Some Results on Strongly Cesáro Ideal Convergent Sequence Spaces

Abstract

Some algebraic properties of Cesáro ideal convergent sequence spaces, and , are studied in this article and some inclusion relations on these spaces are established.

1. Introduction

Consider the space of all real and complex sequences, where and are, respectively, the sets of all real and complex numbers.

Suppose that , c, and are the linear spaces of bounded, convergent, and null sequences, respectively, normed by

being the set of all natural numbers.

A sequence space of complex numbers is said to be (C, 1) summable to if for , lim_k = L. The sequence (C, 1) is also called Cesáro summable sequence of complex numbers over C. Let us denote by C₁ the linear space of all (C, 1) summable sequences of complex numbers over C, i.e.,

Hardy and Littlewood [1] initiated the notion of strong Cesáro convergence for real numbers which is defined as follows.

A sequence () on a normed space (X, |) is said to be strongly Cesáro convergent to L if

In [2–6], the authors have extended the notion of strong Cesáro convergence in various fields. In 1951, Fast [7] introduced the term statistical convergence, while Steinhaus [8] independently introduced the term “ordinary and asymptotic convergences.”

Later on, Fridy [9, 10] also studied the statistical convergence and he linked it with the summability theory. Kostyrko et al. [11] gave the concept of ideal convergence (I- convergence) which was indeed a generalization of statistical convergence. Salat et al. [12] studied some properties of I-convergence, and further investigations in this field are done by Khan [13], Tripathy and Esi [14], Tripathy and Hazarika [15], and many others.

In this article, further interesting properties of Cesáro Ideal Convergent Sequences are established and a few inclusion relations are also proved.

2. Definitions of the Terms Used

Let us first present some definitions and notions that are required in the sequel.(1)A family of subsets I of is called an ideal set in (i)If I(ii)If the sets A, BI, then ABI(iii)If BA and AI, then BI(2)A nontrivial ideal set I is said to be admissible if {{n}: n }}I(3)A nonempty set F is known as a filter in if(a)(b)A, BF  ABF(c)AF with AAB  BF

Remark 1. For every ideal I, there is a filter F(I) (associated with I) defined as follows: A sequence ()X is said to be I-convergent to a number L if, for every >0, the set {x = () X: {k :|−L| ≥ } I}. In this case, we write I–lim  = L. If L = 0, then it is called I-null. A sequence ω is said to be I-Cauchy if, for every  > 0, there exists a number m = m() such that Let If be the class of all finite subsets of . If I = If, then I is admissible ideal set in . A sequence space X is said to be solid (normal) if ()X whenever ()X and () is a sequence of scalars with |αk| ≤ 1, for all k . A sequence space X is a Sequence Algebra if, for every (), () X, () X. Let K =  and X be a sequence space. A K-step space of X is a sequence space . A canonical preimage of a sequence () is a sequence () defined by A sequence space is monotone if it contains the canonical preimages of its step spaces.

3. Result

A canonical preimage of a step space is a set of preimages of all elements in , i.e., y is in the canonical preimage of if and only if y is the canonical preimage of some x .

Let X and Y be two normed linear spaces. An operator T: X ⟶ Y is known as a compact linear operator if [16].

(a) T is linear

(b) If, for every bounded subset D of X, the image M(D) is relatively compact, i.e, the closure is compact

Lemma 1. (see [12]). Every solid space is monotone.

Lemma 2. (see [12]). Let KF(I) and M . If MI, then MK I.

Lemma 3. (see [11]). Let I and M . If MI, then MKI.

4. Main Results

Let us first define C^I, the space of all Cesáro ideal convergent sequences and , the space of all Cesáro ideal null sequences which are given as follows:

Theorem 1. The sequence spaces C^I and are linear.

Proof. Assume that x = (), y = () C^I. Then, one hasLetLet and be some scalers.
By using the properties of norm, one can easily see that Then, from (9) and (10), we have for each ,Therefore, () C^I, for all scalars α, β and (), ()C^I.
Hence, C^I is a linear space.
On the similar manner, one can prove that is also linear.

Theorem 2. Let x = () be any sequence Then, C^I.

Proof. It can be easily observed.

Theorem 3. A sequence x = () C^I is I-convergent if and only if, for every there exists l = l() such that

Proof. Suppose that x = () C^I. Therefore, Then, for all >0 the setFix an l(). Then, we havewhich holds for all k. Hence,Conversely, suppose that, for all , the setThen, for every , we haveFor fixed one has as well as F(I).
Hence, .
This implies that , that is,That is diam P ≤ diam , where the diam P denotes the length of the interval of P.
In this way, by induction, one obtains the sequence of closed intervals:with the property that diam diam for k = 1, 2, 3, …, andfor k = 1,2,3, ….,. Then, there exists a such that showing that x = ()C^I is I-convergent. Hence, the result holds.

Theorem 4. The space is solid and monotone.

Proof. Let () be any element. Then, one hasLet () be a sequence of scalars such that , for all k , and hence .
Then, the result (that is solid) follows from the above equation and inequality:for all k .
The space is monotone which follows from Lemma 1.
Hence, is solid and monotone.

Theorem 5. The space C^I is neither solid nor monotone.

Proof. For this theorem, we provide a counter example for the proof.

5. Counter Example

Let I = I_f, and consider the k-step of defined as follows.

Let () and let () be such that

Let us consider the sequence () defined by for all k . Then, ()C^I, but its K-step preimages do not belong to C^I. Thus, () C^I is not monotone.

Hence, () C^I is not solid.

Theorem 6. Let x = () and y = () be two sequences in such a way that . Then, the space C^I and are sequence algebra.

Proof. Let x = () and y = () be two elements of C^I withFor every  > 0 select β > 0 in such a way that  < β, thenUsing the above and the property of norm, one obtainsTherefore, the setThus, ().() ^ICes. Hence, C^I is a sequence algebra.
On the similar manner, one can prove that space is also sequence algebra.

Data Availability

The data used to support the findings of the study are obtained from the author upon request.

Conflicts of Interest

The author declares no conflicts of interest.

Authors’ Contributions

The author read and approved the final manuscript.

Acknowledgments

The research work was supported by Saudi Electronic University, Deanship of Scientific Research, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh, Kingdom of Saudi Arabia.