# Some Topological and Geometric Properties of Some New Spaces ofλ-Convergent and Bounded Series

Authors: Meltem Kaya; Hasan Furkan
Journal: Journal of Function Spaces
 (2015)

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## Abstract

The main purpose of this study is to introduce the spacescsλ, cs0λ, and bsλ which are BK-spaces of nonabsolute type. We prove that these spaces are linearly isomorphic to the spaces cs, cs0, and bs, respectively, and derive some inclusion relations. Additionally, Schauder bases of the spaces csλ and cs0λ have been constructed and the α-, β-, and γ-duals of these spaces have been computed. Besides, we characterize some matrix classes from the spaces csλ, cs0λ, and bsλ to the spaces lp, c, and c0, where 1≤p≤∞. Finally, we examine some geometric properties of these spaces as Gurarǐ’s modulus of convexity, property m∞, property (M), property WORTH, nonstrict Opial property, and weak fixed point property.


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## Body

## 1. Introduction

By asequence space, we understand a linear subspace of the space w=CN, where N={0,1,2,…} and C denotes the complex field. A sequence space E with a linear topology is called a K-space provided each of the maps pi:E→C defined pi(x)=xi which is continuous for all i∈N. A K-space is called an FK-space provided E is a complete linear metric space. An FK-space whose topology is normable is called a BK-space (see [1, pages 272-273]) which contains ϕ, the set of all finitely nonzero sequences. We write l∞, c, and c0 for the spaces of all bounded, convergent, and null sequences, respectively. Also by lp, we denote the space of all p-absolutely summable sequences, where 1≤p<∞. Moreover, we write bs, cs, and cs0 for the spaces of all bounded, convergent, and null series, respectively.Letμ and ν be two sequence spaces, and let A=(ank) be an infinite matrix of complex numbers ank, where n,k∈N. Then we say that A defines a matrix transformation from μ into ν, and we denote it by writing A:μ→ν if for every sequence x=(xk)∈μ, the sequence Ax={(Ax)n}, the A-transform of x, is in ν, where(1)Axn≔∑kankxk,n∈N,x∈D00A,and by D00(A) denotes the subspace of w consisting of x∈w for which the sum exists as a finite sum. For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞ and we will use the convention that any term with a negative subscript is equal to naught; for example, λ-1=0 and x-1=0.By(μ:ν), we denote the class of all matrices A such that A:μ→ν. Thus A∈(μ:ν) if and only if the series on the right side of (1) converges for each n∈N and each x∈μ and we have Ax={(Ax)n}n∈N∈ν for all x∈μ. For an arbitrary sequence space μ, thematrix domainμA of an infinite matrix A in μ is defined by(2)μA≔x∈w:Ax∈μ,which is a sequence space. If A is triangle, then one can easily observe that the normed sequence spaces μA and μ are norm isomorphic; that is, μA≅μ. If μ is a sequence space, then the continuous dual μA∗ of the space μA is defined by (3)μA∗≔f:f=g∘A,g∈μ∗.We denote the collection of all finite subsets of N by F. Also, we will write e(k) for the sequence whose only nonzero term is 1 in the kth place for each k∈N. Throughout this paper, let λ=(λk) be a strictly increasing sequence of positive real numbers tending to infinity; that is, (4)0<λ0<λ1<λ2<⋯,limk→∞⁡λk=∞.We define the matrix Λ=(λnk) of weighted mean relative to the sequence λ by (5)λnk=λk-λk-1λn,0≤k≤n,0,k>n,for all k; n∈N. With a direct calculation we derive the equality(6)Λxn=1λn∑k=0nλk-λk-1xk;n∈N.It is easy to show that the matrix Λ is regular and is reduced, in the special case λk=k+1 for all k∈N to the matrix C1 of Cesàro means of order one. Introducing the concept of Λ-strong convergence, several results on Λ-strong convergence of numerical sequences and Fourier series were given by Móricz [2]. Since we have in the special case(7)Qn=∑k=0nqk=λn,rnk=qkQn=λk-λk-1λn=λnk,qk=λk-λk-1 for all k∈N, the matrix Λ is also reduced to the Riesz means Rq=(rnk) with respect to the sequence q=(qk).We summarize the knowledge in the existing literature concerning domain of the matrixλ over some sequence spaces. Mursaleen and Noman [3–6] introduced the spaces l∞λ, cλ, c0λ, and lpλ of lambda-bounded, lambda-convergent, lambda-null, and lambda-absolutely p-summable sequences and gave the inclusion relations between these spaces and the classical sequence spaces l∞, c, and c0. Later, Mursaleen and Noman [7] investigated the difference spaces c0λ(Δ) and cλ(Δ) obtained from the spaces c0λ and cλ. Recently, paranormed λ-sequence spaces of nonabsolute type have been studied by Karakaya et al. [8]. More recently, Sönmez and Başar [9] introduce the difference sequence spaces c0λ(B) and cλ(B), which are the generalization of the spaces c0λ(Δ) and cλ(Δ). Quite recently, some new sequence spaces of nonabsolute type and matrix transformations have been studied by Ganie and Sheikh [10]. The same authors have studied the spaces of λ-convergent sequences and almost convergence [11]. Also, the fine spectrum of the operator defined by lambda matrix over the spaces of null and convergent sequences has been studied by Yeşilkayagil and Başar [12].In this work, our purpose is to construct the sequence spacescs0λ, csλ, and bsλ by the domain of the matrix Λ in the spaces cs0, cs, and bs, respectively, of the series whose sequence of partial sums are in the spaces c0, c, and l∞ [3].We define the sequencey=(yk) by the Λ-transform Λx of a sequence x=(xk); that is, y=Λx, and so we have(8)yk≔1λk∑j=0kλj-λj-1xj;k∈N.Also, we say that a sequence x=(xk)∈w is λ-convergent if Λx∈c. In particular, we say that x is λ-null or λ-bounded if Λx∈c0 or l∞, respectively.


## 2. The Sequence Spacescsλ, cs0λ, and bsλ

In the present section, we introduce the sequence spacescsλ, cs0λ, and bsλ as the sets of all sequences whose Λ-transforms are in the spaces cs, cs0, and bs, respectively; that is, (9)csλ=x=xk∈w:limm→∞⁡∑n=0m1λn∑k=0nλk-λk-1xkexists,(10)cs0λ=x=xk∈w:limm→∞⁡∑n=0m1λn∑k=0nλk-λk-1xk=0,(11)bsλ=x=xk∈w:supm∑n=0m1λn∑k=0nλk-λk-1xk<∞.With the notation of (2), we can redefine the spaces csλ, cs0λ, and bsλ as the matrix domains of the triangle Λ in the spaces cs, cs0, and bs by(12)csλ=csΛ,cs0λ=cs0Λ,bsλ=bsΛ.Then, it is immediate by (12) that the sets csλ, cs0λ, and bsλ are linear spaces with coordinatewise addition and scalar multiplication; that is, csλ, cs0λ, and bsλ are the sequence spaces consisting of all sequences which are λ-convergent, λ-null, and λ-bounded series of type λ, respectively.Now, we may begin with the following theorem which is essential in the text.Theorem 1.
The sequence spacescsλ, cs0λ, and bsλ are BK-spaces with the same norm xcsλ=xcs0λ=xbsλ; that is, (13)xbsλ=Λxbs=supm⁡∑n=0mΛxn<∞.Proof.
Since (12) holds and cs, cs0, and bs are BK-spaces with the respect to their natural norms and the matrix Λ is a triangle, Theorem 4.3.12 of Wilansky [13, page 63] gives the fact that csλ, cs0λ, and bsλ are BK-spaces with the given norms. This completes the proof.Remark 2.
One can easily check that the absolute property does not hold on the spacescsλ, cs0λ, and bsλ; that is, xcsλ≠|x|csλ, xcs0λ≠|x|cs0λ, and xbsλ≠|x|bsλ for at least one sequence in the spaces csλ, cs0λ, and bsλ, and this shows that csλ, cs0λ, and bsλ are the sequence spaces of nonabsolute type, where |x|=(|xk|).Now, we give the final theorem of this section.Theorem 3.
The sequence spacescsλ, cs0λ, and bsλ of nonabsolute type are isometrically isomorphic to the spaces cs, cs0, and bs, respectively; that is, csλ≅cs, cs0λ≅cs0, and bsλ≅bs.Proof.
To prove this, we should show the existence of an isometric isomorphism between the spacescs0λ and cs0. Consider the transformation T defined, with the notation of (8), from cs0λ to cs0 by x↦y=Tx. Then, Tx=y=Λx∈cs0 for every x∈cs0λ and the linearity of T is clear. Also, it is trivial that x=θ whenever Tx=θ and hence T is injective. Furthermore, let y=(yk)∈cs0 be given and define the sequence x=(xk) by(14)xk≔∑j=k-1k-1k-jλjλk-λk-1yj;k∈N.Then, by using (8) and (14), we have for every n∈N that(15)Λxn=1λn∑k=0nλk-λk-1xk=1λn∑k=0n∑j=k-1k-1k-jλjyj=1λn∑k=0nλkyk-λk-1yk-1=yn.
This shows thatΛx=y and since y∈cs0, we obtain that Λx∈cs0. Thus, we deduce that x∈cs0λ and Tx=y. Hence T is surjective. Moreover, one can easily see for every x∈cs0λ that (16)Txcs0=xcs0λ,which means that T is norm preserving. Therefore T is isometry. Consequently T is an isometric isomorphism which shows that the spaces cs0λ and cs0 are isometrically isomorphic.
It is clear that if the spacescs0λ and cs0 are replaced by the respective one of the spaces csλ and cs or bsλ and bs, then we obtain the fact that csλ≅cs and bsλ≅bs. This completes the proof.


## 3. The Inclusion Relations

In the present section, we establish some inclusion relations concerning the spacescsλ, cs0λ, and bsλ. We may begin with the following lemma.Lemma 4 (see [3]).
For any sequencex=(xk)∈w, the equality(17)Sxn=xn-Λxn;n∈Nholds, where Sx={(Sx)n} is the sequence defined by (18)Sx0=0,Sxn=1λn∑k=1nλk-1xk-xk-1;n≥1.Theorem 5.
The inclusionscs0λ⊂csλ⊂bsλ strictly hold.Proof.
It is obvious that the inclusionscs0λ⊂csλ⊂bsλ hold. Let us consider the sequence x=(xk) defined by (19)xk=λk1/k+22-λk-11/k+12λk-λk-1;k∈N.In the present case, we obtain for every n∈N that (20)Λxn=1λn∑k=0nλk1k+22-λk-11k+12=1n+22,which shows that Λx∈cs∖cs0. Thus, the sequence x is in csλ but not in cs0λ. Hence cs0λ⊂csλ is a strict inclusion.
To show the strictness of the inclusioncsλ⊂bsλ, we define the sequence y=(yk) by (21)yk=-1kλk+λk-1λk-λk-1;k∈N.Then, we have for every m∈N that (22)∑n=0mΛyn=∑n=0m1λn∑k=0n-1kλk+λk-1=∑n=0m-1n.This shows Λy∈bs∖cs. Thus, the sequence y is in bsλ but not in csλ and hence csλ⊂bsλ is a strict inclusion. This concludes the proof.Lemma 6 (see [3, Theorem 4.1.]).
The inclusionsc0λ⊂cλ⊂l∞λ strictly hold.Theorem 7.
The inclusionscsλ⊂c0λ and bsλ⊂l∞λ strictly hold.Proof.
It is clear that the inclusioncsλ⊂c0λ holds, since x∈csλ implies Λx∈cs and hence Λx∈c0 which means that x∈c0λ. Consider the sequence x=(xk) defined by (23)xk=1k+1;k∈N.Then, x∈c0 and hence x∈c0λ, since the inclusion c0⊂c0λ holds. On the other hand, we have for every n∈N that (24)Λxn=1λn∑k=0nλk-λk-1k+1≥1λnn+1∑k=0nλk-λk-1=1n+1,which shows that Λx∉cs and hence x∉csλ. Thus, the sequence x is in c0λ but not in csλ. Therefore, the inclusion csλ⊂c0λ is strict.
Similarly, it is also trivial that the inclusionbsλ⊂l∞λ holds. To show that this inclusion is strict, we define the sequence y=(yk) by y=e=(1,1,1,…). In the present case, we have for every n∈N that (25)Λyn=1λn∑k=0nλk-λk-1=1,which shows that Λy∈l∞∖bs. Thus, the sequence y is in l∞λ but not in bsλ and hence bsλ⊂l∞λ is a strict inclusion. This completes the proof.Theorem 8.
The inclusioncsλ⊂cs holds if and only if Sx∈cs for every sequence x∈csλ.Proof.
Suppose that the inclusioncsλ⊂cs holds, and take any x=(xk)∈csλ. Then Λx∈cs and x∈cs by the hypothesis. Thus, we deduce from (17) that(26)∑n=0mxn-Λxn=∑n=0mxn-∑n=0mΛxn=∑n=0mSxn.Hence, we obtain from (26) by letting m→∞ that(27)limm∑n=0mSxn=∑nxn-∑nΛxn.As (xn)∈cs and ((Λx)n)∈cs, the right-hand side of equality (27) is convergent as m→∞. Thereby, the series ∑n=0m(Sx)n converges and so Sx∈cs.
Conversely, letx∈csλ be given. Then, we have by the hypothesis that Sx∈cs. Again, it follows by (26) that (28)limm⁡∑n=0mxn=∑nSxn+∑nΛxn,which shows that x∈cs while Λx∈cs and Sx∈cs. Hence, the inclusion csλ⊂cs holds and this concludes the proof.Theorem 9.
The inclusioncs0λ⊂cs0 holds if and only if Sx∈cs0 for every sequence x∈cs0λ.Proof.
One can see by analogy to Theorem8 that the inclusion cs0λ⊂cs0 also holds if and only if Sx∈cs0 for every sequence x∈cs0λ. This completes the proof.Theorem 10.
The inclusionbsλ⊂bs holds if and only if Sx∈bs for every sequence x∈bsλ.Proof.
Suppose that the inclusionbsλ⊂bs holds, and take any x=(xk)∈bsλ. Then, x∈bs by the hypothesis. Thus, we obtain from equality (17)(29)Sxbs≤xbs+Λxbs=xbs+xbsλ<∞,which yields that Sx∈bs.
Conversely, assume thatSx∈bs for every x∈bsλ. Again, we obtain from equality (17) (30)xbs≤Sxbs+Λxbs=Sxbs+xbsλ<∞.This shows that x∈bs. Hence, the inclusion bsλ⊂bs holds. This completes the proof.


## 4. The Basis for the Spacescsλ, cs0λ, and bsλ

In the present section, we give a sequence of the points of the spacescsλ and cs0λ which forms a basis for these spaces. If a normed sequence space X contains a sequence (bn) with the property that for every x∈X there is a unique sequence of scalars (αn) such that (31)limn→∞⁡x-α0b0+α1b1+⋯+αnbn=0,then (bn) is called a Schauder basis (or briefly basis) for X. The series ∑kαkbk which has the sum x is then called the expansion of x with respect to (bn) and is written as x=∑kαkbk.Now, since the transformationT defined from cs0λ to cs0 in the proof of Theorem 3 is an isomorphism, we have the following theorem.Theorem 11.
Define the sequenceeλ(n)={(eλ(n))k} for every fixed k∈N by (32)eλnk=-1k-nλnλk-λk-1,n≤k≤n+1,0,otherwise,for all k∈N.
Then, one has the following:(a)
The sequence(eλ(0),eλ(1),…) is a Schauder basis for the spaces csλ and cs0λ and every x∈csλ or cs0λ has a unique representation of the form(33)x=∑n=0∞Λxneλn.(b)
bsλ has no Schauder basis.Proof.
(a) It is clear thateλ(n) is a basis for csλ since e(n) is a basis for cs and Λ(eλ(n))=e(n) [14, Corollary 2.3]. Let x∈csλ be given. Then, y=Λx∈cs and (34)ym=∑n=0mynen⟶ym⟶∞for a unique sequence (yn)n=0∞ of scalars. Therefore, we obtain that (35)Λ^ym=∑n=0mynΛ^en=∑n=0myneλn,where Λ^ is the inverse of the matrix Λ. Since yn=(Λx)n, we get that (36)xm=∑n=0mΛxneλn.Consequently, (37)xm-xcsλ=Λxm-xcs=Λxm-Λxcs=ym-ycs⟶0m⟶∞.Thus, we deduce that limm→∞x[m]-xcsλ=0, which shows that x∈csλ is represented as in (33).
Finally, let us show the uniqueness of the representation (33) of x∈csλ. Suppose on the contrary that there exists another representation x=∑nαneλ(n). Since the linear transformation T defined from csλ to cs, in the proof of Theorem 3, is continuous, we have (38)Λxk=∑nαnΛeλnk=∑nαnδkn=αk;k∈N.Therefore, the representation (33) of x∈csλ is unique. It can be proved similarly for cs0λ. This completes the proof.
(b) As a direct consequence of Remark 2.2. of Malkowsky and Rakocevic [14], since bs has no Schauder basis bsλ also has no Schauder basis.As a result, it easily follows from Theorem1 that csλ and cs0λ are the Banach spaces with their natural norms. Then, by Theorem 11 we obtain the following corollary.Corollary 12.
The sequence spacescsλ and cs0λ of nonabsolute type are separable.


## 5. Theα-, β-, and γ-Duals of the Spaces csλ, cs0λ, and bsλ

In this section, we state and prove the theorems determining theα-, β-, and γ-duals of the sequence spaces csλ, cs0λ, and bsλ of nonabsolute type. For arbitrary sequence spaces X and Y, the set M(X,Y) defined by(39)MX,Y=a=ak∈w:ax=akxk∈Y∀x=xk∈Xis called themultiplier space of X and Y. One can easily observe for a sequence space Z with Y⊂Z and Z⊂X that the inclusions M(X,Y)⊂M(X,Z) and M(X,Y)⊂M(Z,Y) hold, respectively. With the notation of (39), the α-, β-, and γ-duals of a sequence space X, which are, respectively, denoted by Xα, Xβ, and Xγ, are defined by (40)Xα=MX,l1,Xβ=MX,cs,Xγ=MX,bs.It is clear that Xα⊂Xβ⊂Xγ. Also, it can be obviously seen that the inclusions Xα⊂Yα, Xβ⊂Yβ, and Xγ⊂Yγ hold whenever Y⊂X.The following known results [15] are fundamental for this section.Lemma 13.
ConsiderA=(ank)∈(cs:l1) if and only if(41)supN,K∈F⁡∑n∈N∑k∈Kank-an,k-1<∞.Lemma 14.
ConsiderA=(ank)∈(cs0:l1) if and only if(42)supN,K∈F⁡∑n∈N∑k∈Kank-an,k+1<∞.Lemma 15.
ConsiderA=(ank)∈(bs:l1) if and only if (42) holds and(43)limk⁡ank=0,∀n∈N.Lemma 16.
ConsiderA=(ank)∈(cs:c) if and only if(44)supn⁡∑kank-an,k+1<∞,(45)limn⁡ankexists∀k∈N.Lemma 17.
ConsiderA=(ank)∈(cs0:c) if and only if (44) holds:(46)limn⁡ank-an,k+1exists∀k∈N.Lemma 18.
ConsiderA=(ank)∈(bs:c) if and only if (43) and (45) hold and(47)∑kank-an,k-1converges.Lemma 19.
ConsiderA=(ank)∈(cs:l∞) if and only if(48)supn⁡∑kank-an,k-1<∞.Lemma 20.
ConsiderA=(ank)∈(cs0:l∞) if and only if (44) holds.Lemma 21.
ConsiderA=(ank)∈(bs:l∞) if and only if (43) and (44) hold.Now, we prove the following result.Theorem 22.
Define the setsm1λ and m2λ as follows:(49)m1λ=a=an∈w:supN,K∈F⁡∑n∈N∑k∈Kbnkλ-bn,k-1λ<∞,m2λ=a=an∈w:supN,K∈F⁡∑n∈N∑k∈Kbnkλ-bn,k+1λ<∞,where the matrix Bλ=(bnkλ) is defined via the sequence a=(an)∈w by (50)bnkλ=-1n-kλkλn-λn-1anifn-1≤k≤n,0ifn-1>kork>n,for all n, k∈N. Then {csλ}α=m1λ and {cs0λ}α={bsλ}α=m2λ.Proof.
Leta=(an)∈w. Then, by bearing in mind relations (8) and (14), it is immediate that the equality(51)anxn=∑k=n-1n-1n-kλkλn-λn-1anyk=Bλynholds for all n∈N. We therefore observe by (51) that ax=(anxn)∈l1 whenever x=(xk)∈csλ if and only if Bλy∈l1 whenever y=(yk)∈cs. This means that the sequence a=(an)∈{csλ}α if and only if Bλ∈(cs:l1). Hence, we obtain by Lemma 13 with Bλ instead of A that a=(an)∈{csλ}α if and only if (52)supN,K∈F⁡∑n∈N∑k∈Kbnkλ-bn,k-1λ<∞which yields the result that {csλ}α=m1λ.
Similarly, we deduce from Lemma15 with (51) that a=(an)∈{bsλ}α if and only if Bλ∈(bs:l1). Then, it is clear that the columns of the matrix B are in the space c0, since (53)limk⁡bnkλ=0for all n∈N. Therefore, we derive from (42) that(54)supN,K∈F∑n∈N∑k∈Kbnkλ-bn,k+1λ<∞.This shows that {cs0λ}α={bsλ}α=m2λ. This completes the proof.Theorem 23.
Define the setsm3λ, m4λ, and m5λ as follows: (55)m3λ=a=ak∈w:∑k=0∞Δ¯Δ¯akλk-λk-1λk<∞,m4λ=a=ak∈w:supk⁡λkλk-λk-1ak<∞,m5λ=a=ak∈w:limk→∞⁡λkλk-λk-1ak=0,where(56)Δ¯akλk-λk-1=akλk-λk-1-ak+1λk+1-λkfor all k∈N. Then {csλ}β={cs0λ}β=m3λ∩m4λ and {bsλ}β=m3λ∩m5λ.Proof.
Because the proof may also be obtained for the spacebsλ in a similar way, we omit it. Take any a=(ak)∈w and consider the equation(57)∑k=0nakxk=∑k=0n∑j=k-1k-1k-jλjλk-λk-1yjak=∑k=0n-1Δ¯akλk-λk-1λkyk+λnλn-λn-1anyn=Tλyn,where the matrix Tλ=(tnkλ) is defined by (58)tnkλ=Δ¯akλk-λk-1λkif0≤k≤n-1,λnλn-λn-1anifk=n,0ifk>n,for all n,k∈N. Thus, we deduce by (57) that ax=(akxk)∈cs where x=(xk)∈csλ if and only if Tλy∈c whenever y=(yk)∈cs. This means that a=(ak)∈{csλ}β if and only if Tλ∈(cs:c). Therefore, by using Lemma 16, we derive from (44) and (45) that (59)∑k=0∞Δ¯Δ¯akλk-λk-1λk<∞,supn⁡λnλn-λn-1an<∞,(60)limn⁡tnkλ=Δ¯akλk-λk-1λk,respectively. Thereby, we conclude that {csλ}β={cs0λ}β=m3λ∩m4λ.Theorem 24.
Theγ-dual of the spaces csλ, cs0λ, and bsλ is the set m3λ∩m4λ.Proof.
The proof of this result follows the same lines as those in the proof of Theorem23 using Lemmas 19, 20, and 21 instead of Lemma 16.


## 6. Certain Matrix Mappings on the Spacescsλ, cs0λ, and bsλ

In this present section, we characterize the matrix classes(csλ:lp), (cs0λ:lp), (bsλ:lp), (csλ:c0), (cs0λ:c0), (bsλ:c0), (csλ:c), (cs0λ:c), and (bsλ:c), where 1≤p≤∞.For an infinite matrixA=(ank), we write for brevity that (61)a~nk=Δ¯ankλk-λk-1λk=ankλk-λk-1-an,k+1λk+1-λkλkn,k∈N.The following lemmas [15] will be needed in proving our results.Lemma 25.
ConsiderA=(ank)∈(cs:c0) if and only if (44) holds and(62)limn⁡ank=0∀k∈N.Lemma 26.
ConsiderA=(ank)∈(cs0:c0) if and only if (44) holds and(63)limn⁡ank-an,k+1=0∀k∈N.Lemma 27.
ConsiderA=(ank)∈(bs:c0) if and only if (43) holds and(64)limn∑kank-an,k+1=0.Lemma 28.
ConsiderA=(ank)∈(cs0:lp) if and only if(65)supK∑n∑k∈Kank-an,k+1p<∞1<p<∞.Lemma 29.
ConsiderA=(ank)∈(cs:lp) if and only if(66)supK⁡∑n∑k∈Kank-an,k-1p<∞1<p<∞.Lemma 30.
ConsiderA=(ank)∈(bs:lp) if and only if (43) and (65) hold.Now, we give the following results on the matrix transformations.Theorem 31.
(i) ConsiderA=(ank)∈(csλ:l∞) if and only if(67)supn⁡∑k=0∞a~nk-a~n,k-1<∞,(68)supk⁡λkλk-λk-1ank<∞.
(ii) ConsiderA=(ank)∈(cs0λ:l∞) if and only if (68) holds and(69)supn⁡∑k=0∞a~nk-a~n,k+1<∞.
(iii) ConsiderA=(ank)∈(bsλ:l∞) if and only if (69) holds and(70)limk→∞⁡λkλk-λk-1ank=0,limk⁡a~nk=0.Proof.
Suppose that conditions (67) and (68) hold and take any x=(xk)∈csλ. Then, we have by Theorem 23 that (ank)k=0∞∈(csλ)β for all n∈N and this implies the existence of the A-transform of x; that is, Ax exists. Further, it is clear that the associated sequence y=(yk) is in cs and hence y∈c0.
Let us now consider the following equality derived by using relation (8) from the mth partial sum of the series ∑kankxk:(71)∑k=0mankxk=∑k=0m-1a~nkyk+λmλm-λm-1anmym,∀n,m∈N.Therefore, by using (67) and (68), from (71) as m→∞, we obtain that(72)∑kankxk=∑ka~nkyk∀n∈N.Further, since the matrix A~=(a~nk) is in the class (cs:l∞) by Lemma 19 and (67), we have A~y∈l∞. Therefore, we deduce from (1) and (72) that Ax∈l∞ and hence A∈(csλ,l∞).
Conversely, suppose thatA∈(csλ,l∞). Then (ank)k=0∞∈(csλ)β for all n∈N and this, with Theorem 23, implies both (68) and (73)∑k=0∞a~nk-a~k+1<∞∀n∈Nwhich together imply that relation (72) holds for all sequences x∈csλ and y∈cs. Further, since Ax∈l∞ by the hypothesis, we obtain by (72) that A~y∈l∞ which shows that A~∈(cs:l∞), where A~=(a~nk). Hence, the necessity of (67) is immediate by (48). This concludes the proof of part (i).
Since parts (ii) and (iii) can be proved similarly, we omit their proofs.Corollary 32.
(i) ConsiderA=(ank)∈(csλ:c) if and only if (68) and (69) hold and(74)limn→∞⁡a~nkexists.
(ii) ConsiderA=(ank)∈(cs0λ:c) if and only if (68) and (69) hold and(75)limn⁡a~nk-a~n,k+1exists.
(iii) ConsiderA=(ank)∈(bsλ:c) if and only if (70) and (74) hold and(76)∑ka~nk-a~n,k-1convergent.Corollary 33.
(i) ConsiderA=(ank)∈(csλ:c0) if and only if (68) and (69) hold and(77)limn⁡a~nk=0.
(ii) ConsiderA=(ank)∈(cs0λ:c0) if and only if (68) and (69) hold and(78)limn⁡a~nk-a~n,k+1=0.
(iii) ConsiderA=(ank)∈(bsλ:c0) if and only if (70) hold and(79)limn⁡∑ka~nk-a~n,k+1=0.Corollary 34.
(i) ConsiderA=(ank)∈(csλ:l1) if and only if (68) holds and(80)∑ka~nk-a~n,k+1<∞,(81)supN,K∈F⁡∑n∈N∑k∈Ka~nk-a~n,k-1<∞.
(ii) ConsiderA=(ank)∈(cs0λ:l1) if and only if (68) and (80) hold and(82)supN,K∈F⁡∑n∈N∑k∈Ka~nk-a~n,k+1<∞.
(iii) ConsiderA=(ank)∈(bsλ:l1) if and only if (70), (80), and (82) hold.Corollary 35.
(i) ConsiderA=(ank)∈(csλ:lp) if and only if (68) and (80) hold and(83)supK∈F⁡∑n∑k∈Ka~nk-a~n,k-1p<∞.
(ii) ConsiderA=(ank)∈(cs0λ:lp) if and only if (68) and (80) hold and(84)supK∈F⁡∑n∑k∈Ka~nk-a~n,k+1p<∞.
(iii) ConsiderA=(ank)∈(bsλ:lp) if and only if (70), (80), and (84) hold.Since Corollaries32, 33, 34, and 35 can be proved similarly with Theorem 31, we omit their proofs.


## 7. Some Geometric Properties of the Spacescsλ, cs0λ, and bsλ

In this section, we investigate some geometric properties for the sequence spacescsλ and bsλ.Let(X,·) be a normed linear space, and let S(X) and B(X) be the unit sphere and unit ball of X (for the brevity X=(X,·)), respectively. Consider Clarkson’smodulus of convexity (Clarkson [16] and Day [17]) defined by (85)δXϵ=inf⁡1-x+y2;x,y∈SX,x-y=ϵ,where 0≤ϵ≤2. The inequality δX>0 for all ϵ∈(0,2] characterizes the uniformly convex spaces.In [18], Gurarǐ’s modulus of convexity is defined by (86)βxϵ=inf⁡1-infα∈0,1⁡αx+1-αy;x,y∈SX,x-y=ϵ,where 0≤ϵ≤2. It is easily shown that δX(ϵ)≤βx(ϵ)≤2δX(ϵ) for any 0≤ϵ≤2. Also if 0<βx(ϵ)<1, then X is uniformly convex, and if βx(ϵ)<1, then X is strictly convex.Opial property [19] states that(87)ifxn⟶w0,thenlimsupn→∞⁡xn<limsupn→∞⁡xn-x∀x∈X,x≠0.If the strict inequality becomes≤, this condition becomes a nonstrict Opial property.The coefficientR(X), introduced by García-Falset [20], is defined as(88)RX≔sup⁡liminfn→∞⁡xn-x:xn⟶w0,xn≤1∀n∈N,x≤1.So 1≤R(X)≤2 and it is not hard to see that, in the definition of R(X), “liminf” can be replaced by “limsup.” Some values of R(X) are R(c0)=1 and R(lp)=21/p, 1<p<∞.A Banach spaceX has property (M) if whenever xn→w0, then limsupn→∞xn-x is a function of x only. Property (M) which is introduced by Kalton [21] is equivalent to(89)ifxn⟶w0,u≤v,thenlimsupn→∞⁡xn+u≤limsupn→∞⁡xn+v.Sims [22] introduced a property calledweak orthogonality (WORTH) for Banach spaces. A Banach space X is said to have property WORTH if(90)for  everyxn⟶w0,x∈X,limsupn→∞⁡xn+x=limsupn→∞⁡xn-x.It remains unknown if property WORTH implies fixed point property. In many situations, the fixed point property can be easily obtained when we assume, in addition, that the spaces are considered to have the property WORTH.The following result will be used in our results.Proposition 36 (see [23, Proposition 2.1]).
For the following conditions on a Banach spaceX, one has (i) ⇒ (ii) ⇒ (iii) ⇒ (iv).(i)
X has property (M).(ii)
X has property WORTH.(iii)
Ifxn→w0, then for each x∈X we have that limsupn→∞xn-tx is an increasing function of t on [0,∞).(iv)
X satisfies the nonstrict Opial property.In [24, 25] it has been shown that R(X)=1 implies X has property (M).It has been shown that Banach spaceX has property mp (resp., m∞) [26] if for all x∈X, whenever xn→w0,(91)limsupn→∞⁡x+xnp=xp+limsupn→∞⁡xnpresp.,limsupn→∞⁡x+xn=max⁡x,limsupn→∞⁡xn.Clearly the above properties imply property (M) and property m1 implies Opial property.Theorem 37 (see [27]).
A Banach spaceX has property m∞ if and only if R(X)=1.Remark 38 (see [28]).
A Banach spaceX with R(X)<2 has the weak fixed point property.Now, let us give our first theorem in this section.Theorem 39.
Gurarǐ’s modulus of convexity for the normed spacescsλ and bsλ is(92)βcsλϵ=βbsλϵ≤1-1-ϵ,where 0≤ϵ≤2.Proof.
Assumex∈csλ. Then we have(93)xcsλ=xbsλ=Λxbs=supm⁡∑n=0mΛxn.Let 0≤ϵ≤2 and consider the following sequences:(94)x=xn=Λ^1,Λ^-ϵ2,Λ^ϵ2,0,0,…,y=yn=Λ^1,0,Λ^-ϵ2,Λ^-ϵ2,0,…,where Λ^ is the inverse of the matrix Λ. Since zn=(Λx)n and tn=(Λy)n, we have(95)z=zn=1,-ϵ2,ϵ2,0,0,…,y=yn=1,0,-ϵ2,-ϵ2,0,….By using sequences given above, we obtain the following equalities: (96)xbsλ=Λxbs=supm⁡∑n=0mΛxn=1,ybsλ=Λybs=supm⁡∑n=0mΛyn=1,x-ybsλ=Λx-Λybs=supm⁡∑n=0mΛxn-Λyn=ϵ.To complete the conditions of csλ or bsλ for Gurarǐ’s modulus of convexity, it remains to evaluate the infimum of αx+(1-α)ybsλ for 0≤α≤1. We have (97)inf0≤α≤1⁡αx+1-αybsλ=inf0≤α≤1⁡αΛx+1-αΛybs=inf0≤α≤1⁡1,-αϵ2,αϵ-ϵ2,-ϵ2+αϵ2,0,0,…bs=inf0≤α≤1⁡1-ϵ=1-ϵ.Consequently we get (98)βcsλϵ=βbsλϵ≤1-1-ϵ.This is the desired result. Thereby the proof is completed.Corollary 40.
Sinceδcsλ(ϵ)≤βcsλ(ϵ)≤2δcsλ(ϵ) and δcsλ(ϵ)=δbsλ(ϵ)=0 for 0≤ϵ≤2, βcsλ(ϵ)=0<1 and hence csλ and bsλ are strictly convex.Theorem 41.
ConsiderR(X)=1 if X∈{csλ,cs0λ}.Proof.
Incsλ (or cs0λ), we have eλ(n)∈S(X) and eλ(n)→w0, where (eλ(n)) is the standard basis. Since eλ(1)∈S(X), we have(99)eλn-eλ1bsλ→n→∞1;thus R(X)=1.Now, by Proposition36 and Theorem 41, we obtain the following results.Corollary 42.
The spacescsλ and cs0λ have property m∞ and so the spaces csλ and cs0λ have property (M).Corollary 43.
The spacescsλ and cs0λ have property WORTH.Corollary 44.
Ifxn→w0, then for each x∈csλ or cs0λ we have limsupn→∞xn-tx which is an increasing function of t on [0,∞).Corollary 45.
The spacescsλ and cs0λ satisfy the nonstrict Opial property.Hence, by Remark38 and Theorem 41, we have the following result.Corollary 46.
The spacescsλ and cs0λ have the weak fixed point property.



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