Journal of Applied Sciences

Year: 2017 | Volume: 17 | Issue: 9 | Page No.: 467-474
DOI: 10.3923/jas.2017.467.474

Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator

Vakeel Ahmad Khan , Rami K.A. Rababah, Ayhan Esi , Sameera Ameen Ali Abdullah and Kamal Mohammed Ali Suleiman Alshlool

Abstract: Background and Objective: This study comes from the notion of I-convergence which is the generalization of statistical convergence. The idea of I-convergent sequence spaces was motivated by the statistical convergence for double sequence. The I-convergence for double sequence in real line and general metric space. Methodology: This has motivated us to study the ideal convergence for double sequences using compact operator and define these spaces , and . Further, inclusion relations between these spaces and some properties such as solidity and monotonic was investigated. Results: This study consists of two sections. In the first section, it was given the basic definitions related to double sequences, compact operator, ideal, filter etc. The second section deals with the main results like , and are linear spaces and normed spaces. It was defined a real valued function on as h(x) = I-lim x and proved that this function is Lipschitz continuous and hence uniformly continuous. At the last, topological properties of these spaces like monotonicity and solidity are discussed. Conclusion: This study has used compact linear operator to define spaces of ideal convergent double sequence , and which provide better tool to study a more general spaces of sequences.

Keywords: Lipschitz function, solid space, monotonic, I-convergence of double sequence and compact operator

INTRODUCTION

Fast¹ and Steinhaus² were the first who introduced the notion of the statistical convergence independently. Later on it was further investigated from a sequence space point of view and linked with suitability theory by Fridy³, Salat⁴ and Tripath⁵. The above concept is extended to double sequences by using the idea of a two dimensional analogue of natural density⁶. Kostyrko et al.^7,8 defined I-convergence for single sequences which is a natural generalization of statistical convergence.

The idea of I-convergence is based on the notion of the ideal I of subsets of ℕ, the set of positive integers. Salat et al.⁹ studied the concept of I-convergence and I-Cauchy for sequences and proved some properties. Tripathy and Hazarika^10,11 introduced some spaces of I-convergence of single sequences and proved some properties related to the solidity, symmetrically, completeness and denseness. Das et al.¹² defined the concept of I-convergence for double sequences.

Let us denote ₂ω for the space of all real or complex double sequences x = (x_i,j), where i, j ∈ ℕ. A double sequence x = (x_i,j) ∈ ₂ω is said to be Pringsheims convergent to some number j ∈ ℂ (or P-convergent) if for given ε>0, there exists an integer N such that:

It will be written as:

where, i and j tending to infinity and independent of each other¹³.

Definition 1: A normal linear space X is said to be a Banach space if it is complete, that is if every Cauchy sequence in X is convergent in X¹⁴.

Definition 2: An operator T is said to be linear operator if the domain D(T) of T is a vector space and the rang R(T) lies in a vector space over the same field¹⁵:

T(αx+βy) = αT(x)+βT(y)∀α, β∈K, for all x, y∈D(T)

Definition 3: Let X and Y be two normed linear spaces and T: D(T)→Y be a linear operator, where D(T)⊂X. Then, the operator T is said to be bounded, if there exists a positive real k>0 such that:

||Tx||<k||x||, for all x∈D(T)

the set of all bounded linear operators B(X, Y) is a linear space normed by and B(X, Y) is a Banach space if Y is Banach space¹⁵.

Definition 4: A linear space T: X→Y is said to be a compact linear operator (or completely continuous linear operator) if T maps every bounded sequence x_k in X onto a sequence (Tx_k) in Y which has a convergence subsequence¹⁵. The set of all compact linear operators C(X,Y) is closed subspace of B(X, Y) and it is Banach space if Y is Banach space.

Throughout the paper, it was denoted ₂l_∞, ₂c and ₂c₀ as the Banach spaces of bounded, convergent and null double sequences of reals respectively with norm:

Following Basar and Altay¹⁶ and Sengonul¹⁷, it was introduced the double sequence spaces ₂S and ₂S₀ with help of compact operator T on ℝ as follows:

₂S = {x = (x_{i, j}) ∈ ₂l_∞: Tx ∈ ₂c},
₂S₀ = {x = (x_{i, j}) ∈ ₂l_∞: Tx ∈ ₂c₀}

Definition 5: A sequence x = (x_k) ∈ ω is said to be statistically convergent to limit L if for every ε>0, the density of the set {k ∈ ℕ: |x_k-L|>ε} equal zero.

Definition 6: Let ℕ×ℕ be a non-empty set. A family of sets I⊆2^ℕ×ℕ is said to be an ideal if:

Definition 7: Let ℕ×ℕ be a non-empty set. Then a family of sets F⊆2^ℕ×ℕ is said to be a filter on ℕ×ℕ if and only if¹⁸:

Remark 1: For each ideal I there is a filter F(I) which correspponding to I (filter associate with ideal I), that is¹⁸:

Definition 8: A double sequence (x_i,j) ∈ ₂ω is said to be I-convergent to a number L ∈ ℝ if, for every ε>0, the set:

And write I-lim_(i,j)x_i,j = L. In case L = 0 then (x_i,j) ∈ ₂ω is said to be I-null.

Definition 9: A double sequence (x_i,j) ∈ ₂ω is said to be I-Cauchy if, for each ε>0, there exists a numbers s = s(ε) and t = t(ε) such that the set:

{(i, j) ∈ ℕ×ℕ: |(x_i,j)-(x_s,t)|>ε} ∈ I

Definition 10: A double sequence (x_i,j) ∈ ₂ω is said to be I-bounded if there exists M>0,such that, the set:

{(i, j) ∈ ℕ×ℕ: |(x_i,j)|>M} ∈ I

Definition 11: Let x = (x_i,j) and y = (y_i,j) be two double sequences. It can say that x_i,j = y_i,j for almost all i and j relative to I (in short a.a.i and j.r.I) if the set:

{(i, j) ∈ ℕ×ℕ: x_i,j≠y_i,j} ∈ I

Definition 12: Let x = (x_i,j) be a double sequence and I be an ideal in ℕ×ℕ. A subset D of ℂ, the field of complex numbers, is said to contain x_i,j for a.a.i and j.r.I if the set:

{(i,j) ∈ ℕ×ℕ: x_i,j ∉ D} ∈ I

Definition 13: A double sequence space E is said to be solid or normal, if (α_i,jx_i,j) ∈ E whenever (x_i,j) ∈ E and for any double sequence of scalars (α_i,j) with |(α_i,j) |<1, for all (i,j) ∈ ℕ×ℕ.

Definition 14: A double sequence space E is said to be symmetric, if (x_π(i,j)) ∈ E whenever (x_i,j) ∈ E, where π(i,j) is a permutation on ℕ×ℕ.

Definition 15: A double sequence space E is said to be sequence algebra, if (x_i,j)×(y_i,j) = (x_i,j.y_i,j) ∈ E whenever x_i,j, y_i,j ∈ E.

Definition 16: A double sequence space E is said to be convergent free, if (y_i,j) ∈ E whenever (x_i,j) ∈ E and x_i,j = 0 implies y_i,j = 0, for all (i,j) ∈ ℕ×ℕ.

Definition 17: Let K = {(i_n, j_n) ∈ ℕ×ℕ: i₁<i₂<… and j₁<j₂<…} ⊆ N×N and let E be a double sequence space. A K-step space of E is a sequence space:

is a sequence y_i,j ∈ ₂ω defined as follows:

A canonical pre-image of a step space is a set of canonical pre-images of all elements in iff pre-image of all element in i.e., y is in the canonical pre-image of iff y is a canonical pre-image of same element x ∈ .

Definition 18: A double sequence space E is said to be monotone, if it is contains the canonical pre-images of it is step space.

Definition 19: A map h: D⊂X→ℝ is said to satisfy Lipschitz condition if:

|h(x)-h(y)|<k|x-y|

where, k is known as Lipschitz constant.

Definition 20: The I-convergence can be considered as a summability methods that (In case of admissibility of I are regular). Denoted by F(I) the convergence field of I-convergence, that is:

F(I) = {x = x_k) ∈ l_∞: there exists I-lim x ∈ ℝ}

The convergence field F(I) is a closed linear subspace of l_∞ with respect to the supremum norm:

The function h: F(I)→ℝ defined by h(x) = I-lim x, for all x ∈ F(I) is said to be Lipschitz function¹⁷.

Following lemmas were used to establish some results of this article:

Throughout this article, T is considered as a compact operator on the space ℝ.

RESULTS

In this article the following classes of double sequences are studied:

Theorem 1: The classes of double sequences , and are linear spaces.

Proof: Let x = (x_i,_j), y = (y_i,_j) be two arbitrary elements of the space ₂S¹ and α, β are scalars. Now, since (x_i,_j), (y_i,_j) ∈ ₂S¹ then for given ε>0, there exist L₁, L₂ ∈ ℂ, such that:

and:

Now, let:

be such that Therefore, the set:

Thus, the sets on right hand side of Eq. 8 belongs to F(I). By definition of filter associate with ideal, the complement of the set on left hand side of (Eq. 8) belongs to I. This implies that α(x_i,_j)+β(y_i,_j) ∈ ₂S¹. Hence ₂S¹ is linear space.

Theorem 2: The spaces ₂S¹ and are ₂S¹ normed spaces normed by:

Proof: The proof of the result is easy in view of existing techniques and hence omitted.

Theorem 3: A double sequence = (x_i,_j) ∈ ₂l_∞ is I-converges if and only if for every ε>0, there exists s = s(ε), t = t(ε) ∈ ℕ×ℕ, such that:

{(s, t) ∈ ℕ×ℕ: |T(x_s,t)-L|<ε} ∈ F (I)

Proof: Suppose that the double sequence x = (x_i,j) ∈ ₂l_∞ is I-convergent to some number L ∈ ℂ, then for a given ε>0, the set:

Fix an integers s = s(ε), t = t(ε) ∈ B_ε. Then:

which holds for all (i,j) ∈ B_ε. Hence the set:

{(i,j) ∈ ℕ×ℕ: |T(x_i,_j)-T(x_s,_t)|<ε} ∈ F(I)

Conversely, suppose that:

{(i,j) ∈ ℕ×ℕ: |T(x_i,_j)-T(x_s,_t)|<ε} ∈ F(I)

Then the set:

C_ε = {(i,j) ∈ ℕ×ℕ: T(x_i,_j) ∈ [T(x_i,_j)-ε, T(x_i,_j)+ε]} ∈ F(I)

for all ε>0. Let J_ε = [T(x_i,_j) ∈ [T(x_i,_j)-ε, T(x_i,_j)+ε]. If fix ε>0, then it was C_ε ∈ F(I)as well as ∈ F(I). Hence C_ε ∩ ∈ F(I). This implies that:

That is the set:

{(i,j) ∈ ℕ×ℕ: T(x_i,_j)} ∈ F(I)

That is:

diam J< diam J_ε

where, the diam J denote the length of interval J. Proceeding in this way, it will have a sequence of closed intervals:

J_ε = I0 ⊇ I₁ ⊇…⊇ I_k ⊇…

with the property that:

diam I_k < diam L_k+1, for k = (1,2,3 …)

and:

{(i,j) ∈ ℕ×ℕ: T(x_i,_j) ∈ I_k} ∈ F(I), for k = (1, 2, 3…)

Then there exists a number L ∈∩ I_k where k ∈ ℕ such that L = 1-lim T(x_i,j). Hence the result holds.

Theorem 4: Let I be an admissible ideal. Then the following are equivalent:

(i,j) ∈ ℕ×ℕ and {(i,j) ∈ ℕ×ℕ: |T(x_i,j)-L|>ε} ∈ I

There exists a subset K = {(i_s, j_t): s, t ∈ ℕ, i₁<i₂<i₃<⋯ and j₁<j₂<j₃<⋯} of ℕ×ℕ, such that K ∈ F(I) and lim_i,j|T (x_i,j)-L| = 0.

Proof: (i) Implies (ii): Let x_i,j ∈ ₂S^I, then for any ε > 0, there exists a number L ∈ ℂ such that the set:

{(i,j) ∈ ℕ×ℕ: |T(x_i,j)-L|>ε} ∈ I

Let (m_s,t) be an increasing double sequence with m_s,t ∈ ℕ×ℕ such that:

{(i,j)<m_s,t: |T(x_i,j)-L|>(st)‾¹} ∈ I

Define a double sequence (y_i,j) as (y_i,j) = (x_i,j) for all (i,j)<m_s,t. For m_s,t<(i,j)<m_s+1,t+1, (s,t)∈ℕ×ℕ. That is:

Then y_i,j ∈ ₂S^I and from the following inclusion:

{(i,j)< m_s,t: x_i,j≠y_i,j}⊆{(i,j) ∈ ℕ×ℕ: |T(x_i,j)-L|> ε} ∈ I

It get for x_i,j = y_i,j for a, a, I and j, r, I.

(ii) Implies(iii): For x_i,j ∈ ₂S^I there exists y_i,j ∈ ₂S^I such that x_i,j = y_i,j for a, a, I and j, r, I.

Let K = {(i,j) ∈ ℕ×ℕ: x_i,j≠y_i,j}, then K ∈ I. Define a double sequence (z_i,j) as:

Then z_i,j ∈ ₂S^I and y_i,j ∈ ₂S^I.

(iii) Implies (iv): Let P_c = {(i,j) ∈ ℕ×ℕ: |T(z_i,j)| > ε} ∈ I and K = P_c = {(i_s,j_t) ∈ N×ℕ: s, t ∈ ℕ, i₁<i₂<i₃<⋯ and j₁<j₂<j₃<⋯} ∈ F(I).

Then .

(iv) Implies (i): Let {(i_s,j_t)∈N×ℕ: s, t ∈ ℕ, i₁<i₂<i₃<⋯ and j₁<j₂<j₃<⋯} ∈ F(I). And let lim_s,t |T(x_is, j_t)-L| = 0. Then for any ε>0 and by lemma (II) have:

{(i,j) ∈ ℕ×ℕ: |T(x_i,j)-L|>ε}⊆K^c∪{(i,j) ∈ K: |T(x_i,j)-L|>ε}

Thus (x_i,j) ∈ ₂S^I.

Theorem 5: The function h: ₂S^I→ℝ defined by h(x) = I-lim x, for all x ∈ ₂S^I is a lipschitz function and hence uniformly continuous.

Proof: Clearly the function is well defined. Let x = (x_i,j), y = (y_i,j) ∈ ₂S^I, x≠y, then the sets:

A_x = {(i,j) ∈ ℕ×ℕ: |T(x)-h(x)|>∥x-y∥} ∈ I
A_y = {(i,j) ∈ ℕ×ℕ: |T(y)-h(y)|>∥x-y∥} ∈ I

where, ∥x-y∥ = sup_i,j |T(x_i,j-y_i,j)|. Thus the sets:

B_x = {(i,j) ∈ ℕ×ℕ: |T(x)-h(x)|<∥x-y∥} ∈ F(I)
B_y = {(i,j) ∈ ℕ×ℕ: |T(y)-h(y)|<∥x-y∥} ∈ F(I)

Hence, B = B_x ∩ B_y ∈ F(I), so that B is non-empty set, can choose (i,j) ∈ B, therefore:

|h(x)-h(y)| <|h(x)-T(x)|+|T(x)-T(y)|+|T(y)-h(y)|< 3∥x-y∥

Thus, h is lipschitz function and hence uniformly continuous.

Theorem 6: If T is an identity operator and h: ₂S^I→ℝ is a function defined by h(x) = I-lim x, for all x ∈ ₂S^I and if x = (x_i,j), y = (y_i,j)x ∈ ₂S^I then (x.y) ∈ ₂S^I and h(x.y) = h(x).h(y).

Proof: For ε>0, the sets:

where, ∥x-y∥ = ε. Now since T is identity operator:

As ₂S^I ⊆ ₂l_∞, there exists an M ∈ ℝ such that |x_i,j|<M and |h(y)|<M.

Therefore, from the Eq. 9, 10 and 11:

|T(x.y)-h(x) h(y)| = |T(x_i,j y_i,j)-h(x) h(y)|
< M ∈+M ∈ = 2M ∈

for all (i,j) ∈ B_x ∩ B_y ∈ F(I). Hence, (x.y) ∈ ₂S^I and h(x.y) = h(x).h(y).

Theorem 7: The space ₂S^I is solid and monotone.

Proof: Let (x_i,j) ∈, for ε>0, the set:

Let (α_i,j) be a double sequence of scalars with |α_i,j|<1 for all (i, j) ∈ ℕ×ℕ. Therefore:

|T (α_i,j, x_i,j)| = |α_i,j T(x_i,j)|
< |α_i,j| |T (x_i,j)| < |T (x_i,j)|, for all (i, j) ∈ ℕ×ℕ

Thus, from the above inequality and Eq. 12:

{(i, j) ∈ ℕ×ℕ: |T (α_i,j x_i,j)|>ε}⊆{(i, j) ∈ ℕΧℕ: |T(x_i,j)|>ε} ∈ I

Implies that:

{(i,j) ∈ ℕ×ℕ: |T(α_i,j x_i,j)|>ε} ∈ I

Therefore, Hence, the space is solid and hence by lemma (I), the space is monotone.

Theorem 8: The inclusions hold.

Proof: Let (x_{i, j}) ∈ ₂S^I. Then there exists a number L ∈ ℝ such that:

That is, the set:

{(i,j) ∈ ℕ×ℕ:|T(x_i,j)-L|>ε} ∈ I

Where:

|T(x_i,j)| = |T(x_i,j)-L+L|<|T(x_i,j))-L|+|L|

Taking the supremum over i and j on both sides, it get

The inclusion:

It was showed that a_p,q converse is obvious. Hence,

Theorem 9: The set is ₂S^I closed subspace of ₂l_∞.

Proof: Let be a Cauchy sequence in ₂S^I. Then it have → x in ₂l_∞ as p, q→∞, since ∈ ₂S^I then for each ε>0 there exist a_{p, q} such that ges, (say) to a.

I-lim x = a

Since be a Cauchy sequence in ₂S^I. Then for each ε>0 there exists n₀ ∈ ℕ, such that:

That is for a given ε>0, it have:

Now, it have sets:

and:

Then, Let where:

B = {(i, j) ∈ ℕ×ℕ: |a_p,q-a_s,t)|<ε}, then B^c ∈ I

Consider n₀ ∈ B^C. Then for each p, q, s, t>n₀ it have:

Thus, (a_{p, q}) is a Cauchy sequence of scalars in ℂ, so there exists a scalar a ∈ ℂ such that (a_{p, q})→a as p, q→∞.

For this step, let 0<δ<1 be given. Then it showed that if:

U = {(i,j) ∈ ℕ×ℕ: |T(x)-a|<δ} then U^c ∈ I

Since → x there exists (p₀, q₀) ∈ ℕ×ℕ such that:

Which implies that P^C ∈ I. The numbers p₀, q₀ can be chosen together with Eq. 13, it have:

Which implies that Q^C ∈ I. Since:

Then it have a subset of such that S^C ∈ I. Where:

Let U^C = P^C∪Q^C∪S^C, where:

U = {(i, j)∈ℕ×ℕ: |T(x)-a|<δ}

Therefore, for (i, j)∈U^C each it have:

The sets on the right hand side of Eq. 14 are belongs to F(I). Therefore, the set on the left hand side of Eq. 14, it is also belongs to F(I). Hence its complement belongs to I. Thus, I-Limx = α.

DISCUSSION

The notion of I-convergence, which is a generalization of of statistical convergence, was introduced by Kostyrko et al.⁸. The notion of statistical convergence of double sequences x = x_ij has been defined and studied by Mursaleen and Edely¹⁹. Motivated by this definition, Das et al.¹² studied the notion of I and I*-convergence of double sequences in R. Recently, a compact operator was used to define the single sequence spaces with the help of the notion I-convergence in more general settings by Khan and Ebadullah²⁰ and Khan et al.^21-23. It keep the same direction up, in order to prove that a compact operator can be also used to define spaces of ideal convergence for double sequences. Indeed, most of our results in this paper are just minor adaptations of results in Khan et al.^23-25, through which it was seen that the compact operator has preserved some topological and algebraic properties for these spaces. As no contradictions are yet found in this study. Regarding this case, it was mentioned some different attempts that have been taken place in some previous works which used different operators and the compact operator as well²⁶. The opportunity is still open for other researchers to study some other different properities by the use of compact operator in an extended way to fugure out whether there is any discrepancy or not.

CONCLUSION

In this study, the compact linear operator is used to define spaces of ideal convergent double sequence , and which provide better tool to study a more general spaces of sequences. Some topological and algebraic properties on these spaces are obtained such as the function defined from ₂S^I to R is Lipschitz function and hence uniformly continuous, solidity, monotonicity. The authors will apply the techniques used in this paper for further investigation on ideal convergence.

SIGNIFICANT STATEMENT

This study discovers the idea of ideal convergence double sequence space defined by compact operator that can be beneficial for scholars working in this field. This study will help the researcher to uncover the critical areas of ideal convergence by using compact operator that many researchers were not able to explore. Thus a new theory on ideal convergence double sequence spaces may be arrived at.

REFERENCES