On Generalized Difference Hahn Sequence Spaces

Abstract

We construct some generalized difference Hahn sequence spaces by mean of sequence of modulus functions. The topological properties and some inclusion relations of spaces ℎ_𝑝(ℱ,𝑢,Δ^𝑟) are investigated. Also we compute the dual of these spaces, and some matrix transformations are characterized.

1. Introduction and Preliminaries

By a sequence space, we understand a linear subspace of the space 𝑤=ℂ^ℕ of all real or complex-valued sequences, where ℂ denotes the complex field and ℕ=0,1,2,…. For 𝑥=(𝑥_𝑘)∈𝑤, we write 𝑙_∞, 𝑐, and 𝑐₀ for the classical spaces of all bounded, convergent, and null sequences, respectively. Also by 𝑏𝑠, 𝑐𝑠, and 𝑙_𝑝 we denote the space of all bounded, convergent, and 𝑝-absolutely convergent series, which are Banach spaces with the following norms: ‖𝑥‖_𝑏𝑠=‖𝑥‖_𝑐𝑠=sup_𝑛|∑^𝑛_𝑘=1𝑥_𝑘| and ‖𝑥‖_𝑙𝑝=(∑_𝑘|𝑥_𝑘|^𝑝)^1/𝑝, respectively. Additionally, the spaces 𝑏𝑣^𝑝 and ∫𝜆 are defined by 𝑏𝑣^𝑝=𝑥=𝑥_𝑘∈𝑤∶^∞_𝑘=1𝑥_𝑘−𝑥_𝑘−1^𝑝,𝑥<∞𝜆=𝑥=_𝑘∈𝑤∶𝑘𝑥_𝑘∈𝜆. (1) A coordinate space (or a 𝐾-space) is a vector space of numerical sequences, where addition and scalar multiplication are defined pointwise. That is, a sequence space 𝜆 with a linear topology is called a 𝐾-space provided that each of the maps 𝑝_𝑖∶𝜆→ℂ defined by 𝑝_𝑖(𝑥)=𝑥_𝑖 is continuous for all 𝑖∈ℕ. A 𝐵𝐾-space is a 𝐾-space, which is also a Banach space with continuous coordinate functionals 𝑓_𝑘(𝑥)=𝑥_𝑘, (𝑘=1,2,…). A 𝐾-space 𝜆 is called an 𝐹𝐾-space provided that 𝜆 is a complete linear metric space. An 𝐹𝐾-space whose topology is normable is called a 𝐵𝐾-space. 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, (2) then (𝑏_𝑛) is called 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. An 𝐹𝐾-space 𝜆 is said to have 𝐴𝐾 property, if 𝜙⊂𝜆 and {𝑒^𝑘} is a basis for 𝜆, where 𝑒^𝑘 is a sequence whose only nonzero term is 1 in 𝑘th place for each 𝑘∈ℕ and 𝜙=span{𝑒^𝑘}, the set of all finitely nonzero sequences. If 𝜙 is dense in 𝜆, then 𝜆 is called an 𝐴𝐷-space, and thus 𝐴𝐾 implies 𝐴𝐷.

The notion of difference sequence spaces was introduced by Kizmaz [1], who defined the sequence spaces as follows: 𝑍(Δ)=𝑥=𝑥_𝑘∈𝑤∶Δ𝑥_𝑘∈𝑍for𝑍=𝑐,𝑐₀,𝑙_∞, (3) where Δ𝑥=(Δ𝑥_𝑘)=(𝑥_𝑘−𝑥_𝑘+1). The notion was further generalized by Et and Çolak [2] by introducing the spaces. Let 𝑟 be a nonnegative integer; then, 𝑍Δ^𝑟=𝑥=𝑥_𝑘∈𝑤∶Δ^𝑟𝑥_𝑘∈𝑍for𝑍=𝑐,𝑐₀,𝑙_∞, (4) where Δ^𝑟𝑥=(Δ^𝑟𝑥_𝑘)=(Δ^𝑟−1𝑥_𝑘−Δ^𝑟−1𝑥_𝑘+1) and Δ⁰𝑥_𝑘=𝑥_𝑘 for all 𝑘∈ℕ. The generalized difference sequence has the following binomial representation: Δ^𝑟𝑥_𝑘=^𝑟_𝑚=0(−1)^𝑚𝑟𝑚𝑥_𝑘+𝑚. (5) Later concept have been studied by Bektaş et al. [3] and Et and Esi [4]. Another type of generalization of the difference sequence spaces is due to Tripathy and Esi [5] who studied the spaces 𝑙_∞(Δ_𝑣), 𝑐(Δ_𝑣), and 𝑐₀(Δ_𝑣). Recently, Esi et al. [6] and Tripathy et al. [7] have introduced a new type of generalized difference operators and unified those as follows.

Let 𝑟, 𝑣 be nonnegative integers; then, for 𝑍 a given sequence space, we have 𝑍Δ^𝑟_𝑣=𝑥=𝑥_𝑘∈𝑤∶Δ^𝑟_𝑣𝑥_𝑘∈𝑍 (6) for 𝑍=𝑐, 𝑐₀ and 𝑙_∞, where Δ^𝑟_𝑣𝑥=(Δ^𝑟_𝑣𝑥_𝑘)=(Δ_𝑣^𝑟−1𝑥_𝑘−Δ_𝑣^𝑟−1𝑥_𝑘+𝑣) and Δ⁰_𝑣𝑥_𝑘=𝑥_𝑘 for all 𝑘∈ℕ.

Let 𝑋 be a linear metric space. A function 𝑝∶𝑋→ℝ is called paranorm, if(1)𝑝(𝑥)≥0 for all 𝑥∈𝑋,(2)𝑝(−𝑥)=𝑝(𝑥) for all 𝑥∈𝑋,(3)𝑝(𝑥+𝑦)≤𝑝(𝑥)+𝑝(𝑦) for all 𝑥, 𝑦∈𝑋,(4)(𝜆_𝑛) is a sequence of scalars with 𝜆_𝑛→𝜆 as 𝑛→∞ and (𝑥_𝑛) is a sequence of vectors with 𝑝(𝑥_𝑛−𝑥)→0 as 𝑛→∞, then 𝑝(𝜆_𝑛𝑥_𝑛−𝜆𝑥)→0 as 𝑛→∞.A paranorm 𝑝 for which 𝑝(𝑥)=0 implies 𝑥=0 is called total paranorm and the pair (𝑋,𝑝) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [8], Theorem 10.4.2, pp. 183). For more details about sequence spaces (see [9, 10]) and the references therein.

A modulus function is a function 𝑓∶[0,∞)→[0,∞) such that(1)𝑓(𝑥)=0 if and only if 𝑥=0,(2)𝑓(𝑥+𝑦)≤𝑓(𝑥)+𝑓(𝑦), for all 𝑥,𝑦≥0,(3)𝑓 is increasing,(4)𝑓 is continuous from the right at 0.It follows that 𝑓 must be continuous everywhere on [0,∞). The modulus function may be bounded or unbounded. For example, if we take 𝑓(𝑥)=𝑥/(𝑥+1), then 𝑓(𝑥) is bounded. If 𝑓(𝑥)=𝑥^𝑝, 0<𝑝<1, then the modulus function 𝑓(𝑥) is unbounded. Subsequentially, modulus function has been discussed in ([11–14]) and references therein.

Let 𝜆 and 𝜇 be two sequence spaces and 𝐴=(𝑎_𝑛𝑘) be an infinite matrix of complex numbers, where 𝑘, 𝑛∈ℕ. Then, we say that 𝐴 defines a matrix mapping from 𝜆 into 𝜇, and we denote it by writing 𝐴∶𝜆→𝜇 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_𝑘foreach𝑛∈ℕ. (7) 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 (7) converges for each 𝑛∈ℕ and each 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 𝑙 if 𝐴𝑥 converges to 𝑙 which is called the 𝐴-limit of 𝑥.

The matrix domain 𝜆_𝐴 of an infinite matrix 𝐴 in a sequence space 𝜆 is defined by 𝜆_𝐴=𝑥=𝑥_𝑘∈𝑤∶𝐴𝑥∈𝜆 (8) which is a sequence space (for several examples of matrix domains, see [15] p. 49–176). In [16], Başar and Altay have defined the sequence space 𝑏𝑣_𝑝 which consists of all sequences such that Δ-transforms of them are in 𝑙_𝑝, where Δ denotes the matrix Δ=(𝛿_𝑛𝑘) as follows: Δ=𝛿_𝑛𝑘=(−1)^𝑛−𝑘,();()𝑛−1≤𝑘≤𝑛0,0≤𝑘<𝑛−1or𝑘>𝑛 (9) for all 𝑘, 𝑛∈ℕ. The space [𝑙(𝑝)]_𝐴^𝑢=𝑏𝑣(𝑢,𝑝) has been studied by Başar et al. [17], where 𝐴^𝑢=𝑎^𝑢_𝑛𝑘=(−1)^𝑛−𝑘𝑢_𝑘,();().𝑛−1≤𝑘≤𝑛0,0≤𝑘<𝑛−1or𝑘>𝑛 (10) Hahn [18] introduced the 𝐵𝐾-space ℎ of all sequences 𝑥=(𝑥_𝑘) such that ℎ=𝑥∶^∞_𝑘=1𝑘Δ𝑥_𝑘<∞,lim_𝑘→∞𝑥_𝑘=0, (11) where Δ𝑥_𝑘=𝑥_𝑘−𝑥_𝑘+1, for all 𝑘∈ℕ. The following norm: ‖𝑥‖_ℎ=_𝑘𝑘Δ𝑥_𝑘+sup_𝑘𝑥_𝑘 (12) was defined on the space ℎ by Hahn [18] (and also [19]). Rao ([20], Proposition 2.1) defined a new norm on ℎ as ‖𝑥‖=∑_𝑘𝑘|Δ𝑥_𝑘|. G. Goes and S. Goes [19] proved that the space ℎ is a 𝐵𝐾-space.

Hahn proved the following properties of the space ℎ.

Lemma 1. (i) ℎ is a Banach space.
(ii) ℎ⊂𝑙₁∩∫𝑐₀.
(iii) ℎ^𝛽=𝜎_∞.

In [19], G. Goes and S. Goes studied functional analytic properties of the 𝐵𝐾-space 𝑏𝑣₀∩𝑑𝑙₁. Additionally, G. Goes and S. Goes considered the arithmetic means of sequences in 𝑏𝑣₀ and 𝑏𝑣₀∩𝑑𝑙₁ and used an important fact which the sequence of arithmetic means (𝑛⁻¹∑^𝑛_𝑘=1𝑥_𝑘) of 𝑥∈𝑏𝑣₀ is a quasiconvex null sequence. And also G. Goes and S. Goes proved that ℎ=𝑙₁∩∫𝑏𝑣=𝑙₁∩∫𝑏𝑣₀.

Rao [20] studied some geometric properties of Hahn sequence space and gave the characterizations of some classes of matrix transformations. Balasubramanian and Pandiarani [21] defined the new sequence space ℎ(𝐹) called the Hahn sequence space of fuzzy numbers and proved that 𝛽 and 𝛾 duals of ℎ(𝐹) is the Cesàro space of the set of all fuzzy bounded sequences. Kirişci [22] compiled to studies on Hahn sequence space and defined a new Hahn sequence space by Cesàro mean in [23].

In [24], Kirişci introduce the sequence space ℎ_𝑝 by ℎ_𝑝=𝑥∶^∞_𝑘=1𝑘Δ𝑥_𝑘^𝑝<∞,lim_𝑘→∞𝑥_𝑘=0,1<𝑝<∞, (13) where Δ𝑥_𝑘=𝑥_𝑘−𝑥_𝑘+1, for all 𝑘∈ℕ. If we take 𝑝=1, ℎ_𝑝=ℎ which are called Hahn sequence spaces. We denote the collection of all finite subsets of ℕ by 𝐹.

Let ℱ=(𝑓_𝑘) be a sequence of modulus functions, 𝑝=(𝑝_𝑘) be a bounded sequence of positive real numbers, and 𝑢=(𝑢_𝑘) be a sequence of strictly positive real numbers. In the present paper we defined the following sequence space: ℎ_𝑝ℱ,𝑢,Δ^𝑟=𝑥∶^∞_𝑘=1𝑓_𝑘𝑘𝑢_𝑘Δ^𝑟𝑥_𝑘^𝑝𝑘<∞,lim_𝑘→∞𝑥_𝑘,,=01<𝑝<∞ (14) where Δ^𝑟𝑥=(Δ^𝑟𝑥_𝑘)=(Δ^𝑟−1𝑥_𝑘−Δ^𝑟−1𝑥_𝑘+1) and Δ⁰𝑥_𝑘=𝑥_𝑘 for all 𝑘∈ℕ. Define the sequence 𝑦=(𝑦_𝑘), which will be frequently used, by 𝑀-transform of a sequence 𝑥=(𝑥_𝑘); that is, 𝑦_𝑘=(𝑀𝑥)_𝑘=𝑓_𝑘𝑢_𝑘𝑘Δ^𝑟−1𝑥_𝑘−Δ^𝑟−1𝑥_𝑘+1, (15) where 𝑀=(𝑚_𝑛𝑘) with 𝑚_𝑛𝑘=();();𝑛,𝑛=𝑘−𝑛,𝑛+1=𝑘0,other (16) for all 𝑘, 𝑛∈ℕ.

If we take 𝑢=(𝑢_𝑘)=1, 𝑟=1 and ℱ=𝑓_𝑘(𝑥)=𝑥 for all 𝑘∈ℕ, then we get the sequence space ℎ_𝑝 defined by [24] Kirişci. By taking 𝑢=(𝑢_𝑘)=1, 𝑟=1,ℱ=𝑓_𝑘(𝑥)=𝑥, and 𝑝=(𝑝_𝑘)=1 for all 𝑘∈ℕ, we obtained a Hahn sequence space defined by Hanh [18].

The following inequality will be used throughout the paper.

Let 𝑝=(𝑝_𝑘) be a sequence of positive real numbers with 0<𝑝_𝑘≤sup_𝑘𝑝_𝑘=𝐻, and let 𝐾=max{1,2^𝐻−1}. Then, for the factorable sequences (𝑎_𝑘) and (𝑏_𝑘) in the complex plane, we have 𝑎_𝑘+𝑏_𝑘^𝑝𝑘≤𝐾𝑎_𝑘^𝑝𝑘+𝑏_𝑘^𝑝𝑘. (17) The main purpose of this paper is to study some difference Hahn sequence spaces by mean of sequence of modulus functions. We will study some topological and algebraic properties of the sequence spaces ℎ_𝑝(ℱ,𝑢,Δ^𝑟) in Section 2. In Section 3 we will determine the 𝛼-, 𝛽-, and 𝛾-duals of the spaces ℎ_𝑝(ℱ,𝑢,Δ^𝑟). Finally, we also made an attempt to characterize some matrix transformations on the spaces ℎ_𝑝(ℱ,𝑢,Δ^𝑟).

2. Main Results

The purpose of this section is to study the properties like linearity, paranorm, and relevant inclusion relations in the spaces ℎ_𝑝(ℱ,𝑢,Δ^𝑟).

Theorem 2. The sequence space ℎ_𝑝(ℱ,𝑢,Δ^𝑟) is a linear space over the complex field ℂ.

Proof. Let 𝑥=(𝑥_𝑘), 𝑦=(𝑦_𝑘)∈ℎ_𝑝(ℱ,𝑢,Δ^𝑟) and 𝜌, 𝜚∈ℂ. Then their exist integers 𝑀_𝜌 and 𝑁_𝜚 such that |𝜌|≤𝑀_𝜌 and |𝜚|≤𝑁_𝜚. By using the inequality (17) and the properties of modulus function, we have ^∞_𝑘=1𝑓_𝑘𝑘𝑢_𝑘Δ^𝑟𝜌𝑥_𝑘+𝜚𝑦_𝑘^𝑝𝑘≤^∞_𝑘=1𝑓_𝑘𝑘𝑢_𝑘𝜌Δ^𝑟𝑥_𝑘+𝜚Δ^𝑟𝑦_𝑘^𝑝𝑘≤𝐾^∞_𝑘=1𝑓_𝑘𝑀_𝜌𝑘𝑢_𝑘Δ^𝑟𝑥_𝑘^𝑝𝑘+𝐾^∞_𝑘=1𝑓_𝑘𝑁_𝜚𝑘𝑢_𝑘Δ^𝑟𝑦_𝑘^𝑝𝑘≤𝐾𝑀^𝐻_𝜌^∞_𝑘=1𝑓_𝑘𝑘𝑢_𝑘Δ^𝑟𝑥_𝑘^𝑝𝑘+𝐾𝑁^𝐻_𝜚^∞_𝑘=1𝑓_𝑘𝑘𝑢_𝑘Δ^𝑟𝑦_𝑘^𝑝𝑘<∞. (18) Thus, 𝜌𝑥+𝜚𝑦∈ℎ_𝑝(ℱ,𝑢,Δ^𝑟). This proves that ℎ_𝑝(ℱ,𝑢,Δ^𝑟) is a linear space over the field of complex number ℂ.

Theorem 3. Let ℱ=(𝑓_𝑘) be a sequence of modulus functions and 𝑢=(𝑢_𝑘) be any sequence of strictly positive real numbers. Then ℎ_𝑝(ℱ,𝑢,Δ^𝑟) is a paranormed space with the paranorm defined by 𝑔(𝑥)=sup_𝑘{𝑓_𝑘𝑘𝑢_𝑘Δ^𝑟𝑥_𝑘)^{𝑝𝑘1/𝑀}, (19) where 𝐻=sup_𝑘𝑝_𝑘<∞ and 𝑀=max(1,𝐻).

Proof. Clearly 𝑔(𝑥)=𝑔(−𝑥) for all 𝑥∈ℎ_𝑝(ℱ,𝑢,Δ^𝑟). It is trivial that 𝑢_𝑘Δ^𝑟𝑥_𝑘=0 for 𝑥=0. Since 𝑝_𝑘/𝑀≤1 using Minkowski’s inequality, we have 𝑓_𝑘𝑘𝑢_𝑘Δ^𝑟𝑥_𝑘+𝑦_𝑘^{𝑝𝑘1/𝑀}≤𝑓_𝑘𝑘𝑢_𝑘Δ^𝑟𝑥_𝑘+𝑢_𝑘Δ^𝑟𝑦_𝑘^{𝑝𝑘1/𝑀}≤𝑓_𝑘𝑘𝑢_𝑘Δ^𝑟𝑥_𝑘^{𝑝𝑘1/𝑀}+𝑓_𝑘𝑘𝑢_𝑘Δ^𝑟𝑦_𝑘^{𝑝𝑘1/𝑀}. (20) Hence 𝑔(𝑥+𝑦)≤𝑔(𝑥)+𝑔(𝑦). Finally to check the continuity of scalar multiplication, let us take a complex number 𝛿 by definition, we have 𝑔(𝛿𝑥)=sup_𝑘𝑓_𝑘𝑘𝑢_𝑘Δ^𝑟𝛿𝑥_𝑘^{𝑝𝑘1/𝑀}≤𝐾_𝛿^𝐻/𝑀𝑔(𝑥), (21) where 𝐾_𝛿 is a positive integer such that |𝛿|≤𝐾_𝛿. Let 𝛿→0 for any fixed 𝑥 with 𝑔(𝑥)=0. By definition for |𝛿|<1, we have sup_𝑘𝑓_𝑘𝑘𝑢_𝑘Δ^𝑟𝑥_𝑘^{𝑝𝑘1/𝑀}<𝜖for𝑛>𝑁(𝜖). (22) Also for 1≤𝑛≤𝑁, taking 𝛿 small enough, since 𝑓_𝑘 is continuous for each 𝑘, we have sup_𝑘𝑓_𝑘𝑘𝑢_𝑘Δ^𝑟𝑥_𝑘^{𝑝𝑘1/𝑀}<𝜖. (23) Equations (22) and (23) imply that 𝑔(𝛿𝑥)→0 as 𝛿→0. This completes the proof.

Theorem 4. Let ℱ=(𝑓_𝑘) be a sequence of modulus functions and 𝜙=lim_𝑡→∞(𝑓_𝑘(𝑡)/𝑡)>0. Then ℎ_𝑝(ℱ,𝑢,Δ^𝑟)⊂ℎ_𝑝(𝑢,Δ^𝑟).

Proof. Let 𝜙>0. By definition of 𝜙, we have 𝑓_𝑘(𝑡)≥𝜙⋅𝑡, for all 𝑡≥0. Since 𝜙>0, we have 𝑡≤(1/𝜙)𝑓_𝑘(𝑡) for all 𝑡≥0. Let 𝑥=(𝑥_𝑘)∈ℎ_𝑝(ℱ,𝑢,Δ^𝑟). Thus, we have ^∞_𝑘=1𝑘𝑢_𝑘Δ^𝑟𝑥_𝑘^𝑝𝑘≤1𝜙^∞_𝑘=1𝑓_𝑘𝑘𝑢_𝑘Δ^𝑟𝑥_𝑘^𝑝𝑘<∞. (24) Which implies that 𝑥=(𝑥_𝑘)∈ℎ_𝑝(𝑢,Δ^𝑟). This completes the proof.

Theorem 5. ℎ_𝑝(ℱ,𝑢,Δ^𝑟)=𝑙_𝑝(ℱ,𝑢,Δ^𝑟)∩∫𝑏𝑣^𝑝(ℱ,𝑢,Δ^𝑟)=𝑙_𝑝(ℱ,𝑢,Δ^𝑟)∩∫𝑏𝑣^𝑝₀(ℱ,𝑢,Δ^𝑟).

Proof. We consider 𝑓_𝑘𝑢_𝑘𝑘Δ^𝑟𝑥_𝑘≤𝑓_𝑘𝑢_𝑘𝑥_𝑘+𝑓_𝑘𝑢_𝑘Δ^𝑟𝑘𝑥_𝑘. (25) Then, for 𝑥∈𝑙_𝑝(ℱ,𝑢,Δ^𝑟)∩∫𝑏𝑣^𝑝(ℱ,𝑢,Δ^𝑟), ^𝑛_𝑘=1𝑓_𝑘𝑢_𝑘𝑘Δ^𝑟𝑥_𝑘≤^𝑛_𝑘=1𝑓_𝑘𝑢_𝑘𝑥_𝑘+^𝑛_𝑘=1𝑓_𝑘𝑢_𝑘Δ^𝑟𝑘𝑥_𝑘, (26) and from |𝑎+𝑏|^𝑝𝑘≤2^𝑝𝑘(|𝑎|^𝑝𝑘+|𝑏|^𝑝𝑘), (1≤𝑝_𝑘<∞), we obtain ^𝑛_𝑘=1𝑓_𝑘𝑢_𝑘𝑘^𝑝𝑘Δ^𝑟𝑥_𝑘^𝑝𝑘≤2^𝑝𝑘𝑓_𝑘𝑢_𝑘^𝑛_𝑘=1𝑥_𝑘^𝑝𝑘+^𝑛_𝑘=1Δ^𝑟𝑘𝑥_𝑘^𝑝𝑘. (27) For each positive integer 𝑠, we get ^𝑠_𝑘=1𝑓_𝑘𝑢_𝑘𝑘^𝑝𝑘Δ^𝑟𝑥_𝑘|^𝑝𝑘≤2^𝑝𝑘𝑓_𝑘𝑢_𝑘^𝑠_𝑘=1𝑥_𝑘^𝑝𝑘+^𝑠_𝑘=1Δ^𝑟𝑘𝑥_𝑘^𝑝𝑘, (28) and as 𝑠→∞, ^∞_𝑘=1𝑓_𝑘𝑢_𝑘𝑘^𝑝𝑘Δ^𝑟𝑥_𝑘^𝑝𝑘≤2^𝑝𝑘𝑓_𝑘𝑢_𝑘^∞_𝑘=1𝑥_𝑘^𝑝𝑘+^∞_𝑘=1Δ^𝑟𝑘𝑥_𝑘^𝑝𝑘 (29) and lim_𝑘→∞𝑥_𝑘=0. Then 𝑥∈ℎ_𝑝(ℱ,𝑢,Δ^𝑟) and 𝑙_𝑝ℱ,𝑢,Δ^𝑟∩𝑏𝑣^𝑝ℱ,𝑢,Δ^𝑟⊂ℎ_𝑝ℱ,𝑢,Δ^𝑟. (30) Let 𝑥∈ℎ_𝑝(ℱ,𝑢,Δ^𝑟), and we consider ^∞_𝑘=1𝑓_𝑘𝑢_𝑘𝑥_𝑘+1^𝑝𝑘−^∞_𝑘=1𝑓_𝑘𝑢_𝑘Δ^𝑟𝑘𝑥_𝑘^𝑝𝑘≤^∞_𝑘=1𝑓_𝑘𝑢_𝑘𝑘^𝑝𝑘Δ^𝑟𝑘𝑥_𝑘^𝑝𝑘. (31) Then the series ∑^∞_𝑘=1𝑓_𝑘𝑢_𝑘|𝑥_𝑘+1|^𝑝𝑘 is convergent from the definition of 𝑙_𝑝. Also, ∑^∞_𝑘=1𝑓_𝑘𝑢_𝑘|Δ^𝑟(𝑘𝑥_𝑘)|^𝑝𝑘<∞, and therefore 𝑥∈𝑙_𝑝(ℱ,𝑢,Δ^𝑟)∩∫𝑏𝑣^𝑝(ℱ,𝑢,Δ^𝑟).
Then, ℎ_𝑝ℱ,𝑢,Δ^𝑟⊂𝑙_𝑝ℱ,𝑢,Δ^𝑟∩𝑏𝑣^𝑝ℱ,𝑢,Δ^𝑟. (32) From (30) and (32), we have ℎ_𝑝ℱ,𝑢,Δ^𝑟=𝑙_𝑝ℱ,𝑢,Δ^𝑟∩𝑏𝑣^𝑝ℱ,𝑢,Δ^𝑟. (33)

Theorem 6. The sequence space ℎ_𝑝(ℱ,𝑢,Δ^𝑟) is a BK-space with AK.

Proof. If 𝑥 is any sequence, we write 𝜎_𝑛(𝑥)=𝑀_𝑛𝑥. Let 𝜖>0 and 𝑥∈ℎ_𝑝(ℱ,𝑢,Δ^𝑟). Then, their exists 𝑁 such that 𝜎_𝑛(𝑥)<𝜖2 (34) for all 𝑛≥𝑁. Now let 𝑚≥𝑁 be given. Then, we have for all 𝑛≥𝑚+1 by (34) 𝜎_𝑛𝑥−𝑥^[𝑚]≤^∞_𝑘=𝑚+1𝑓_𝑘𝑘𝑢_𝑘Δ^𝑟𝑥_𝑘^{𝑝𝑘1/𝑝𝑘}≤𝜎_𝑛(𝑥)+𝜎_𝑚(𝑥<𝜖)2+𝜖2=𝜖, (35) whence ‖𝑥−𝑥^[𝑚]‖_{ℎ𝑝(ℱ,𝑢,Δ}^𝑟₎≤𝜖 for all 𝑚>𝑁. This shows that 𝑥=lim_𝑚→∞𝑥^[𝑚].

Since ℎ_𝑝(ℱ,𝑢,Δ^𝑟) is an AK-space and every AK-space is AD, we can give the following corollary.

Corollary 7. The sequence space ℎ_𝑝(ℱ,𝑢,Δ^𝑟) is AD.

3. Duals of Hahn Sequence Space ℎ_𝑝(ℱ,𝑢,Δ^𝑟)

In this section, we determining the 𝛼, 𝛽, and 𝛾-duals of the sequence space ℎ_𝑝(ℱ,𝑢,Δ^𝑟). Let 𝑥 and 𝑦 be sequences, 𝑋 and 𝑌 be subsets of 𝑤, and 𝐴=(𝑎_𝑛𝑘)^∞_𝑛,𝑘=0 be an infinite matrix of complex numbers. We write 𝑥𝑦=(𝑥_𝑘𝑦_𝑘)^∞_𝑘=0,𝑥⁻¹∗𝑌={𝑎∈𝑤∶𝑎𝑥∈𝑌}, and 𝑀(𝑋,𝑌)=∩_𝑥∈𝑋𝑥⁻¹∗𝑌 = {𝑎∈𝑤∶𝑎𝑥∈𝑌forall𝑥∈𝑋} for the multiplier space of 𝑋 and 𝑌. In the special cases of 𝑌={𝑙₁,𝑐𝑠,𝑏𝑠}, we write 𝑥^𝛼=𝑥⁻¹∗𝑙₁, 𝑥^𝛽=𝑥⁻¹∗𝑐𝑠, 𝑥^𝛾=𝑥⁻¹∗𝑏𝑠 and 𝑋^𝛼=𝑀(𝑋,𝑙₁), 𝑋^𝛽=𝑀(𝑋,𝑐𝑠), 𝑋^𝛾=𝑀(𝑋,𝑏𝑠) for the 𝛼-dual, 𝛽-dual, and 𝛾-dual of 𝑋. By 𝐴_𝑛=(𝑎_𝑛𝑘)^∞_𝑘=0, we denote the sequence in the 𝑛th-row of 𝐴, and we write 𝐴_𝑛(𝑥)=∑^∞_𝑘=0𝑎_𝑛𝑘𝑥_𝑘𝑛=(0,1,…) and 𝐴(𝑥)=(𝐴_𝑛(𝑥))^∞_𝑛=0, provided that 𝐴_𝑛∈𝑥^𝛽 for all 𝑛.

Given an 𝐹𝐾-space 𝑋 containing 𝜙, its conjugate is denoted by 𝑋^′ and its 𝑓-dual or sequential dual is denoted by 𝑋^𝑓 and is given by 𝑋^𝑓={allsequences(𝑓(𝑒^𝑘))∶𝑓∈𝑋^′}. Let 𝜆 be a sequence space. Then 𝜆 is called perfect if 𝜆=𝜆^𝛼𝛼, normal if 𝑦∈𝜆 whenever |𝑦_𝑘|≤|𝑥_𝑘|,𝑘≥1 for some 𝑥∈𝜆, and monotone if 𝜆 contains the canonical preimages of all its stepspace.

Lemma 8. (i) 𝐴∈(ℎ∶𝑙₁) if and only if ^∞_𝑛=1𝑎_𝑛𝑘𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑠,(𝑘=1,2,3,…),sup_𝑘1𝑘^∞_𝑛=1^𝑘_𝑣=1𝑎_𝑛𝑣<∞. (36)
(ii) 𝐴∈(𝑙_𝑝∶𝑙₁) if and only if sup_{𝐾∈𝐹𝑘𝑛∈𝐾}𝑎_𝑛𝑘^𝑞<∞. (37)

Lemma 9. (i) 𝐴∈(ℎ∶𝐶) if and only if sup_𝑛,𝑘1𝑘^𝑘_𝑣=1𝑎_𝑛𝑣<∞,lim_𝑛→∞𝑎_𝑛𝑘𝑒𝑥𝑖𝑠𝑡𝑠,(𝑘=1,2,3,…). (38)(39)
(ii) 𝐴∈(𝑙_𝑝∶𝐶) if and only if (39) holds and sup_𝑛𝑘𝑎_𝑛𝑘^𝑞<∞1<𝑝≤∞. (40)

Lemma 10. (i) 𝐴∈(ℎ∶𝑙_∞) if and only if (38) holds.
(ii) 𝐴∈(𝑙_𝑝∶𝑙_∞) if and only if (40) holds with 1<𝑝≤∞.

Lemma 11. (i) 𝐴∈(ℎ∶𝐶₀) if and only if (38) holds and lim_𝑛→∞𝑎_𝑛𝑘=0. (41)

Lemma 12. (i) 𝐴∈(ℎ∶ℎ) if and only if (41) holds and ^∞_𝑛=1𝑛𝑎_𝑛𝑘−𝑎_𝑛+1,𝑘𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑠,(𝑘=1,2,3,…),sup_𝑘1𝑘^∞_𝑛=1𝑛^𝑘_𝑣=1𝑎_𝑛𝑣−𝑎_𝑛+1,𝑣<∞. (42)

Theorem 13. We define the set 𝑑₁=𝑎=𝑎_𝑘∈𝑤∶sup_{𝐾∈𝐹𝑘𝑛∈𝐾}1𝑘𝑓_𝑘𝑢_𝑘Δ^𝑟𝑥_𝑘𝑎_𝑛^𝑝𝑘<∞. (43) Then, [ℎ_𝑝(ℱ,𝑢,Δ^𝑟)]^𝛼=𝑑₁.

Proof. Let us take any 𝑎=(𝑎_𝑘)∈𝑤. We define the matrix 𝐷=𝑑_𝑛𝑘 by 𝑑_𝑛𝑘=1𝑘𝑓_𝑘𝑢_𝑘Δ^𝑟𝑎_𝑛,𝑘≥𝑛;0,𝑘<𝑛 (44) for all, 𝑘, 𝑛∈ℕ.
Consider the equation 1𝑘𝑓_𝑘𝑢_𝑘Δ^𝑟𝑎_𝑛𝑥_𝑛=^∞_𝑗=𝑛1𝑗𝑓_𝑗𝑢_𝑗Δ^𝑟𝑎_𝑛𝑦_𝑗=𝐷𝑦_𝑛(𝑛∈ℕ). (45) It follows from (45) with Lemma 8(ii) that 𝑎𝑥=(𝑎_𝑛𝑥_𝑛)∈𝑙₁ whenever 𝑥=(𝑥_𝑘)∈ℎ_𝑝(ℱ,𝑢,Δ^𝑟) if and only if 𝐷𝑦∈𝑙₁, whenever 𝑦=(𝑦_𝑘)∈𝑙_𝑝. This means that 𝑎=(𝑎_𝑛)∈[ℎ_𝑝(ℱ,𝑢,Δ^𝑟)]^𝛼, whenever 𝑥=(𝑥_𝑘)∈ℎ_𝑝(ℱ,𝑢,Δ^𝑟) if and only if 𝐷∈(ℎ_𝑝(ℱ,𝑢,Δ^𝑟)∶𝑙₁). This gives the result that [ℎ_𝑝(ℱ,𝑢,Δ^𝑟)]^𝛼=𝑑₁.

Theorem 14. Let 1<𝑝<∞. Then [ℎ_𝑝(ℱ,𝑢,Δ^𝑟)]^𝛽=𝑑₂, where 𝑑₂=𝑎=𝑎_𝑘∈𝑤∶sup_𝑛∈ℕ𝑛^−1𝑞×_𝑘𝑓_𝑘𝑢_𝑘Δ^𝑟𝑥_𝑘^𝑛_𝑗=𝑘𝑎_𝑗^𝑞<∞. (46)

Proof. Consider the equation ^𝑛_𝑘=11𝑘𝑓_𝑘𝑢_𝑘Δ^𝑟𝑎_𝑘𝑥_𝑘=^𝑛_𝑘=11𝑘𝑓_𝑘𝑢_𝑘Δ^𝑟𝑎_𝑘^𝑘_𝑗=1𝑦_𝑗𝑗=^𝑛_𝑘=1^𝑘_𝑗=11𝑘𝑓_𝑗𝑢_𝑗Δ^𝑟𝑎_𝑗𝑦_𝑘=𝐵𝑦_𝑛(𝑛∈ℕ), (47) where 𝐵=(𝑏_𝑛𝑘) are defined by 𝑏_𝑛𝑘^𝑘_𝑗=11𝑘𝑓_𝑗𝑢_𝑗Δ^𝑟𝑎_𝑗,𝑘≥𝑛;0,𝑘<𝑛 (48) for all, 𝑘,𝑛∈ℕ. Thus, we deduce from Lemma 9(ii) with (47) that 𝑎𝑥=(𝑎_𝑘𝑥_𝑘)∈𝑐𝑠 whenever 𝑥=(𝑥_𝑘)∈ℎ_𝑝(ℱ,𝑢,Δ^𝑟) if and only if 𝐵𝑦∈𝑐 whenever 𝑦=(𝑦_𝑘)∈𝑙_𝑝. Thus (𝑎_𝑘)∈𝑐𝑠 and (𝑎_𝑘)𝑖𝑛𝑑₂ by (39) and (40), respectively. Nevertheless, the inclusion 𝑑₂⊂𝑐𝑠 holds, and thus, we have (𝑎_𝑘)∈𝑑₂, whence [ℎ_𝑝(ℱ,𝑢,Δ^𝑟)]^𝛽=𝑑₂.

Lemma 15. Let 𝑋 be FK-space with 𝑋⊃𝜑. Then,(i)𝑋^𝛽⊂𝑋^𝛾⊂𝑋^𝑓; (ii)If 𝑋 has AK, 𝑋^𝛽=𝑋^𝑓;(iii)If 𝑋 has AD, 𝑋^𝛽=𝑋^𝛾.

From Theorem 6, Corollary 7, and Lemma 15, we can write the following corollary.

Corollary 16. (i) [ℎ_𝑝(ℱ,𝑢,Δ^𝑟)]^𝛽=[ℎ_𝑝(ℱ,𝑢,Δ^𝑟)]^𝑓;
(ii) [ℎ_𝑝(ℱ,𝑢,Δ^𝑟)]^𝛽=[ℎ_𝑝(ℱ,𝑢,Δ^𝑟)]^𝛾.

Lemma 17. Let 𝜆 be a sequence space. Then, the following assertions are true:(i)𝜆 is perfect ⇒𝜆 is normal ⇒𝜆 is monotone;(ii)𝜆 is normal ⇒𝜆^𝛼=𝜆^𝛾;(iii)𝜆 is monotone ⇒𝜆^𝛼=𝜆^𝛽.

Combining Theorem 13, Theorem 14, and Lemma 17, we can give the following corollary.

Corollary 18. The space ℎ_𝑝(ℱ,𝑢,Δ^𝑟) is not monotone and so it is neither normal nor perfect.

4. Matrix Transformations

In this section we characterize some matrix transformations on the space ℎ_𝑝(ℱ,𝑢,Δ^𝑟).

Lemma 19. Let 𝜆, 𝜇 be any two sequence spaces, 𝐴 be an infinite matrix, and 𝑈 be a triangle matrix. Then, 𝐴∈(𝜆∶𝜇_𝑈) if and only if 𝑈𝐴∈(𝜆∶𝑈).

If we define 𝑎_𝑛𝑘=𝑛(𝑎_𝑛𝑘−𝑎_𝑛+1,𝑘), then we can give the following corollary from Lemma 19 with 𝑈=𝑀 defined by (16).

Corollary 20. (i) 𝐴∈(𝑙₁∶ℎ) if and only if sup_𝑘𝑛𝑎_𝑛𝑘<∞. (49)
(ii) 𝐴∈(𝑐∶ℎ)=(𝑐₀∶ℎ)=(𝑙_∞∶ℎ) if and only if sup_{𝐾∈𝐹𝑛}sup_𝑘∈𝐾𝑎_𝑛𝑘<∞. (50)

Theorem 21. Suppose that the entries of the infinite matrices 𝐴=(𝑎_𝑛𝑘) and 𝐸=(𝑒_𝑛𝑘) are connected with the relation 𝑒_𝑛𝑘=𝑎_𝑛𝑘 (51) for all 𝑘,𝑛∈ℕ, where 𝑎_𝑛𝑘=∑^∞_𝑗=𝑘(𝑎_𝑛𝑗/𝑗)[𝑓_𝑗(|𝑢_𝑗Δ^𝑟𝑥_𝑗|)] and 𝜇 is any sequence space. Then 𝐴∈(ℎ_𝑝(ℱ,𝑢,Δ^𝑟)∶𝜇) if and only if {𝑎_𝑛𝑘}_𝑘∈ℕ∈[ℎ_𝑝(ℱ,𝑢,Δ^𝑟)]^𝛽, for all 𝑛∈ℕ and 𝐸∈(ℎ∶𝜇).

Proof. Let 𝜇 be any given sequence spaces. Suppose that (51) holds between 𝐴=(𝑎_𝑛𝑘) and 𝐸=(𝑒_𝑛𝑘), and take into account that the spaces ℎ_𝑝(ℱ,𝑢,Δ^𝑟) and ℎ are norm isomorphic. Let 𝐴∈(ℎ_𝑝(ℱ,𝑢,Δ^𝑟)∶𝜇) and take any 𝑦=𝑦_𝑘∈ℎ. Then, 𝐸𝑀 exists and {𝑎_𝑛𝑘}_𝑘∈𝕂∈[ℎ_𝑝(ℱ,𝑢,Δ^𝑟)]^𝛽, which yields that {𝑒_𝑛𝑘}_𝑘∈ℕ for each 𝑛∈ℕ. Hence, 𝐸𝑦 exists, and thus, _𝑘𝑓_𝑘𝑢_𝑘Δ^𝑟𝑒_𝑛𝑘𝑦_𝑘=_𝑘𝑓_𝑘𝑢_𝑘Δ^𝑟𝑎_𝑛𝑘𝑥_𝑘 (52) for all 𝑛∈ℕ. We have 𝐸𝑦=𝐴𝑥 which leads us to the consequence 𝐸∈(ℎ∶𝜇). Conversely, let {𝑎_𝑛𝑘}_𝑘∈𝕂∈𝑑₁ for all 𝑛∈ℕ, and 𝐸∈(ℎ∶𝜇) hold, and take any 𝑥=𝑥_𝑘∈ℎ_𝑝(ℱ,𝑢,Δ^𝑟). Then, 𝐴𝑥 exists. Therefore, we obtain from the equality that _𝑘𝑓_𝑘𝑢_𝑘Δ^𝑟𝑎_𝑛𝑘𝑥_𝑘=^∞_𝑗=𝑘𝑎_𝑛𝑗𝑗_𝑘𝑓_𝑘𝑢_𝑘Δ^𝑟𝑦_𝑘 (53) for all 𝑛∈ℕ. Thus, 𝐴𝑥=𝐸𝑦 and this shows that 𝐴∈(ℎ_𝑝(ℱ,𝑢,Δ^𝑟)∶𝜇).

If we use the Corollary 20 and change the roles of the spaces ℎ_𝑝(ℱ,𝑢,Δ^𝑟) with 𝜇 in Theorem 21, we can give the following theorem.

Theorem 22. Suppose that the entries of the infinite matrices 𝐴=(𝑎_𝑛𝑘) and 𝐴=(𝑎_𝑛𝑘) are connected with the relation 𝑎_𝑛𝑘=𝑛𝑓_𝑘(|𝑢_𝑘Δ^𝑟𝑥_𝑘|)(𝑎_𝑛𝑘−𝑎_𝑛+1,𝑘) for all 𝑘,𝑛∈ℕ and 𝜇 is any sequence space. Then 𝐴∈(𝜇∶ℎ_𝑝(ℱ,𝑢,Δ^𝑟)) if and only if 𝐴∈(𝜇∶ℎ).

Proof. Let 𝑧=𝑧_𝑘∈𝜇 and consider the following equality: ^𝑚_𝑘=0𝑎_𝑛𝑘𝑧_𝑘=^𝑚_𝑘=0𝑛𝑓_𝑘𝑢_𝑘Δ^𝑟𝑎_𝑛𝑘−𝑎_𝑛+1,𝑘𝑧_𝑘,∀𝑚,𝑛∈ℕ (54) which yields that as 𝑚→∞(𝐴𝑧)_𝑛=(𝑀(𝐴𝑧))_𝑛 for all 𝑛∈ℕ. Therefore, one can observe from here that 𝐴𝑧∈ℎ_𝑝(ℱ,𝑢,Δ^𝑟) whenever 𝑧∈𝜇 if and only if 𝐴𝑧∈ℎ whenever 𝑧∈𝜇.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors express their sincere gratitude to the referees for the careful and detailed reading of the paper and for the very helpful suggestions that improved the paper substantially. The authors also gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the GP-IBT Grant Scheme having Project no. GP-IBT/2013/9420100.