Some Topological and Geometrical Properties of a New Difference Sequence Space

Abstract

We introduce the new difference sequence space 𝑎^𝑟𝑝(Δ) . Further, it is proved that the space 𝑎^𝑟𝑝(Δ) is the BK-space including the space 𝑏𝑣_𝑝, which is the space of sequences of pbounded variation. We also show that the spaces 𝑎^𝑟𝑝(Δ), and ℓ_𝑝 are linearly isomorphic for 1≤𝑝<∞. Furthermore, the basis and the 𝛼-, 𝛽- and 𝛾-duals of the space 𝑎^𝑟𝑝(Δ) are determined. We devote the final section of the paper to examine some geometric properties of the space 𝑎^𝑟𝑝(Δ).

1. Preliminaries, Background, and Notation

By 𝜔, we will denote the space of all real valued sequences. Any vector subspace of 𝜔 is called as a sequence space. We will write ℓ_∞,𝑐, and 𝑐₀ for the spaces of all bounded, convergent and null sequences, respectively. Also, by 𝑏𝑠,𝑐𝑠,ℓ₁ and ℓ_𝑝; we denote the spaces of all bounded, convergent, absolutely, and 𝑝-absolutely convergent series, respectively, where 1<𝑝<∞.

A sequence space 𝜆 with a linear topology is called a 𝐾-space, provided each of the maps 𝑝_𝑖∶𝜆→ℂ defined by 𝑝_𝑖(𝑥)=𝑥_𝑖 is continuous for all 𝑖∈ℕ, where ℂ denotes the complex field and ℕ={0,1,2,…}. A 𝐾-space 𝜆 is called an FK-space, provided 𝜆 is a complete linear metric space. An 𝐹𝐾-space whose topology is normable is called a BK-space (see [1, pages 272-273]).

Let 𝜆,𝜇 be two sequence spaces and 𝐴=(𝑎_𝑛𝑘) an infinite matrix of real or complex numbers 𝑎_𝑛𝑘, where 𝑛,𝑘∈ℕ. Then, we say that 𝐴 defines a matrix mapping from 𝜆 into 𝜇, and we denote it by writing 𝐴∶𝜆→𝜇; if for every sequence 𝑥=(𝑥_𝑘)∈𝜆, the sequence 𝐴𝑥={(𝐴𝑥)_𝑛}, the 𝐴-transform of 𝑥, is in 𝜇, where (𝐴𝑥)_𝑛=_𝑘𝑎_𝑛𝑘𝑥_𝑘(𝑛∈ℕ).(1.1) For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞. By (𝜆∶𝜇) we denote the class of all matrices 𝐴 such that 𝐴∶𝜆→𝜇. Thus, 𝐴∈(𝜆∶𝜇) if and only if the series on the right side of (1.1) converges for each 𝑛∈ℕ and every 𝑥∈𝜆, and we have 𝐴𝑥={(𝐴𝑥)_𝑛}_𝑛∈ℕ∈𝜇 for all 𝑥∈𝜆. A sequence 𝑥 is said to be 𝐴-summable to 𝛼 if 𝐴𝑥 converges to 𝛼 which is called as the 𝐴-limit of 𝑥.

If a normed sequence space 𝜆 contains a sequence (𝑏_𝑛) with the property that for every 𝑥∈𝜆, there is a unique sequence of scalars (𝛼_𝑛) such that lim_𝑛⟶∞‖‖𝑥−𝛼₀𝑏₀+𝛼₁𝑏₁+⋯+𝛼_𝑛𝑏_𝑛‖‖=0,(1.2) then (𝑏_𝑛) is called a Schauder basis (or briefly basis) for 𝜆. The series ∑𝛼_𝑘𝑏_𝑘 which has the sum 𝑥 is then called the expansion of 𝑥 with respect to (𝑏_𝑛) and written as 𝑥=∑𝛼_𝑘𝑏_𝑘.

For a sequence space 𝜆, the matrix domain 𝜆_𝐴 of an infinite matrix 𝐴 is defined by𝜆_𝐴=𝑥=𝑥_𝑘∈𝜔∶𝐴𝑥∈𝜆,(1.3) which is a sequence space. The new sequence space 𝜆_𝐴 generated by the limitation matrix 𝐴 from the space 𝜆 either includes the space 𝜆 or is included by the space 𝜆, in general; that is, the space 𝜆_𝐴 is the expansion or the contraction of the original space 𝜆.

We will define 𝐵_𝑟=(𝑏^𝑟𝑛𝑘) by𝑏^𝑟𝑛𝑘=⎧⎪⎪⎨⎪⎪⎩1−𝑟𝑛+1𝑟^𝑘𝑟,0≤𝑘≤𝑛−1,_𝑛+10𝑛,+1,𝑘𝑘>=𝑛𝑛,,(0<𝑟<1),(1.4) for all 𝑛,𝑘∈ℕ and denote the collection of all finite subsets of ℕ by ℱ. We will also use the convention that any term with negative subscript is equal to naught.

The approach constructing a new sequence space by means of the matrix domain of a particular limitation method has been recently employed by Wang [2], Ng and Lee [3], Malkowsky [4], and Altay et al. [5]. They introduced the sequence spaces (ℓ_𝑝)_𝑁𝑞 in [2], (ℓ_𝑝)_𝐶1=𝑋_𝑝 in [3], (ℓ_∞)_𝑅^𝑡=𝑟^𝑡∞,𝑐_𝑅^𝑡=𝑟^𝑡𝑐 and (𝑐₀)_𝑅^𝑡=𝑟^𝑡0 in [4] and (ℓ_𝑝)_𝐸^𝑟=𝑒^𝑟𝑝 in [5]; where 𝑁_𝑞,𝐶₁,𝑅^𝑡 and 𝐸^𝑟 denote the Nörlund, arithmetic, Riesz and Euler means, respectively, and 1≤𝑝≤∞.

Recently, there has been a lot of interest in investigating geometric properties of sequence spaces besides topological and some other usual properties. In literature, there are many papers concerning the geometric properties of different sequence spaces. For example, in [6], Mursaleen et al. studied some geometric properties of normed Euler sequence space. Şimşek and Karakaya [7] investigated the geometric properties of sequence space ℓ_𝜌(𝑢,𝑣,𝑝) equipped with Luxemburg norm. Further information on geometric properties of sequence space can be found in [8, 9].

The main purpose of the present paper is to introduce the difference sequence space 𝑎^𝑟𝑝(Δ) together with matrix domain and is to derive some inclusion relations concerning with 𝑎^𝑟𝑝(Δ). Also, we investigate some topological properties of this new space and furthermore characterize geometric properties concerning Banach-Saks type 𝑝.

2. 𝑎^𝑟𝑝(Δ) Difference Sequence Space

In the present section, we introduce the difference sequence space 𝑎^𝑟𝑝(Δ) and emphasize its some properties. Although the difference sequence space 𝜆(Δ) corresponding to the space 𝜆 was defined by Kızmaz [10] as follows: 𝜆(Δ)=𝑥=𝑥_𝑘∈𝜔∶𝑥_𝑘−𝑥_𝑘+1∈𝜆,(2.1) the difference sequence space corresponding to the space ℓ_𝑝 was not examined, where 𝜆 denotes the anyone of the spaces 𝑐₀,𝑐 or ℓ_∞. So, Başar and Altay have recently studied the sequence space 𝑏𝑣_𝑝, the space of 𝑝-bounded variation, in [11] defined by 𝑏𝑣_𝑝=𝑥=𝑥_𝑘∈𝜔∶𝑥_𝑘−𝑥_𝑘−1∈ℓ_𝑝,1≤𝑝<∞,(2.2) which fills up the gap in the existing literature. Recently, Aydín and Başar [12] studied the sequence spaces 𝑎^𝑟0 and 𝑎^𝑟𝑐, defined by 𝑎^𝑟0=𝑥=𝑥_𝑘∈𝜔∶lim_𝑛⟶∞1𝑛+1^𝑛_𝑘=01+𝑟^𝑘𝑥_𝑘𝑎=0_𝑟𝑐,=𝑥=𝑥_𝑘∈𝜔∶lim_𝑛⟶∞1𝑛+1^𝑛_𝑘=01+𝑟^𝑘𝑥_𝑘exists.(2.3) Aydín and Başar [13] introduced the difference sequence spaces 𝑎^𝑟0(Δ) and 𝑎^𝑟𝑐(Δ), defined by 𝑎^𝑟0(Δ)=𝑥=𝑥_𝑘∈𝜔∶lim_𝑛⟶∞1𝑛+1^𝑛_𝑘=01+𝑟^𝑘𝑥_𝑘𝑎−𝑥_𝑘−1=0_𝑟𝑐,(Δ)=𝑥=𝑥_𝑘∈𝜔∶lim_𝑛⟶∞1𝑛+1^𝑛_𝑘=01+𝑟^𝑘𝑥_𝑘−𝑥_𝑘−1exists.(2.4) Aydín [14] introduced 𝑎^𝑟𝑝 sequence space, defined by 𝑎^𝑟𝑝=⎧⎪⎨⎪⎩𝑥=𝑥_𝑘∈𝜔∶_𝑛|||||1𝑛+1^𝑛_𝑘=01+𝑟^𝑘𝑥_𝑘|||||^𝑝<∞⎫⎪⎬⎪⎭;1≤𝑝<∞.(2.5) Define the matrix Δ=(𝛿_𝑛𝑘) by 𝛿_𝑛𝑘=(−1)^𝑛−𝑘0,,𝑛0−≤1𝑘≤<𝑘𝑛≤−𝑛1,or𝑘>𝑛.(2.6) As was made by Başar and Altay in [11], we treat slightly more different than Kızmaz and the other authors following him and employ the technique obtaining a new sequence space by the matrix domain of a triangle limitation method. We will introduce the sequence space 𝑎^𝑟𝑝(Δ) which is a natural continuation of Aydín and Başar [13], as follows: 𝑎^𝑟𝑝(Δ)=⎧⎪⎨⎪⎩𝑥=𝑥_𝑘∈𝜔∶_𝑛|||||1𝑛+1^𝑛_𝑘=01+𝑟^𝑘𝑥_𝑘−𝑥_𝑘−1|||||^𝑝<∞⎫⎪⎬⎪⎭;1≤𝑝<∞.(2.7) With the notation of (1.3), we may redefine the space 𝑎^𝑟𝑝(Δ) by𝑎^𝑟𝑝(Δ)=𝑎^𝑟𝑝_Δ.(2.8)

Define the sequence 𝑦={𝑦_𝑛(𝑟)} which will be frequently used as the 𝐵_𝑟-transform of a sequence 𝑥=(𝑥_𝑘), that is,𝑦_𝑛(𝑟)=^𝑛−1_𝑘=0(1−𝑟)𝑟^𝑘1+𝑛𝑥^𝑘+1+𝑟^𝑛1+𝑛𝑥^𝑛,𝑛∈ℕ.(2.9) Now, we may begin with the following theorem which is essential in the text.

Theorem 2.1. The set 𝑎^𝑟𝑝(Δ) becomes the linear space with the coordinatewise addition and scalar multiplication which is the BK-space with the norm ‖𝑥‖_𝑎^𝑟𝑝_(Δ)=‖𝑦‖_ℓ𝑝,(2.10) where 1≤𝑝<∞.

Proof. Since the proof is routine, we omit the details of the proof.

Theorem 2.2. The space 𝑎^𝑟𝑝(Δ) is linearly isomorphic to the space ℓ_𝑝; that is, 𝑎^𝑟𝑝(Δ)≅ℓ_𝑝, where 1≤𝑝<∞.

Proof. It is enough to show the existence of a linear bijection between the spaces 𝑎^𝑟𝑝(Δ) and ℓ_𝑝 for 1≤𝑝<∞. Consider the transformation 𝑇 defined, with the notation of (2.9), from 𝑎^𝑟𝑝(Δ) to ℓ_𝑝 by 𝑥↦𝑦=𝑇𝑥. The linearity of 𝑇 is clear. Furthermore, it is trivial that 𝑥=𝜃 whenever 𝑇𝑥=𝜃, and hence, 𝑇 is injective.
We assume that 𝑦∈ℓ_𝑝 for 1≤𝑝<∞ and define the sequence 𝑥=(𝑥_𝑘) by 𝑥_𝑛(𝑟)=^𝑛_𝑘𝑘=0_{𝑗=𝑘−1}(−1)^𝑘−𝑗1+𝑗1+𝑟^𝑘𝑦_𝑗;𝑘∈ℕ.(2.11) Then, since (Δ𝑥)_𝑛=^𝑛_𝑗=_𝑛−1(−1)^𝑛−𝑗1+𝑗1+𝑟^𝑛𝑦_𝑗;𝑛∈ℕ,(2.12) we get that _𝑛|||||11+𝑛^𝑛_𝑘=01+𝑟^𝑘^𝑘_{𝑗=𝑘−1}(−1)^𝑘−𝑗1+𝑗1+𝑟^𝑘𝑦_𝑗|||||^𝑝=_𝑛||𝑦_𝑛||^𝑝<∞.(2.13) Thus, we have that 𝑥∈𝑎^𝑟𝑝(Δ). In addition, one can derive that ‖𝑥‖_𝑎^𝑟𝑝_(Δ)=⎛⎜⎜⎝_𝑛|||||11+𝑛^𝑛_𝑘=01+𝑟^𝑘𝑥_𝑘−𝑥_𝑘−1|||||^𝑝⎞⎟⎟⎠^1/𝑝=⎛⎜⎜⎝_𝑛|||||11+𝑛^𝑛_𝑘=01+𝑟^𝑘^𝑘_{𝑗=𝑘−1}(−1)^𝑘−𝑗1+𝑗1+𝑟^𝑘𝑦_𝑗|||||^𝑝⎞⎟⎟⎠^1/𝑝=‖𝑦‖_ℓ𝑝(2.14) which means that 𝑇 is surjective and is norm preserving. Hence, 𝑇 is a linear bijection.

We wish to exhibit some inclusion relations concerning with the space 𝑎^𝑟𝑝(Δ).

Theorem 2.3. The inclusion 𝑏𝑣_𝑝⊂𝑎^𝑟𝑝(Δ) strictly holds for 1<𝑝<∞.

Proof. To prove the validity of the inclusion 𝑏𝑣_𝑝⊂𝑎^𝑟𝑝(Δ) for 1<𝑝<∞, it suffices to show the existence of a number 𝐾>0 such that ‖𝑥‖_𝑎^𝑟𝑝_(Δ)≤𝐾‖𝑥‖_𝑏𝑣𝑝 for every 𝑥∈𝑏𝑣_𝑝.
Let 𝑥∈𝑏𝑣_𝑝 and 1<𝑝<∞. Then, we obtain _𝑛|||||11+𝑛^𝑛_𝑘=01+𝑟^𝑘𝑥_𝑘−𝑥_𝑘−1|||||^𝑝≤_𝑛2^𝑛_𝑘=0||𝑥_𝑘−𝑥_𝑘−1||1+𝑛^𝑝<2^𝑝𝑝−𝑝1^𝑝_𝑛||𝑥_𝑛−𝑥_𝑛−1||^𝑝,(2.15) as expected, ‖𝑥‖_𝑎^𝑟𝑝_(Δ)≤2𝑝𝑝−1‖𝑥‖^𝑏𝑣𝑝,(2.16) for 1<𝑝<∞.
Furthermore, let us consider the sequence 𝑥={𝑥_𝑛(𝑟)} defined by 𝑥_𝑛(𝑟)=^𝑛_𝑘=0(−1)^𝑛1+𝑟^𝑘,𝑛∈ℕ.(2.17) Then, the sequence 𝑥 is in 𝑎^𝑟𝑝(Δ)−𝑏𝑣_𝑝, as asserted.

Lemma 2.4 (see [11, Theorem  2.4]). The inclusion ℓ_𝑝⊂𝑏𝑣_𝑝 strictly holds for 1<𝑝<∞.

Combining Lemma 2.4 and Theorem 2.3, we get the following corollary.

Corollary 2.5. The inclusion ℓ_𝑝⊂𝑎^𝑟𝑝(Δ) strictly holds for 1<𝑝<∞.

3. The Basis for the Space 𝑎^𝑟𝑝(Δ)

In the present section, we will give a sequence of the points of the space 𝑎^𝑟𝑝(Δ) which forms a basis for the space 𝑎^𝑟𝑝(Δ), where 1≤𝑝<∞.

Theorem 3.1. Define the matrix 𝐵_𝑟={𝑏^𝑛(𝑘)(𝑟)}_𝑛∈ℕ of elements of the space 𝑎^𝑟𝑝(Δ) for every fixed 𝑘∈ℕ by 𝑏^𝑛(𝑘)(𝑟)=⎧⎪⎪⎨⎪⎪⎩1+𝑘1+𝑟^𝑘−1+𝑘1+𝑟^𝑘+11+𝑛,0≤𝑘≤𝑛−1,1+𝑟^𝑛0,,𝑘𝑘=>𝑛𝑛,,(3.1) for every fixed 𝑘∈ℕ. Then, the sequence {𝑏^(𝑘)(𝑟)}_𝑘∈ℕ is a basis for the space 𝑎^𝑟𝑝(Δ), and any 𝑥∈𝑎^𝑟𝑝(Δ) has a unique representation of the form 𝑥=_𝑘𝜆_𝑘(𝑟)𝑏^(𝑘)(𝑟),(3.2) where 𝜆_𝑘(𝑟)=(𝐵_𝑟𝑥)_𝑘 for all 𝑘∈ℕ and 1≤𝑝<∞.

Proof. It is clear that {𝑏^(𝑘)(𝑟)}⊂𝑎^𝑟𝑝(Δ), since 𝐵_𝑟𝑏^(𝑘)(𝑟)=𝑒^(𝑘)∈ℓ_𝑝,𝑘=0,1,2,…,(3.3) for 1≤𝑝<∞; here, 𝑒^(𝑘) is the sequence whose only nonzero term is 1 in the 𝑘th place for each 𝑘∈ℕ. Let 𝑥∈𝑎^𝑟𝑝(Δ) be given. For every nonnegative integer 𝑚, we set 𝑥^[𝑚]=^𝑚_𝑘=0𝜆_𝑘(𝑟)𝑏^(𝑘)(𝑟).(3.4) Then, by applying 𝐵_𝑟 to (3.4), we obtain with (3.3) that 𝐵_𝑟𝑥^[𝑚]=^𝑚_𝑘=0𝜆_𝑘(𝑟)𝐵_𝑟𝑏^(𝑘)(𝑟)=^𝑚_𝑘=0𝐵_𝑟𝐵_𝑟𝑥_𝑘𝑒^(𝑘),𝑥−𝑥^[𝑚]_𝑖=0𝐵,0≤𝑖≤𝑚,_𝑟𝑥_𝑖,𝑖>𝑚,(3.5) where 𝑖,𝑚∈ℕ. For a given 𝜀>0, there is an integer 𝑚₀ such that ^∞_𝑖=𝑚||𝐵_𝑟𝑥_𝑖||^𝑝^1/𝑝<𝜀2,(3.6) for all 𝑚≥𝑚₀. Hence, ‖‖𝑥−𝑥^[𝑚]‖‖_{𝑎𝑟𝑝(Δ)}=^∞_𝑖=𝑚||𝐵_𝑟𝑥_𝑖||^𝑝^1/𝑝≤^∞_𝑖_=𝑚0||𝐵_𝑟𝑥_𝑖||^𝑝^1/𝑝<𝜀2<𝜀,(3.7) for all 𝑚≥𝑚₀, which proves that 𝑥∈𝑎^𝑟𝑝(Δ) is represented as in (3.2).
Let us show the uniqueness of representation for 𝑥∈𝑎^𝑟𝑝(Δ) given by (3.2). Assume, on the contrary, that there exists a representation 𝑥=∑_𝑘𝜇_𝑘(𝑟)𝑏^(𝑘)(𝑟). Since the linear transformation 𝑇, from 𝑎^𝑟𝑝(Δ) to ℓ_𝑝, used in Theorem 2.2 is continuous, at this stage, we have 𝐵_𝑟𝑥_𝑛=_𝑘𝜇_𝑘(𝑟)𝐵_𝑟𝑏^(𝑘)(𝑟)_𝑛=_𝑘𝜇_𝑘(𝑟)𝑒^𝑛(𝑘)=𝜇_𝑛(𝑟);𝑛∈ℕ,(3.8) which contradicts the fact that (𝐵_𝑟𝑥)_𝑛=𝜆_𝑛(𝑟) for all 𝑛∈ℕ. Hence, the representation (3.2) of 𝑥∈𝑎^𝑟𝑝(Δ) is unique. This step concludes the proof.

4. The 𝛼-, 𝛽-, and 𝛾-Duals of the Space 𝑎^𝑟𝑝(Δ)

In this section, we state and prove theorems determining the 𝛼-, 𝛽-, and 𝛾-duals of the space 𝑎^𝑟𝑝(Δ). Since the case 𝑝=1 can be proved by the same analogy and can be found in the literature, we omit the proof of that case and consider only the case 1<𝑝<∞ in the proof of Theorems 4.4–4.6.

For the sequence spaces 𝜆 and 𝜇, define the set 𝑆(𝜆,𝜇) by𝑆(𝜆,𝜇)=𝑧=𝑧_𝑘∈𝜔∶𝑥𝑧=𝑥_𝑘𝑧_𝑘∈𝜇,∀𝑥∈𝜆.(4.1) With the notation of (4.1), 𝛼-, 𝛽- and 𝛾-duals of a sequence space 𝜆, which are, respectively, denoted by 𝜆^𝛼,𝜆^𝛽 and 𝜆^𝛾, are defined by 𝜆^𝛼=𝑆𝜆,ℓ₁,𝜆^𝛽=𝑆(𝜆,𝑐𝑠),𝜆^𝛾=𝑆(𝜆,𝑏𝑠).(4.2) It is well-known for the sequence spaces 𝜆 and 𝜇 that 𝜆^𝛼⊆𝜆^𝛽⊆𝜆^𝛾 and 𝜆^𝜂⊃𝜇^𝜂 whenever 𝜆⊂𝜇, where 𝜂∈{𝛼,𝛽,𝛾}.

We begin with to quoting the lemmas due to Stieglitz and Tietz [15], which are needed in the proof of the following theorems.

Lemma 4.1. 𝐴∈(ℓ_𝑝∶ℓ₁) if and only if sup_𝐾∈ℱ_𝑘|||||_𝑛∈𝐾𝑎_𝑛𝑘|||||^𝑞<∞(1<𝑝≤∞).(4.3)

Lemma 4.2. 𝐴∈(ℓ_𝑝∶𝑐) if and only if sluimp_{𝑛⟶∞𝑛∈ℕ}𝑎_𝑛𝑘existsforeach𝑘∈ℕ,(4.4)_𝑘||𝑎_𝑛𝑘||^𝑞<∞1<𝑝<∞.(4.5)

Lemma 4.3. 𝐴∈(ℓ_𝑝∶ℓ_∞) if and only if (4.5) holds.

Theorem 4.4. Define the set 𝑎^𝑟𝑞 by 𝑎^r_𝑞=⎧⎪⎨⎪⎩𝑎=𝑎_𝑘∈𝜔∶sup_𝐾∈ℱ_𝑘|||||_𝑛∈𝐾𝑐^𝑟𝑛𝑘|||||^𝑞<∞⎫⎪⎬⎪⎭,(4.6) where 𝐶^𝑟=(𝑐^𝑟𝑛𝑘) is defined via the sequence 𝑎=(𝑎_𝑛) by 𝑐^𝑟𝑛𝑘=⎧⎪⎪⎨⎪⎪⎩1+𝑘1+𝑟^𝑘−1+𝑘1+𝑟^𝑘+1𝑎^𝑛1+𝑛,0≤𝑘≤𝑛−1,1+𝑟^𝑛0,𝑎_𝑛,𝑘𝑘=>𝑛𝑛,.(4.7) for all 𝑛,𝑘∈ℕ. Then, {𝑎^r_𝑝(Δ)}^𝛼=𝑎^𝑟𝑞, where 1<𝑝<∞.

Proof. Bearing in mind the relation (2.9), we immediately derive that 𝑎_𝑛𝑥_𝑛=^𝑛_𝑘𝑘=0_{𝑗=𝑘−1}(−1)^𝑘−𝑗1+𝑗1+𝑟^𝑘𝑎_𝑛𝑦_𝑗=(𝐶^𝑟𝑦)_𝑛,𝑛∈ℕ.(4.8) It follows from (4.8) that 𝑎𝑥=(𝑎_𝑛𝑥_𝑛)∈ℓ₁ whenever 𝑥∈𝑎^𝑟𝑝(Δ) if and only if 𝐶^𝑟𝑦∈ℓ₁ whenever 𝑦∈ℓ_𝑝. This means that 𝑎=(𝑎_𝑛)∈{𝑎^𝑟𝑝(Δ)}^𝛼 if and only if 𝐶^𝑟∈(ℓ_𝑝∶ℓ₁). Then, we derive by Lemma 4.1 with 𝐶^𝑟 instead of 𝐴 that sup_𝐾∈ℱ_𝑘|||||_𝑛∈𝐾𝑐^𝑟𝑛𝑘|||||^𝑞<∞.(4.9) This yields the desired consequence that {𝑎^𝑟𝑝(Δ)}^𝛼=𝑎^𝑟𝑞.

Theorem 4.5. Define the sets 𝑎^𝑟1,𝑎^𝑟2, and 𝑎^𝑟3 by 𝑎^𝑟1=𝑎=𝑎_𝑘∈𝜔∶sup_𝑛∈ℕ_𝑘𝑎||𝑒^𝑟𝑛𝑘||^𝑞<∞_𝑟2,=𝑎=𝑎_𝑘∈𝜔∶^∞_𝑗=𝑘𝑎𝑎_𝑗existsforeachfixed𝑘∈ℕ_𝑟3,=𝑎=𝑎_𝑘∈𝜔∶1+𝑛1+𝑟^𝑛𝑎_𝑛∈𝑐𝑠,(4.10) where 𝐸=(𝑒^𝑟𝑛𝑘) is defined by 𝑒^𝑟𝑛𝑘=⎧⎪⎪⎪⎨⎪⎪⎪⎩(𝑘+1)𝑎^𝑘1+𝑟^𝑘+11+𝑟^𝑘−11+𝑟^𝑘+1^𝑛_𝑗=𝑘+1𝑎_𝑗1+𝑛,0≤𝑘≤𝑛−1,1+𝑟^𝑛0,𝑎_𝑛,𝑘𝑘=>𝑛𝑛,,(4.11) for all 𝑛,𝑘∈ℕ. Then, {𝑎^𝑟𝑝(Δ)}^𝛽=𝑎^𝑟1∩𝑎^𝑟2∩𝑎^𝑟3, where 1<𝑝<∞.

Proof. Consider the equation ^𝑛_𝑘=0𝑎_𝑘𝑥_𝑘=^𝑛_𝑘=0^𝑘_𝑗=0^𝑗_{𝑖=𝑗−1}(−1)^𝑖−𝑗1+𝑖1+𝑟^𝑗𝑦_𝑗𝑎^𝑘=^𝑛−1_𝑘=0(𝑘+1)𝑎^𝑘1+𝑟^𝑘+11+𝑟^𝑘−11+𝑟^𝑘+1^𝑛_𝑗=𝑘+1𝑎_𝑗𝑦^𝑘+1+𝑛1+𝑟^𝑛𝑎_𝑛𝑦_𝑛=(𝐸𝑦)_𝑛𝑛∈ℕ.(4.12) Thus, we deduce from Lemma 4.2 with (4.12) that 𝑎𝑥=(𝑎_𝑘𝑥_𝑘)∈𝑐𝑠 whenever 𝑥=(𝑥_𝑘)∈𝑎^𝑟𝑝(Δ) if and only if 𝐸𝑦∈𝑐 whenever 𝑦=(𝑦_𝑘)∈ℓ_𝑝. That is to say that 𝑎=(𝑎_𝑘)∈{𝑎^𝑟𝑝(Δ)}^𝛽 if and only if 𝐸∈(ℓ_𝑝∶𝑐). Therefore, we derive from (4.4) and (4.5) that {𝑎^𝑟𝑝(Δ)}^𝛽=𝑎^𝑟1∩𝑎^𝑟2∩𝑎^𝑟3.

Theorem 4.6. {𝑎^𝑟𝑝(Δ)}^𝛾=𝑎^𝑟1, where 1<𝑝<∞.

Proof. It is natural that the present theorem may be proved by the same technique used in the proof of Theorems 4.4 and 4.5, above. But, we prefer here the following classical way.
Let 𝑎=(𝑎_𝑘)∈𝑎^𝑟1 and 𝑥=(𝑥_𝑘)∈𝑎^𝑟𝑝(Δ). Then, we obtain by applying Hölder’s inequality that |||||_𝑛_𝑘=0𝑎_𝑘𝑥_𝑘|||||=|||||_𝑛_𝑘=0^𝑘_𝑗=0^𝑗_{𝑖=𝑗−1}(−1)^𝑖−𝑗1+𝑖1+𝑟^𝑗𝑦_𝑗𝑎^𝑘|||||=|||||_𝑛_𝑘=0𝑒^𝑟𝑛𝑘𝑦_𝑘≤_𝑛|||||_𝑘=0||𝑒^𝑟𝑛𝑘||^𝑞^1/𝑞^𝑛_𝑘=0||𝑦_𝑘||^𝑝^1/𝑝,(4.13) which gives us by taking supremum over 𝑛∈ℕ that sup_𝑛∈ℕ|||||_𝑛_𝑘=0𝑎_𝑘𝑥_𝑘|||||≤sup_𝑛∈ℕ⎡⎢⎢⎣^𝑛_𝑘=0||𝑒^𝑟𝑛𝑘||^𝑞^1/𝑞^𝑛_𝑘=0||𝑦_𝑘||^𝑝^1/𝑝≤‖𝑦‖_ℓ𝑝⎤⎥⎥⎦⋅sup_{𝑛𝑛∈ℕ}_𝑘=0||𝑒^𝑟𝑛𝑘||^𝑞^1/𝑞<∞.(4.14) This means that 𝑎=(𝑎_𝑘)∈{𝑎^𝑟𝑝(Δ)}^𝛾. Hence, 𝑎^𝑟1⊂𝑎^𝑟𝑝(Δ)^𝛾.(4.15) Conversely, let 𝑎=(𝑎_𝑘)∈{𝑎^𝑟𝑝(Δ)}^𝛾 and 𝑥=(𝑥_𝑘)∈𝑎^𝑟𝑝(Δ). Then, one can easily see that (∑^𝑛𝑘=0𝑒^𝑟𝑛𝑘𝑦_𝑘)_𝑛∈ℕ∈ℓ_∞ whenever (𝑎_𝑘𝑥_𝑘)∈𝑏𝑠. This shows that the triangle matrix 𝐸=(𝑒^𝑟𝑛𝑘), defined by (4.11), is in the class (ℓ_𝑝∶ℓ_∞). Hence, the condition (4.5) holds with 𝑒^𝑟𝑛𝑘 instead of 𝑎_𝑛𝑘 which yields that 𝑎=(𝑎_𝑘)∈𝑎^𝑟1. That is to say that 𝑎^𝑟𝑝(Δ)^𝛾⊂𝑎^𝑟1.(4.16) Therefore, by combining the inclusions (4.15) and (4.16), we deduce that the 𝛾-dual of the space 𝑎^𝑟𝑝(Δ) is the set 𝑎^𝑟1, and this step completes the proof.

5. Some Geometric Properties of the Space 𝑎^𝑟𝑝(Δ)

In this section, we study some geometric properties of the space 𝑎^𝑟𝑝(Δ).

A Banach space 𝑋 is said to have the Banach-Saks property if every bounded sequence (𝑥_𝑛) in 𝑋 admits a subsequence (𝑧_𝑛) such that the sequence {𝑡_𝑘(𝑧)} is convergent in the norm in 𝑋 [16], where 𝑡_𝑘(𝑧)=1𝑘+1𝑧⁰+𝑧₁+⋯+𝑧_𝑘(𝑘∈ℕ).(5.1) A Banach space 𝑋 is said to have the weak Banach-Saks property whenever given any weakly null sequence (𝑥_𝑛)⊂𝑋 and there exists a subsequence (𝑧_𝑛) of (𝑥_𝑛) such that the sequence {𝑡_𝑘(𝑧)} strongly convergent to zero.

In [17], García-Falset introduce the following coefficient: 𝑅(𝑋)=supliminf_𝑛⟶∞‖‖𝑥_𝑛−𝑥‖‖∶𝑥_𝑛⊂𝐵(𝑋),𝑥−→_𝑛𝜔0,𝑥∈𝐵(𝑋),(5.2) where 𝐵(𝑋) denotes the unit ball of 𝑋.

Remark 5.1. A Banach space 𝑋 with 𝑅(𝑋)<2 has the weak fixed point property, [18].

Let 1<𝑝<∞. A Banach space is said to have the Banach-Saks type 𝑝 or property (BS)_𝑝, if every weakly null sequence (𝑥_𝑘) has a subsequence (𝑥_𝑘𝑙) such that for some 𝐶>0, ‖‖‖‖^𝑛_𝑙=0𝑥_𝑘𝑙‖‖‖‖<𝐶(𝑛+1)_1/𝑝,(5.3) for all 𝑛∈ℕ (see [19]).

Now, we may give the following results related to the some geometric properties, mentioned above, of the space 𝑎^𝑟𝑝(Δ).

Theorem 5.2. The space 𝑎^𝑟𝑝(Δ) has the Banach-Saks type 𝑝.

Proof. Let (𝜀_𝑛) be a sequence of positive numbers for which ∑𝜀_𝑛≤1/2, and also let (𝑥_𝑛) be a weakly null sequence in 𝐵(𝑎^𝑟𝑝(Δ)). Set 𝑏₀=𝑥₀=0 and 𝑏₁=𝑥_𝑛1=𝑥₁. Then, there exists 𝑚₁∈ℕ such that ‖‖‖‖^∞_𝑖=_𝑚1+1𝑏₁(𝑖)𝑒^(𝑖)‖‖‖‖_{𝑎𝑟𝑝(Δ)}<𝜀₁.(5.4) Since (𝑥_𝑛) is a weakly null sequence implies 𝑥_𝑛→0 coordinatewise, there is an 𝑛₂∈ℕ such that ‖‖‖‖^𝑚1_𝑖=0𝑥_𝑛(𝑖)𝑒^(𝑖)‖‖‖‖_{𝑎𝑟𝑝(Δ)}<𝜀₁,(5.5) where 𝑛≥𝑛₂. Set 𝑏₂=𝑥_𝑛2. Then, there exists an 𝑚₂>𝑚₁ such that ‖‖‖‖^∞_𝑖=_𝑚2+1𝑏₂(𝑖)𝑒^(𝑖)‖‖‖‖_{𝑎𝑟𝑝(Δ)}<𝜀₂.(5.6) By using the fact that 𝑥_𝑛→0 coordinatewise, there exists an 𝑛₃>𝑛₂ such that ‖‖‖‖^𝑚2_𝑖=0𝑥_𝑛(𝑖)𝑒^(𝑖)‖‖‖‖_{𝑎𝑟𝑝(Δ)}<𝜀₂,(5.7) where 𝑛≥𝑛₃.
If we continue this process, we can find two increasing subsequences (𝑚_𝑖) and (𝑛_𝑖) such that ‖‖‖‖^𝑚𝑗_𝑖=0𝑥_𝑛(𝑖)𝑒^(𝑖)‖‖‖‖_{𝑎𝑟𝑝(Δ)}<𝜀_𝑗,(5.8) for each 𝑛≥𝑛_𝑗+1 and ‖‖‖‖^∞_𝑖=_𝑚𝑗+1𝑏_𝑗(𝑖)𝑒^(𝑖)‖‖‖‖_{𝑎𝑟𝑝(Δ)}<𝜀_𝑗,(5.9) where 𝑏_𝑗=𝑥_𝑛𝑗. Hence, ‖‖‖‖^𝑛_𝑗=0𝑏_𝑗‖‖‖‖_{𝑎𝑟𝑝(Δ)}=‖‖‖‖^𝑛_𝑗=0⎛⎜⎜⎝^𝑚𝑗−1_𝑖=0𝑏_𝑗(𝑖)𝑒^(𝑖)+^𝑚𝑗_𝑖=𝑚_𝑗−1+1𝑏_𝑗(𝑖)𝑒^(𝑖)+^∞_𝑖=_𝑚𝑗+1𝑏_𝑗(𝑖)𝑒^(𝑖)⎞⎟⎟⎠‖‖‖‖_{𝑎𝑟𝑝(Δ)}≤‖‖‖‖^𝑛_𝑗=0^𝑚𝑗−1_𝑖=0𝑏_𝑗(𝑖)𝑒^(𝑖)‖‖‖‖_{𝑎𝑟𝑝(Δ)}+‖‖‖‖^𝑛_𝑗=0⎛⎜⎜⎝^𝑚𝑗_𝑖=𝑚_𝑗−1+1𝑏_𝑗(𝑖)𝑒^(𝑖)⎞⎟⎟⎠‖‖‖‖_{𝑎𝑟𝑝(Δ)}+‖‖‖‖^𝑛_𝑗=0⎛⎜⎜⎝^∞_𝑖=_𝑚𝑗+1𝑏_𝑗(𝑖)𝑒^(𝑖)⎞⎟⎟⎠‖‖‖‖_{𝑎𝑟𝑝(Δ)}≤‖‖‖‖^𝑛_𝑗=0⎛⎜⎜⎝^𝑚𝑗_𝑖=𝑚_𝑗−1+1𝑏_𝑗(𝑖)𝑒^(𝑖)⎞⎟⎟⎠‖‖‖‖_{𝑎𝑟𝑝(Δ)}+2^𝑛_𝑗=0𝜀_𝑗.(5.10) On the other hand, it can be seen that ‖𝑥_𝑛‖_{𝑎𝑟𝑝(Δ)}<1. Therefore, ‖𝑥_𝑛‖^𝑝𝑎_𝑟𝑝(Δ)<1. We have ‖‖‖‖^𝑛_𝑗=0⎛⎜⎜⎝^𝑚𝑗_𝑖=𝑚_𝑗−1+1𝑏_𝑗(𝑖)𝑒^(𝑖)⎞⎟⎟⎠‖‖‖‖^𝑝𝑎_𝑟𝑝(Δ)=^𝑛_𝑚𝑗=0^𝑗_{𝑖=𝑚𝑗−1+1}|||||_𝑖−1_𝑘=0(1−𝑟)𝑟^𝑘1+𝑖𝑥^𝑗(𝑘)+1+𝑟^𝑖1+𝑖𝑥^𝑖(𝑘)|||||^𝑝≤^𝑛_∞𝑗=0_𝑖=0|||||_i−1_𝑘=0(1−𝑟)𝑟^𝑘1+𝑖𝑥^𝑗(𝑘)+1+𝑟^𝑖1+𝑖𝑥^𝑖(𝑘)|||||^𝑝≤(𝑛+1).(5.11) Hence, we obtain ‖‖‖‖^𝑛_𝑗=0⎛⎜⎜⎝^𝑚𝑗_𝑖=𝑚_𝑗−1+1𝑏_𝑗(𝑖)𝑒^(𝑖)⎞⎟⎟⎠‖‖‖‖_{𝑎𝑟𝑝(Δ)}≤(𝑛+1)^1/𝑝.(5.12) By using the fact 1≤(𝑛+1)^1/𝑝 for all 𝑛∈ℕ, we have ‖‖‖‖^𝑛_𝑗=0𝑏_𝑗‖‖‖‖_{𝑎𝑟𝑝(Δ)}≤(𝑛+1)^1/𝑝+1≤2(𝑛+1)^1/𝑝.(5.13) Hence, 𝑎^𝑟𝑝(Δ) has the Banach-Saks type 𝑝. This completes the proof of the theorem.

Remark 5.3. Note that 𝑅(𝑎^𝑟𝑝(Δ))=𝑅(ℓ_𝑝)=2^1/𝑝, since 𝑎^𝑟𝑝(Δ) is linearly isomorphic to ℓ_𝑝.

Hence, by the Remarks 5.1 and 5.3, we have the following.

Theorem 5.4. The space 𝑎^𝑟𝑝(Δ) has the weak fixed point property, where 1<𝑝<∞.