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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
 
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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.

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  How to cite this article:

Vakeel Ahmad Khan, Rami K.A. Rababah, Ayhan Esi, Sameera Ameen Ali Abdullah and Kamal Mohammed Ali Suleiman Alshlool, 2017. Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator. Journal of Applied Sciences, 17: 467-474.

DOI: 10.3923/jas.2017.467.474

URL: https://scialert.net/abstract/?doi=jas.2017.467.474
 
Received: August 27, 2017; Accepted: November 22, 2017; Published: November 29, 2017


Copyright: © 2017. This is an open access article distributed under the terms of the creative commons attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.

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 Hazarika10,11 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 = (xi,j), where i, j ∈ ℕ. A double sequence x = (xi,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:

Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator
(1)

It will be written as:

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

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 Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator 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 xk in X onto a sequence (Txk) 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:

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

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 = (xi, j) ∈ 2l: Tx ∈ 2c},
2S0 = {x = (xi, j) ∈ 2l: Tx ∈ 2c0}

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

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

(i) φ ∈ I
(ii) A, B ∈ I⇒A∪B ∈ I, (additivity)
(iii) A ∈ I, B⊆A⇒B ∈ I, (hereditary)

An ideal I⊆2ℕ×ℕ is said to be non-trivial if I≠2ℕ×ℕ
A non-trivial ideal I⊆2ℕ×ℕ is said to be admissible if I⊇{{x}: x ∈ ℕ×ℕ}
A non-trivial ideal I⊆2ℕ×ℕ is said to be maximal if there cannot exist any non-trivial ideal J≠I containing I as a subset18

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

(i) φ ∉ F
(ii) A, B ∈ F⇒A∩B ∈ F
(iii) A ∈ F with A⊆B⇒B ∈ F

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

Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator
(2)

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

Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator
(3)

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

Definition 9: A double sequence (xi,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: |(xi,j)-(xs,t)|>ε} ∈ I

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

{(i, j) ∈ ℕ×ℕ: |(xi,j)|>M} ∈ I

Definition 11: Let x = (xi,j) and y = (yi,j) be two double sequences. It can say that xi,j = yi,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: xi,j≠yi,j} ∈ I

Definition 12: Let x = (xi,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 xi,j for a.a.i and j.r.I if the set:

{(i,j) ∈ ℕ×ℕ: xi,j ∉ D} ∈ I

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

Definition 14: A double sequence space E is said to be symmetric, if (xπ(i,j)) ∈ E whenever (xi,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 (xi,j)×(yi,j) = (xi,j.yi,j) ∈ E whenever xi,j, yi,j ∈ E.

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

Definition 17: Let K = {(in, jn) ∈ 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:

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

Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator is a sequence yi,j2ω defined as follows:

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

A canonical pre-image of a step space is a set of canonical pre-images of all elements in Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator iff pre-image of all element in Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator i.e., y is in the canonical pre-image of Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator iff y is a canonical pre-image of same element x ∈ Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator.

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 = xk) ∈ 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:

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

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:

(i) Every solid is monotone17
(ii) Let K ∈ F(I) and M ⊆ ℕ. If M ∉ I, then M ∩ K ∉ I
(iii) If I ⊂ 2 and M ⊆ ℕ. If M ∉ I, then M ∩ ℕ ∉ I18

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:

Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator
(4)

Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator
(5)

Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator
(6)

Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator
(7)

Theorem 1: The classes of double sequences Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator, Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator and Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator are linear spaces.

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

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

and:

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

Now, let:

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

be such that Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact OperatorTherefore, the set:

Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator
(8)

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(xi,j)+β(yi,j) ∈ 2S1. Hence 2S1 is linear space.

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

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

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

Theorem 3: A double sequence = (xi,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(xs,t)-L|<ε} ∈ F (I)

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

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

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

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

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

{(i,j) ∈ ℕ×ℕ: |T(xi,j)-T(xs,t)|<ε} ∈ F(I)

Conversely, suppose that:

{(i,j) ∈ ℕ×ℕ: |T(xi,j)-T(xs,t)|<ε} ∈ F(I)

Then the set:

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

for all ε>0. Let Jε = [T(xi,j) ∈ [T(xi,j)-ε, T(xi,j)+ε]. If fix ε>0, then it was Cε ∈ F(I)as well as Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator ∈ F(I). Hence CεImage for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator ∈ F(I). This implies that:

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

That is the set:

{(i,j) ∈ ℕ×ℕ: T(xi,j)} ∈ F(I)

That is:

diam J< diam Jε

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

Jε = I0 ⊇ I1 ⊇…⊇ Ik ⊇…

with the property that:

diam Ik < diam Lk+1, for k = (1,2,3 …)

and:

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

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

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

(i) (Xi,j) ∈ 2S1
(ii) There exists (yi,j) ∈ 2S1 such that xi,j = yi,j, for a, a, i and j, r, I
(iii) There exists (yi,j) ∈ 2S1 and (zi,j) ∈Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator such that xi,j = yi,j+zi,j, for all:

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

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

Proof: (i) Implies (ii): Let xi,j2SI, then for any ε > 0, there exists a number L ∈ ℂ such that the set:

{(i,j) ∈ ℕ×ℕ: |T(xi,j)-L|>ε} ∈ I

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

{(i,j)<ms,t: |T(xi,j)-L|>(st)‾1} ∈ I

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

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

Then yi,j2SI and from the following inclusion:

{(i,j)< ms,t: xi,j≠yi,j}⊆{(i,j) ∈ ℕ×ℕ: |T(xi,j)-L|> ε} ∈ I

It get for xi,j = yi,j for a, a, I and j, r, I.

(ii) Implies(iii): For xi,j2SI there exists yi,j2SI such that xi,j = yi,j for a, a, I and j, r, I.

Let K = {(i,j) ∈ ℕ×ℕ: xi,j≠yi,j}, then K ∈ I. Define a double sequence (zi,j) as:

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

Then zi,j2SI and yi,j2SI.

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

Then Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator.

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

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

Thus (xi,j) ∈ 2SI.

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

Proof: Clearly the function is well defined. Let x = (xi,j), y = (yi,j) ∈ 2SI, x≠y, then the sets:

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

where, ∥x-y∥ = supi,j |T(xi,j-yi,j)|. Thus the sets:

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

Hence, B = Bx ∩ By ∈ 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: 2SI→ℝ is a function defined by h(x) = I-lim x, for all x ∈ 2SI and if x = (xi,j), y = (yi,j)x ∈ 2SI then (x.y) ∈ 2SI and h(x.y) = h(x).h(y).

Proof: For ε>0, the sets:

Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator
(9)

Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator
(10)

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

Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator
(11)

As 2SI2l, there exists an M ∈ ℝ such that |xi,j|<M and |h(y)|<M.

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

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

for all (i,j) ∈ Bx ∩ By ∈ F(I). Hence, (x.y) ∈ 2SI and h(x.y) = h(x).h(y).

Theorem 7: The space 2SI is solid and monotone.

Proof: Let (xi,j) ∈Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator, for ε>0, the set:

Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator
(12)

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

|T (αi,j, xi,j)| = |αi,j T(xi,j)|
<i,j| |T (xi,j)| < |T (xi,j)|, for all (i, j) ∈ ℕ×ℕ

Thus, from the above inequality and Eq. 12:

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

Implies that:

{(i,j) ∈ ℕ×ℕ: |T(αi,j xi,j)|>ε} ∈ I

Therefore, Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator Hence, the space Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator is solid and hence by lemma (I), the space Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator is monotone.

Theorem 8: The inclusions Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator hold.

Proof: Let (xi, j) ∈ 2SI. Then there exists a number L ∈ ℝ such that:

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

That is, the set:

{(i,j) ∈ ℕ×ℕ:|T(xi,j)-L|>ε} ∈ I

Where:

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

Taking the supremum over i and j on both sides, it get Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator

The inclusion:

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

It was showed that ap,q converse Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator is obvious. Hence, Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator

Theorem 9: The set is 2SI closed subspace of 2l.

Proof: Let Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator be a Cauchy sequence in 2SI. Then it have Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator→ x in 2l as p, q→∞, since Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator2SI then for each ε>0 there exist ap, q such that ges, (say) to a.

I-lim x = a

Since Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operatorbe a Cauchy sequence in 2SI. Then for each ε>0 there exists n0 ∈ ℕ, such that:

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

That is for a given ε>0, it have:

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

Now, it have sets:

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

and:

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

Then, Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact OperatorLet Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator where:

B = {(i, j) ∈ ℕ×ℕ: |ap,q-as,t)|<ε}, then Bc ∈ I

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

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

Thus, (ap, q) is a Cauchy sequence of scalars in ℂ, so there exists a scalar a ∈ ℂ such that (ap, 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 Uc ∈ I

Since Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator→ x there exists (p0, q0) ∈ ℕ×ℕ such that:

Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator
(13)

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

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

Which implies that QC ∈ I. Since:

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

Then it have a subset of such that SC ∈ I. Where:

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

Let UC = PC∪QC∪SC, where:

U = {(i, j)∈ℕ×ℕ: |T(x)-a|<δ}

Therefore, for (i, j)∈UC each it have:

Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator
(14)

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 = xij 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 Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator, Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator and Image for - Some New Spaces of Ideal Convergent Double Sequences by Using Compact Operator 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 2SI 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
1:  Fast, H., 1951. Sur la convergence statistique. Colloq. Math., 2: 241-244.
Direct Link  |  

2:  Steinhaus, H., 1951. Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math., 2: 73-74.

3:  Fridy, J.A., 1985. On statistical convergence. Analysis, 5: 301-313.
CrossRef  |  Direct Link  |  

4:  Salat, T., 1980. On statistically convergent sequences of real numbers. Math. Slovaca, 30: 139-150.
Direct Link  |  

5:  Tripath, B.C., 1998. On statistical convergence. Proc. Estonian Acad. Sci. Bhy. Math. Anal., 47: 299-303.

6:  Mursaleen and O.H.H. Edely, 2003. Statistical convergence of double sequences. J. Math. Anal. Applic., 288: 223-231.
CrossRef  |  Direct Link  |  

7:  Kostyrko, P., W. Wilczynski and T. Salat, 2000. I-convergence. Real Anal. Exchange, 26: 669-686.
Direct Link  |  

8:  Kostyrko, P., M. Macij, T. Salat and M. Sleziak, 2005. I-convergence and extremal I*-limit point. Math. Slovaca, 55: 413-464.
Direct Link  |  

9:  Salat, T., B.C. Tripathy and M. Ziman, 2004. On some properties of I-convergence. Tatra Mt. Math. Publ., 28: 284-286.

10:  Tripathy, B. and B. Hazarika, 2009. Paranorm I-convergent sequence spaces. Math. Slovaca, 59: 485-494.
CrossRef  |  Direct Link  |  

11:  Tripathy, B.C. and B. Hazarika, 2011. Some I-convergent sequence spaces defined by Orlicz functions. Acta Math. Appl. Sin., 27: 149-154.
CrossRef  |  Direct Link  |  

12:  Das, P., P. Kostyrko, W. Wilczynski and P. Malik, 2008. I and I*-convergence of double sequences. Math. Slovaca, 58: 605-620.
CrossRef  |  Direct Link  |  

13:  Pringsheim, A., 1900. Zur theorie der zweifach unendlichen Zahlenfolgen. Math. Ann., 53: 289-321.
CrossRef  |  Direct Link  |  

14:  Kreyszig, E., 1978. Introductory Functional Analysis with Aplications. John Wiley and Sons Inc., New York.

15:  Maddox, I.G., 1988. Elements of Functional Analysis. Cambridge University Press, Cambridge, UK., ISBN: 9780521358682, Pages: 242.

16:  Basar, F. and B. Altay, 2003. On the space of sequences of p-bounded variation and related matrix mappings. Ukrainian Math. J., 55: 136-147.
CrossRef  |  Direct Link  |  

17:  Sengonul, M., 2007. On the Zweier sequence space. Demonstratio Math., 40: 181-196.
Direct Link  |  

18:  Salat, T., B.C. Tripathy and M. Ziman, 2005. On I-convergence field. Ital. J. Pure Appl. Math., 17: 45-54.

19:  Mursaleen and O.H.H. Edely, 2003. Statistical convergence of double sequences. J. Math. Anal. Applic., 288: 223-231.
CrossRef  |  Direct Link  |  

20:  Khan, V.A. and K. Ebadullah, 2011. On zweier I-convergent sequence spaces defined by modullus function. Theory Aplic. Math. Comput. Sci., 2: 22-30.

21:  Khan, V.A., K. Ebadullah and Y. Aligarh, 2014. On zweier I-convergent sequence spaces. Proyecciones J. Math., 33: 259-276.
CrossRef  |  Direct Link  |  

22:  Khan, V.A., M. Shafiq and R.K.A. Rababah, 2015. On some \(I\)-convergent sequence spaces defined by a compact operator and an Orlicz function. J. Adv. Stud. Topol., 6: 28-37.
CrossRef  |  Direct Link  |  

23:  Khan, V.A., M. Shafiq, R.K.A. Rababah and A. Esi, 2016. On some I-convergent sequence spaces defined by a compact operator. Ann. Univ. Craiova-Math. Comput. Sci. Ser., 43: 141-150.
Direct Link  |  

24:  Khan, V.A., Yasmeen, H. Fatima, H. Altaf and Q.D. Lohani, 2016. Intuitionistic fuzzy I-convergent sequence spaces defined by compact operator. Cogent Math., Vol. 3. 10.1080/23311835.2016.1267904

25:  Khan, V.A., Y. Khan, H. Altaf, A. Esi and A. Ahamd, 2017. On paranorm intuitionistic fuzzy I-convergent sequence spaces defined by compact operator. Int. J. Adv. Applied Sci., 5: 138-143.

26:  Khan, V.A., H. Fatima, A. Esi, S.A. Abdullah and K.M.A.S. Alshlool, 2017. On some new I-convergent double sequence spaces defined by a compact operator. Int. J. Adv. Applied Sci., 4: 43-48.
CrossRef  |  Direct Link  |  

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