩d1. Indeed, if Γ(Δ) is the space of all difference of entire sequences, then by Lemma 4.3, (Γ(Δ))f = cs∩d1.

Lemma 4.3 (Wilansky [1, Theorem 4.3.7]). Let z be a sequence. Then (zβ, P) is an AK space with P = (Pk : k = 0,1, 2, …), where , and Pn(x) = |xn|. For any k such that zk ≠ 0, Pk may be omitted. If z ∈ ϕ, P0 may be omitted.

Proposition 4.4. Let z be a sequence. zβ is a semi-difference entire sequence space cs∩d1 if and only if z is in cs∩d1.

Proof. Suppose that zβ is a semi-difference entire sequence space cs∩d1. zβ has AK by Lemma 4.3. Therefore by Lemma 1 [1]. So zβ is semi-difference entire sequence space cs∩d1 if and only if zββ ⊂ cs∩d1. But then z ∈ zββ ⊂ cs∩d1. Hence, z is in cs∩d1.

Conversely, suppose that z is in cs∩d1. Then and . But. Hence, . Therefore zβ is semi-difference entire sequence space cs∩d1. This completes the proof.

Proposition 4.5. Every semi-difference entire sequence space cs∩d1 contains Γ.

Proof. Let ΔX be any semi-difference entire sequence space cs∩d1. Hence, (ΔX)f ⊂ cs∩d1. Therefore f(δ(k)) ∈ cs∩d1 for all f ∈ (ΔX)′. So, {δ(k)} is weakly converges in cs∩d1 with respect to ΔX. Hence, ΔX⊃Γ(Δ) by Proposition 3.8. But Γ(Δ)⊃Γ. Hence, ΔX⊃Γ. This completes the proof.

Proposition 4.6. ΔX is semi-difference entire sequence space cs∩d1.

Proof. Let . Then ΔX is an FK-space which contains ϕ. Also every f ∈ (ΔX)′ can be written as f = g1 + g2 + …+gm, where for some n and for 1 ≤ k ≤ m. But then f(δk) = g1(δk) + g2(δk) + ⋯+gm(δk). Since ΔXn  (n = 1,2, …) are semi-difference entire sequence space cs∩d1, it follows that gi(δk) ∈ cs∩d1 for all i = 1,2, …m. Therefore f(δk) ∈ cs∩d1 for all k and for all f. 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 space cs∩d1 is and Γ(Δ) ⊂ 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

(4.1)

By Proposition 4.6 it follows that I is semi-difference entire sequence space cs∩d1. By Proposition 4.5, consequently

(4.2)

From (4.1) and (4.2), we get and Γ(Δ) ⊂ I. This completes the proof.

Corollary 4.8. The smallest semi-difference entire sequence space cs ∩ d1 is and Γ(Δ) ⊂ 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.

      The full text of this article hosted at iucr.org is unavailable due to technical difficulties.