Some Results on Strongly Cesáro Ideal Convergent Sequence Spaces

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 (Iconvergence) 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.


Introduction
Consider the space ω � x � (x k ): x k ∈ R or C of all real and complex sequences, where R and C are, respectively, the sets of all real and complex numbers.
Suppose that ℓ ∞ , c, and c 0 are the linear spaces of bounded, convergent, and null sequences, respectively, normed by N being the set of all natural numbers. A sequence space x � (x k ) of complex numbers is said to be (C, 1) summable to L ∈ C if for ρ k � 1/k k i�1 x i , lim k ρ k � L. e sequence (C, 1) is also called Cesáro summable sequence of complex numbers over C. Let us denote by C 1 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 (x k ) on a normed space (X, ‖ · ‖|) is said to be strongly Cesáro convergent to L if In [2][3][4][5][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 (Iconvergence) 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.

Definitions of the Terms Used
Let us first present some definitions and notions that are required in the sequel. (2) A nontrivial ideal set I is said to be admissible if For every ideal I, there is a filter F(I) (associated with I) defined as follows: A sequence (x k )∈X is said to be I-convergent to a number L if, for every ∈>0, the set {x � (x k ) ∈X: {k∈N: Let If be the class of all finite subsets of N.
A sequence space is monotone if it contains the canonical preimages of its step spaces.

Result
A canonical preimage of a step space λ X K is a set of preimages of all elements in λ X K , i.e., y is in the canonical preimage of λ X K if and only if y is the canonical preimage of some x ∈λ X K . 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 T(D)is compact Lemma 1 (see [12]). Every solid space is monotone.

Main Results
Let us first define C I , the space of all Cesáro ideal convergent sequences and C I 0 , the space of all Cesáro ideal null sequences which are given as follows: Theorem 1.
e sequence spaces C I and C I 0 are linear.
Proof. Assume that x � (x k ), y � (y k ) ∈C I . en, one has Let Let α and β be some scalers. By using the properties of norm, one can easily see that en, from (9) and (10), we have for each ε > 0, erefore, (αx k + βy k ) ∈C I , for all scalars α, β and (x k ), (y k )∈C I .
Hence, C I is a linear space.
On the similar manner, one can prove that C I 0 is also linear. Proof. It can be easily observed.

Theorem 3. A sequence x � (x k ) ∈C I is I-convergent if and only if, for every ε > 0, there exists l � l(ε)∈N such that
Proof. Suppose that x � (x k ) ∈C I . erefore, I − lim n⟶∞ 1/n n k�1 ‖x k − L‖ � 0. en, for all ε>0 the set Fix an l(ε)∈∈C ε . en, we have which holds for all k ∈∈C ε . Hence, Conversely, suppose that, for all ε > 0, the set en, for every ε > 0, we have For fixed ε > 0,one has B ε ∈ F(I)as well as B ε/2 ∈F(I). Hence, is implies that B ε ∩ B ε/2 ≠ ϕ, that is, (19) at is diam P ≤ diam P ε , 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 J k ≤ 1/2diam J k−1 for k � 1, 2, 3, . . ., and for k � 1,2,3, . . ..,. en, there exists a L ∈ ∩ J k such that L � I − lim n⟶∞ 1/n n k�1 ‖x k ‖showing that x � (x k )∈C I is I-convergent. Hence, the result holds.

Theorem 4.
e space C I 0 is solid and monotone.
Proof. Let (x k ) ∈C I 0 be any element. en, one has Let (α k ) be a sequence of scalars such that |α k | ≤ 1, for all k∈N, and hence 1/n n k�1 |a k | ≤ 1. en, the result (that C I 0 is solid) follows from the above equation and inequality: for all k∈N. e space C I 0 is monotone which follows from Lemma 1. Hence, C I 0 is solid and monotone. □ Theorem 5. e space C I is neither solid nor monotone.
Proof. For this theorem, we provide a counter example for the proof.

Counter Example
Let I � I f , and consider the k-step χ k of χ defined as follows.
Let (x k )∈χ and let (y k ) ∈χ k be such that Let us consider the sequence (x k ) defined by x k � 1 for all k ∈N. en, (x k )∈C I , but its K-step preimages do not belong to C I . us, (x k ) ∈C I is not monotone.
Hence, (x k ) ∈C I is not solid.

Journal of Mathematics
Theorem 6. Let x � (x k ) and y � (y k ) be two sequences in such a way that T(x · y) � T(x)T(y). en, the space C I and C I 0 are sequence algebra.
Proof. Let x � (x k ) and y � (y k ) be two elements of C I with For every ε > 0 select β > 0 in such a way that ε < β, then (26) Using the above and the property of norm, one obtains erefore, the set k ∈ N: us, (x k ).(y k ) ∈ I Ces. Hence, C I is a sequence algebra. On the similar manner, one can prove that space C I 0 is also sequence algebra.

Data Availability
e data used to support the findings of the study are obtained from the author upon request.

Conflicts of Interest
e author declares no conflicts of interest.

Authors' Contributions
e author read and approved the final manuscript.