# Some Results on Strongly Cesáro Ideal Convergent Sequence Spaces

Authors: Mohammad Faisal Khan
Journal: Journal of Mathematics
 (2020)

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## Abstract

Some algebraic properties of Cesáro ideal convergent sequence spaces,CI and C0I, are studied in this article and some inclusion relations on these spaces are established.


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## Body

## 1. Introduction

Consider the spaceω=x=xk:xk∈ℝorℂ of all real and complex sequences, where ℝ and ℂ are, respectively, the sets of all real and complex numbers.Suppose thatℓ∞, c, and c0 are the linear spaces of bounded, convergent, and null sequences, respectively, normed by (1)x∞=supkxk,where,k∈N,ℕ being the set of all natural numbers.A sequence spacex=xk of complex numbers is said to be (C, 1) summable to L∈C if for ρk=1/k∑i=1kxi, limkρk = 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.,(2)C1=x=xk∈ω:1k∑i=1kxi∈c.Hardy and Littlewood [1] initiated the notion of strong Cesáro convergence for real numbers which is defined as follows.A sequence (xk) on a normed space (X, ⋅|) is said to be strongly Cesáro convergent to L if(3)limn⟶∞1n∑k=1nxk−L=0.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 subsetsI of ℕ is called an ideal set in ℕ(i)
Ifϕ/∈I(ii)
If the setsA, B∈I, then A∪B∈I(iii)
IfB⊆A and A∈I, then B∈I(2)
A nontrivial ideal setI is said to be admissible if{{n}: n∈ℕ}}⊂I(3)
A nonempty setF∈2ℕ is known as a filter in ℕ if(a)
ϕ/∉F(b)
A, B∈F ⇒ A∩B∈F(c)
A∈F with A⊆AB ⇒ B∈FRemark 1.
For every idealI, there is a filter F(I) (associated with I) defined as follows:(4)FI=P⊆ℕ:ℕpεI.A sequence (xk)∈X is said to be I-convergent to a number L if, for every ∈>0, the set {x = (xk) ∈X: {k∈ℕ:|xk−L| ≥ ε} ∈I}. In this case, we write I–lim xk = L. If L = 0, then it is called I-null.A sequencexk∈ω is said to be I-Cauchy if, for every ε > 0, there exists a number m = m(ε) such that(5)x=xk∈X:xn−xm≥ε∈I.LetIf be the class of all finite subsets of ℕ. If I = IfIf, then I is admissible ideal set in ℕ.A sequence spaceX is said to be solid (normal) if (αkxk)∈X whenever (xk)∈X and (αk) is a sequence of scalars with |αk| ≤ 1, for all k∈ℕ.A sequence spaceX is a Sequence Algebra if, for every (xk), (yk) ∈X, (sk,yk) ∈X.LetK =  k1<k2<k3,....⊂ℕ and X be a sequence space. A K-step space of X is a sequence space λKX=xkn∈ω:xk∈X.Acanonical preimage of a sequence (xkn)∈λKXis a sequence (yk)∈ω defined by(6)yk=xk,ifk∈K,0,otherwise,A sequence space ismonotone if it contains the canonical preimages of its step spaces.


## 3. Result

A canonical preimage of a step spaceλKX is a set of preimages of all elements in λKX, i.e., y is in the canonical preimage of λKXif and only if y is the canonical preimage of some x∈λKX.LetX 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 TD¯is compactLemma 1.
(see [12]). Every solid space is monotone.Lemma 2.
(see [12]). Let K∈F(I) and M⊆ℕ. If M∉I, then M∩K ∉I.Lemma 3.
(see [11]). Let I⊂2ℕ and M⊆ℕ. If M∉I, then M∩K∉I.


## 4. Main Results

Let us first defineCI, the space of all Cesáro ideal convergent sequences and C0I, the space of all Cesáro ideal null sequences which are given as follows:(7)CI=x=xk∈ω:I−limn⟶∞1n∑k=1nxk−L=0,for someL∈C,C0I=x=xk∈ω:I−limn⟶∞1n∑k=1nxk=0,.Theorem 1.
The sequence spacesCI and C0I are linear.Proof.
Assume thatx = (xk), y = (yk) ∈CI. Then, one has(8)I−limn⟶∞1n∑k=1nxk−L1=0,for someL1∈C,I−limn⟶∞1n∑k=1nyk−L2=0,for someL2∈C.
Let(9)A1=k∈N:1n∑k=1nxk−L1,(10)A2=k∈N:1n∑k=1nyk−L2.
Letα and ​β be some scalers.
By using the properties of norm, one can easily see that(11)limn⟶∞1n∑k=1naxk+βyk−αL1+βL2≤limn⟶∞1n∑k=1nαxk−L1+limn⟶∞1n∑k=1nβyk−L2.
Then, from (9) and (10), we have for each ε>0,(12)k∈N:limn⟶∞1n∑k=1nαxk+βyk−αL1+βL2>ε⊂A1∪A2.
Therefore, (αxk+βyk) ∈CI, for all scalars α, β and (xk), (yk)∈CI.
Hence,CI is a linear space.
On the similar manner, one can prove thatC0Iis also linear.Theorem 2.
Letx = (xk) ∈ω be any sequence Then, C0I⊂CI.Proof.
It can be easily observed.Theorem 3.
A sequencex = (xk) ∈CI is I-convergent if and only if, for every ε>0, there exists l = l(ε)∈ℕ such that(13)k∈N:1n∑k=1nxk−xl<ε∈FI.Proof.
Suppose thatx = (xk) ∈CI. Therefore, I−limn⟶∞1/n∑k=1nxk−L=0. Then, for all ε>0 the set(14)Cε=k∈N:1n∑k=1nxk−L<ε2∈FI.
Fix anl(ε)∈Cε. Then, we have(15)1n∑k=1nxk−xl≤1n∑k=1nxk−L+1n∑k=1nxl−L<ε2+ε2=ε,which holds for all k∈Cε. Hence,(16)k∈N:1n∑k=1nxk−xl<ε∈FI.
Conversely, suppose that, for allε>0, the set(17)k∈N:1n∑k=1nxk−xl<ε∈FI.
Then, for everyε>0, we have(18)Bε=k∈N:1n∑k=1nxk∈1n∑k=1nxl−ε,1n∑k=1nxl+ε∈FI,Let,Pε=1n∑k=1nxl−ε,1n∑k=1nxl+ε.
For fixedε>0,one has Bε∈FIas well as Bε/2∈F(I).
Hence,Bε∩Bε/2∈FI.
This implies thatBε∩Bε/2≠ϕ, that is,(19)k∈N:1n∑k=1nxk∈P∈FI.
That is diamP ≤ diam Pε, where the diam P denotes the length of the interval of P.
In this way, by induction, one obtains the sequence of closed intervals:(20)Pε=J0⊇J1⊇J2,⋯,⊇Jk⊇,⋯,with the property that diam Jk≤1/2diam Jk−1 for k = 1, 2, 3, …, and(21)k∈N:1n∑k=1nxk∈Jk∈FI,for k = 1,2,3, ….,. Then, there exists a L∈∩Jk such that L=I−limn⟶∞1/n∑k=1nxkshowing that x = (xk)∈CI is I-convergent. Hence, the result holds.Theorem 4.
The spaceC0Iis solid and monotone.Proof.
Let (xk) ∈C0Ibe any element. Then, one has(22)k∈N:I−limn⟶∞1n∑k=1nxk=0.
Let (αk) be a sequence of scalars such that αk≤1, for all k∈ℕ, and hence 1/n∑k=1nak≤1.
Then, the result (thatC0I is solid) follows from the above equation and inequality:(23)1n∑k=1nαkxk=1n∑k=1nαkxk=1n∑k=1nαk1n∑k=1nxk≤1n∑k=1nxk,for all k∈ℕ.
The spaceC0Iis monotone which follows from Lemma 1.
Hence,C0Iis solid and monotone.Theorem 5.
The spaceCI is neither solid nor monotone.Proof.
For this theorem, we provide a counter example for the proof.


## 5. Counter Example

LetI = If, and consider the k-step χk of χ defined as follows.Let (xk)∈χ and let (yk) ∈χk be such that(24)yk=xk,ifkis even,0,otherwise.Let us consider the sequence (xk) defined by xk=1 for all k∈ℕ. Then, (xk)∈CI, but its K-step preimages do not belong to CI. Thus, (xk) ∈CI is not monotone.Hence, (xk) ∈CI is not solid.Theorem 6.
Letx = (xk) and y = (yk) be two sequences in such a way that Tx·y=TxTy. Then, the space CI and C0Iare sequence algebra.Proof.
Letx = (xk) and y = (yk) be two elements of CI with(25)Tx·y=TxTy.
For everyε > 0 select β > 0 in such a way that ε < β, then(26)k∈N:1n∑k=1nTxk−L1<ε2β∈FI,k∈N:1n∑k=1nTyk−L2<ε2L1∈FI.
Using the above and the property of norm, one obtains(27)1n∑k=1nTxk.yk−L1L2=1n∑k=1nTxkTyk−L1L2=1n∑k=1nTxkTyk−L1Tyk+L1Tyk−L1L2≤1n∑k=1nTyk1n∑k=1nTxk−L1+L11n∑k=1nTyk−L2<ε22β+L1ε2L1<ε.
Therefore, the set(28)k∈N:1n∑k=1nTxk.yk−L1L2≥ε∈I.
Thus, (xk).(yk) ∈ICes. Hence, CI is a sequence algebra.
On the similar manner, one can prove that spaceC0I is also sequence algebra.



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*Source: 8897155-2020-11-06.xml*