Almost Sequence Spaces Derived by the Domain of the Matrix 𝐴^𝑟

Abstract

By using 𝐴^𝑟, we introduce the sequence spaces 𝑎^𝑟_𝑓, 𝑎^𝑟_𝑓0, and 𝑎^𝑟_𝑓𝑠 of normed space and 𝐵𝐾-space and prove that 𝑎^𝑟_𝑓,𝑎^𝑟_𝑓0, and 𝑎^𝑟_𝑓𝑠 are linearly isomorphic to the sequence spaces 𝑓, 𝑓₀, and 𝑓𝑠 , respectively. Further, we give some inclusion relations concerning the spaces 𝑎^𝑟_𝑓, 𝑎^𝑟_𝑓0, and the nonexistence of Schauder basis of the spaces 𝑓𝑠 and 𝑎^𝑟_𝑓𝑠 is shown. Finally, we determine the 𝛽- and 𝛾-duals of the spaces 𝑎^𝑟_𝑓 and 𝑎^𝑟_𝑓𝑠. Furthermore, the characterization of certain matrix classes on new almost convergent sequence and series spaces has exhaustively been examined.

1. Preliminaries, Background and Notation

By 𝑤, we will denote the space of all real or complex valued sequences. Any vector subspace of 𝑤 is called sequence space. We will write ℓ_∞, 𝑐₀, 𝑐, and ℓ_𝑝 for the spaces of all bounded, null, convergent, and absolutely 𝑝-summable sequences, respectively, which are 𝐵𝐾-space with the usual sup-norm defined by ‖𝑥‖_∞=sup_𝑘|𝑥_𝑘| and ‖𝑥‖_ℓ𝑝=(∑_𝑘|𝑥_𝑘|^𝑝)^1/𝑝, for 1<𝑝<∞, where, here and in what follows, the summation without limits runs from 0 to ∞. Further, we will write 𝑏𝑠, 𝑐𝑠 for the spaces of all sequences associated with bounded and convergent series, respectively, which are 𝐵𝐾-spaces with their natural norm [1].

Let 𝜇 and 𝛾 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 𝜇 into 𝛾 and we denote it by writing that 𝐴∶𝜇→𝛾 and if for every sequence 𝑥=(𝑥_𝑘)∈𝜇 the sequence 𝐴𝑥=(𝐴𝑥)_𝑛, the 𝐴-transform of 𝑥 is in 𝛾, where(𝐴𝑥)_𝑛=_𝑘𝑎_𝑛𝑘𝑥_𝑘,(𝑛∈ℕ). (1)

The notation (𝜇∶𝛾) denotes the class of all matrices 𝐴 such that 𝐴∶𝜇→𝛾. Thus, 𝐴∈(𝜇∶𝛾) if and only if the series on the right hand side of (1) converges for each 𝑛∈ℕ and every 𝑥∈𝜇 and we have 𝐴𝑥={(𝐴𝑥)_𝑛}_𝑛∈ℕ∈𝛾 for all 𝑥∈𝜇. The matrix domain 𝜇_𝐴 of an infinite matrix 𝐴 in a sequence space 𝜇 is defined by 𝜇_𝐴=𝑥=𝑥_𝑘∈𝜔∶𝐴𝑥∈𝜇. (2)

The approach constructing a new sequence space by means of the matrix domain of a particular triangle has recently been employed by several authors in many research papers. For example, they introduced the sequence spaces (𝑐)_𝐶1=𝑐̃ in [2], (ℓ_𝑝)_𝐴^𝑟=𝑎^𝑟_𝑝 and (ℓ_∞)_𝐴^𝑟=𝑎^𝑟_∞ in [3], 𝜇_𝐺=𝑍(𝑢,𝑣;𝜇) in [4], (𝑐₀)_Λ=𝑐^𝜆₀ and 𝑐_Λ=𝑐^𝜆 in [5], and (ℓ_𝑝)_𝐸^𝑟=𝑒^𝑟_𝑝 and (ℓ_∞)_𝐸^𝑟=𝑒^𝑟_∞ in [6]. Recently, matrix domains of the generalized difference matrix 𝐵(𝑟,𝑠) and triple band matrix 𝐵(𝑟,𝑠,𝑡) in the sets of almost null and almost convergent sequences have been investigated by Başar and Kirişçi [7] and Sönmez [8], respectively. Later, Kayaduman and Şengönül introduced some almost convergent spaces which are the matrix domains of the Riesz matrix and Cesàro matrix of order 1 in the sets of almost null and almost convergent sequences (see [9, 10]).

We now focus on the sets of almost convergent sequences. A continuous linear functional 𝜙 on ℓ_∞ is called a Banach limit if (i) 𝜙(𝑥)⩾0 for 𝑥=(𝑥_𝑘) and 𝑥_𝑘⩾0 for every 𝑘, (ii) 𝜙(𝑥_𝜎(𝑘))=𝜙(𝑥_𝑘), where 𝜎 is shift operator which is defined on 𝜔 by 𝜎(𝑘)=𝑘+1, and (iii) 𝜙(𝑒)=1, where 𝑒=(1,1,1,…). A sequence 𝑥=(𝑥_𝑘)∈ℓ_∞ is said to be almost convergent to the generalized limit 𝛼 if all Banach limits of 𝑥 are 𝛼 [11] and denoted by 𝑓−lim𝑥=𝛼. In other words, 𝑓−lim𝑥_𝑘=𝛼 uniformly in 𝑛 if and only if lim_𝑚→∞1𝑚+1^𝑚_𝑘=0𝑥_𝑘+𝑛uniformlyin𝑛. (3) The characterization given above was proved by Lorentz in [11]. We denote the sets of all almost convergent sequences 𝑓 and series 𝑓𝑠 by 𝑓=𝑥=𝑥_𝑘∈𝜔∶lim_𝑚→∞𝑡_𝑚𝑛(𝑥)=𝛼uniformlyin𝑛, (4) where 𝑡_𝑚𝑛(𝑥)=^𝑚_𝑘=01𝑥𝑚+1^𝑘+𝑛,𝑡_−1,𝑛𝑥=0,𝑓𝑠=𝑥=_𝑘∈𝜔∶∃𝑙∈ℂ∋lim^𝑚_{𝑚→∞𝑘=0}^𝑛+𝑘_𝑗=0𝑥_𝑗𝑚+1=𝑙uniformlyin𝑛. (5) We know that the inclusions 𝑐⊂𝑓⊂ℓ_∞ strictly hold. Because of these inclusions, norms ‖⋅‖_𝑓 and ‖⋅‖_∞ of the spaces 𝑓 and ℓ_∞ are equivalent. So the sets 𝑓 and 𝑓₀ are BK-spaces with the norm ‖𝑥‖_𝑓=sup_𝑚,𝑛|𝑡_𝑚𝑛(𝑥)|.

The rest of this paper is organized, as follows. We give foreknowledge on the main argument of this study and notations in this section. In Section 2, we introduce the almost convergent sequence and series spaces 𝑎^𝑟_𝑓𝑠 and 𝑎^𝑟_𝑓 which are the matrix domains of the 𝐴^𝑟 matrix in the almost convergent sequence and series spaces 𝑓𝑠 and 𝑓, respectively. In addition, we give some inclusion relations concerning the spaces 𝑎^𝑟_𝑓, 𝑎^𝑟_𝑓0, and the non-existence of Schauder basis of the spaces 𝑓𝑠 and 𝑎^𝑟_𝑓𝑠 is shown to give certain theorems related to behavior of some sequences. In Section 3, we determine the beta- and gamma-duals of the spaces 𝑎^𝑟_𝑓 and 𝑎^𝑟_𝑓𝑠 and characterize the classes (𝛾∶𝑎^𝑟_𝑓), (𝑎^𝑟_𝑓∶𝜇), (𝛿∶𝑎^𝑟_𝑓𝑠) and (𝑎^𝑟_𝑓∶𝜃), where 𝛾∈{𝑐(𝑝),𝑐₀(𝑝),ℓ_∞(𝑝),𝑐𝑠,𝑏𝑠,𝑓𝑠,𝑓,𝑐,ℓ_∞}, 𝜇∈{𝑐𝑠,𝑐,ℓ_∞}, 𝛿∈{𝑐𝑠,𝑓𝑠,𝑏𝑠}, and 𝜃∈{𝑓,𝑐,𝑓𝑠,ℓ_∞}, where 𝑐(𝑝), 𝑐₀(𝑝), and ℓ_∞(𝑝) denote the space of Maddox convergent, null and bounded sequence spaces defined by Maddox [12].

Lemma 1 (see [13]). The set 𝑓𝑠 has no Schauder basis.

2. The Sequence Spaces 𝑎^𝑟_𝑓, 𝑎^𝑟_𝑓0, and 𝑎^𝑟_𝑓𝑠 Derived by the Domain of the Matrix 𝐴^𝑟

In the present section, we introduce the sequence spaces 𝑎^𝑟_𝑓, 𝑎^𝑟_𝑓0, and 𝑎^𝑟_𝑓𝑠 as the set of all sequences such that 𝐴^𝑟-transforms of them are in the spaces 𝑓, 𝑓₀, and 𝑓𝑠, respectively. Further, this section is devoted to examination of the basic topological properties of the sets 𝑎^𝑟_𝑓, 𝑎^𝑟_𝑓0, and 𝑎^𝑟_𝑓𝑠. Recently, Aydın and Başar [14] studied the sequence spaces 𝑎^𝑟_𝑐 and 𝑎^𝑟₀:𝑎^𝑟_𝑐=𝑥=𝑥_𝑘∈𝜔∶lim_𝑛→∞1𝑛+1^𝑛_𝑘=01+𝑟^𝑘𝑥_𝑘exists,𝑎^𝑟₀=𝑥=𝑥_𝑘∈𝜔∶lim_𝑛→∞1𝑛+1^𝑛_𝑘=01+𝑟^𝑘𝑥_𝑘=0, (6) where 𝐴^𝑟 denotes the matrix 𝐴^𝑟=(𝑎^𝑟_𝑛𝑘) defined by 𝑎^𝑟_𝑛𝑘=1+𝑟^𝑘(),0().𝑛+10⩽𝑘⩽𝑛𝑘>𝑛 (7)

Now we introduce the sequence spaces 𝑎^𝑟_𝑓, 𝑎^𝑟_𝑓, and 𝑎^𝑟_𝑓𝑠 as the sets of all sequences such that their 𝐴^𝑟-transforms are in the spaces 𝑓, 𝑓₀, and 𝑓𝑠, respectively; that is, 𝑎^𝑟_𝑓=𝑥=𝑥_𝑘∈𝜔∶∃𝛼∈ℂ∋lim^𝑚_{𝑚→∞𝑘=0}1𝑚+1^𝑘_𝑖=01𝑘+11+𝑟^𝑖𝑥_𝑛+𝑖=𝛼uniformlyin𝑛,𝑎^𝑟_𝑓0=𝑥=𝑥_𝑘∈𝜔∶lim^𝑚_{𝑚→∞𝑘=0}1𝑚+1^𝑘_𝑖=01+𝑟^𝑖𝑥𝑘+1^𝑛+𝑖=0uniformlyin𝑛,𝑎^𝑟_𝑓𝑠=𝑥=𝑥_𝑘∈𝜔∶∃𝛽∈ℂ∋lim^𝑚_{𝑚→∞𝑘=0}^{𝑛+𝑘𝑗}_𝑗=0𝑖=01+𝑟^𝑗𝑥𝑖+1^𝑛+𝑗=𝛽uniformlyin𝑛. (8) We can redefine the spaces 𝑎^𝑟_𝑓𝑠, 𝑎^𝑟_𝑓, and 𝑎^𝑟_𝑓0 by the notation of (2): 𝑎^𝑟_𝑓0=𝑓_0𝐴𝑟,𝑎^𝑟_𝑓=𝑓_𝐴𝑟,𝑎^𝑟_𝑓𝑠=𝑓𝑠_𝐴𝑟. (9) It is known by Başar [15] that the method is regular for 0<𝑟<1. We assume unless stated otherwise that 0<𝑟<1.

Define the sequence 𝑦=(𝑦_𝑘), which will be frequently used, as the 𝐴^𝑟-transform of a sequence 𝑥=(𝑥_𝑘); that is, 𝑦_𝑘(𝑟)=^𝑘_𝑖=01+𝑟^𝑖𝑥𝑘+1^𝑖(𝑘∈ℕ). (10)

Theorem 2. The spaces 𝑎^𝑟_𝑓 and 𝑎^𝑟_𝑓𝑠 have no Schauder basis.

Proof. Since it is known that the matrix domain 𝜇_𝐴 of a normed sequence space 𝜇 has a basis if and only if 𝜇 has a basis whenever 𝐴=(𝑎_𝑛𝑘) is a triangle [16, Remark 2.4] and the space 𝑓 has no Schauder basis by [7, Corollary 3.3], we have that 𝑎^𝑟_𝑓 has no Schauder basis. Since the set 𝑓𝑠 has no basis in Lemma 1, 𝑎^𝑟_𝑓𝑠 has no Schauder basis.

Theorem 3. The following statements hold.(i)The sets 𝑎^𝑟_𝑓 and 𝑎^𝑟_𝑓0 are linear spaces with the coordinatewise addition and scalar multiplication which are 𝐵𝐾-spaces with the norm‖𝑥‖_𝑎^𝑟_𝑓=sup_𝑚^𝑚_𝑘=01𝑚+1^𝑘_𝑖=01+𝑟^𝑖𝑥𝑘+1^𝑖+𝑛. (11)(ii)The set 𝑎^𝑟_𝑓𝑠 is a linear space with the coordinatewise addition and scalar multiplication which is a 𝐵𝐾-space with the norm‖𝑥‖_𝑎^𝑟_𝑓𝑠=sup_𝑚^𝑚_𝑘=01𝑚+1^{𝑘+𝑛𝑗}_𝑗=0𝑖=01+𝑟^𝑖𝑥𝑗+1^𝑖. (12)

Proof. Since the second part can be similarly proved, we only focus on the first part. Since the sequence spaces 𝑓 and 𝑓₀ endowed with the norm ‖⋅‖_∞ are 𝐵𝐾-spaces (see [1, Example 7.3.2(b)]) and the matrix 𝐴^𝑟=(𝑎^𝑟_𝑛𝑘) is normal, Theorem 4.3.2 of Wilansky [17, p.61] gives the fact that the spaces 𝑎^𝑟_𝑓 and 𝑎^𝑟_𝑓0 are 𝐵𝐾-spaces with the norm in (11).

Now, we may give the following theorem concerning the isomorphism between our spaces and the sets 𝑓, 𝑓₀, and 𝑓𝑠.

Theorem 4. The sequence spaces 𝑎^𝑟_𝑓, 𝑎^𝑟_𝑓0, and 𝑎^𝑟_𝑓𝑠 are linearly isomorphic to the sequence spaces 𝑓, 𝑓₀, and 𝑓𝑠, respectively; that is, 𝑎^𝑟_𝑓≅𝑓, 𝑎^𝑟_𝑓0≅𝑓₀, and 𝑎^𝑟_𝑓𝑠≅𝑓𝑠.

Proof. To prove the fact that 𝑎^𝑟_𝑓≅𝑓, we should show the existence of a linear bijection between the spaces 𝑎^𝑟_𝑓 and 𝑓. Consider the transformation 𝑇 defined with the notation of (2) from 𝑎^𝑟_𝑓 to 𝑓 by 𝑥↦𝑦=𝑇𝑥=𝐴^𝑟𝑥. The linearity of 𝑇 is clear. Further, it is clear that 𝑥=𝜃 whenever 𝑇𝑥=𝜃, and hence, 𝑇 is injective.
Let 𝑦=(𝑦_𝑘)∈𝑎^𝑟_𝑓, and define the sequence 𝑥=(𝑥_𝑘(𝑟)) by 𝑥_𝑘=11+𝑟^𝑘(𝑘+1)𝑦_𝑘−𝑘𝑦_𝑘−1foreach𝑘∈ℕ, (13) whence 𝑓_𝐴𝑟−lim𝑥=lim^𝑚_{𝑚→∞𝑘=0}1×𝑚+1_𝑘𝑖=01+𝑟^𝑖𝑥_𝑖+𝑛1+𝑘uniformlyin𝑛=lim^𝑚_{𝑚→∞𝑘=0}1×𝑚+1_𝑘𝑖=01+𝑟^𝑖1/1+𝑟^𝑖𝑦_𝑖+𝑛(𝑘+1)−𝑦_{𝑖+𝑛−1}𝑘1+𝑘uniformlyin𝑛=lim_𝑚→∞1𝑚+1^𝑚_𝑘=0𝑦_𝑘+𝑛uniformlyin𝑛=𝑓−lim𝑦 (14) which implies that 𝑥∈𝑎^𝑟_𝑓. As a result, 𝑇 is surjective. Hence, 𝑇 is a linear bijection which implies that the spaces 𝑎^𝑟_𝑓 and 𝑓 are linearly isomorphic, as desired. Similarly, the isomorphisms 𝑎^𝑟_𝑓0≅𝑓₀ and 𝑎^𝑟_𝑓𝑠≅𝑓𝑠 can be proved.

Theorem 5. The inclusion 𝑓⊂𝑎^𝑟_𝑓 strictly holds.

Proof. Let 𝑥=(𝑥_𝑘)∈𝑐. Since 𝑐⊂𝑓, 𝑥∈𝑓. Because 𝐴^𝑟 is regular for 0<𝑟<1, 𝐴^𝑟𝑥∈𝑐. Therefore, since lim𝐴^𝑟𝑥=𝑓−lim𝐴^𝑟𝑥, we see that 𝑥∈𝑎^𝑟_𝑓. So we have that the inclusion 𝑓⊂𝑎^𝑟_𝑓 holds. Further, consider the sequence 𝑡=(𝑡_𝑘(𝑟)) defined by 𝑡_𝑘(𝑟)=(2𝑘+1)/(1+𝑟^𝑘)(−1)^𝑘∀𝑘∈ℕ. Then, since 𝐴^𝑟t=(−1)^𝑛∈𝑓, 𝑥∈𝑎^𝑟_𝑓. One can easily see that 𝑡∉𝑓. Thus, 𝑡∈𝑎^𝑟_𝑓⧵𝑓, and this completes the proof.

Theorem 6. The sequence spaces 𝑎^𝑟_𝑓 and ℓ_∞ overlap, but neither of them contains the other.

Proof. Let us consider the sequence 𝑢=(𝑢_𝑘(𝑟)) defined by 𝑢_𝑘(𝑟)=1/(1+𝑟^𝑘) for all ℕ. Then, since 𝐴^𝑟𝑢=𝑒∈𝑓, 𝑢∈𝑎^𝑟_𝑓. It is clear that 𝑢∈ℓ_∞. This means that the sequence spaces 𝑎^𝑟_𝑓 and ℓ_∞ are not disjoint. Now, we show that the sequence space 𝑎^𝑟_𝑓 and ℓ_∞ do not include each other. Let us consider the sequence 𝑡=(𝑡_𝑘(𝑟)) defined as in proof of Theorem 5 above and 𝑧=(𝑧_𝑘(𝑟)) = (0,…,0,1/(1+𝑟¹⁰¹),…,1/(1+𝑟¹¹⁰),0,…,0,1/(1+𝑟²¹¹),…,1/(1+𝑟²³¹),0,…,0,…) where the blocks of 0’s are increasing by factors of 100 and the blocks of 1/(1+𝑟^𝑘)’s are increasing by factors of 10. Then, since 𝐴^𝑟𝑡=(−1)^𝑛∈𝑓, 𝑡∈𝑎^𝑟_𝑓, but 𝑡∉ℓ_∞. Therefore, 𝑡∈𝑎^𝑟_𝑓⧵ℓ_∞. Also, the sequence 𝑧∉𝑎^𝑟_𝑓 since 𝐴^𝑟𝑧=(0,…,0,1,…,1,0,…,0,1,…,1,0,…,0,…)∉𝑓 where the blocks of 0’s are increasing by factors of 100 and the blocks of 1’s are increasing by factors of 10, but 𝑧 is bounded. This means that 𝑧∈ℓ_∞⧵𝑎^𝑟_𝑓. Hence, the sequence spaces 𝑎^𝑟_𝑓 and ℓ_∞ overlap, but neither of them contains the other. This completes the proof.

Theorem 7. Let the spaces 𝑎^𝑟_𝑓0, 𝑎^𝑟_𝑐, and 𝑎^𝑟_𝑓 be given. Then, (i)𝑎^𝑟_𝑓0⊂𝑎^𝑟_𝑓 strictly hold;(ii)𝑎^𝑟_𝑐⊂𝑎^𝑟_𝑓 strictly hold.

Proof. (i) Let 𝑥=(𝑥_𝑘)∈𝑎^𝑟_𝑓0 which means that 𝐴^𝑟𝑥∈𝑓₀. Since 𝑓₀⊂𝑓, 𝐴^𝑟𝑥∈𝑓. This implies that 𝑥∈𝑎^𝑟_𝑓. Thus, we have 𝑎^𝑟_𝑓0⊂𝑎^𝑟_𝑓.
Now, we show that this inclusion is strict. Let us consider the sequence 𝑢=(𝑢_𝑘(𝑟)) defined as in proof of Theorem 6 for all 𝑘∈ℕ. Consider the following: 𝑓_𝐴𝑟−lim𝑢=lim^𝑚_{𝑚→∞𝑘=0}1𝑚+1^𝑘_𝑖=01+𝑟^𝑖𝑢𝑘+1^𝑖+𝑛=lim^𝑚_{𝑚→∞𝑘=0}1()𝑚+1=𝑒=1,1,… (15) which means that 𝑢∈𝑎^𝑟_𝑓⧵𝑎^𝑟_𝑓0; that is to say, the inclusion is strict.(ii) Let 𝑥=(𝑥_𝑘)∈𝑎^𝑟_𝑐 which means that 𝐴^𝑟𝑥∈𝑐. Since 𝑐⊂𝑓, 𝐴^𝑟𝑥∈𝑓. This implies that 𝑥∈𝑎^𝑟_𝑓. Thus, we have 𝑎^𝑟_𝑐⊂𝑎^𝑟_𝑓. Furthermore, let us consider the sequence 𝑡={𝑡_𝑘(𝑟)} defined as in proof of Theorem 5 for all 𝑘∈ℕ. Then, since 𝐴^𝑟𝑡=(−1)^𝑛∈𝑓⧵𝑐, 𝑡∈𝑎^𝑟_𝑓⧵𝑎^𝑟_𝑐. This completes the proof.

3. Certain Matrix Mappings on the Sets 𝑎^𝑟_𝑓, 𝑎^𝑟_𝑓𝑠 and Some Duals

In this section, we will characterize some matrix transformations between the spaces of 𝐴^𝑟 almost convergent sequence and almost convergent series in addition to paranormed and classical sequence spaces after giving 𝛽- and 𝛾-duals of the spaces 𝑎^𝑟_𝑓𝑠 and 𝑎^𝑟_𝑓. We start with the definition of the beta- and gamma-duals.

If 𝑥 and 𝑦 are sequences and 𝑋 and 𝑌 are subsets of 𝜔, then we write 𝑥⋅𝑦=(𝑥_𝑘𝑦_𝑘)^∞_𝑘=0, 𝑥⁻¹∗𝑌={𝑎∈𝜔∶𝑎⋅𝑥∈𝑌} and 𝑀(𝑋,𝑌)=_𝑥∈𝑋𝑥⁻¹∗𝑌={𝑎∶𝑎⋅𝑥∈𝑌∀𝑥∈𝑋} (16) for the multiplier space of 𝑋 and 𝑌. One can easily observe for a sequence space 𝑍 with 𝑌⊂𝑍 and 𝑍⊂𝑋 that inclusions 𝑀(𝑋,𝑌)⊂𝑀(𝑋,𝑍) and 𝑀(𝑋,𝑌)⊂𝑀(𝑍,𝑌) hold, respectively. The 𝛼-,𝛽-, and 𝛾-duals of a sequence space, which are, respectively, denoted by 𝑋^𝛼,𝑋^𝛽, and 𝑋^𝛾, are defined by 𝑋^𝛼=𝑀𝑋,ℓ₁,𝑋^𝛽=𝑀(𝑋,𝑐𝑠),𝑋^𝛾=𝑀(𝑋,𝑏𝑠). (17) It is obvious that 𝑋^𝛼⊂𝑋^𝛽⊂𝑋^𝛾. Also, it can easily be seen that the inclusions 𝑋^𝛼⊂𝑌^𝛼, 𝑋^𝛽⊂𝑌^𝛽, and 𝑋^𝛾⊂𝑌^𝛾 hold whenever 𝑌⊂𝑋.

Lemma 8 (see [18]). 𝐴=(𝑎_𝑛𝑘)∈(𝑓∶ℓ_∞) if and only if sup_𝑛𝑘𝑎_𝑛𝑘<∞. (18)

Lemma 9 (see [18]). 𝐴=(𝑎_𝑛𝑘)∈(𝑓∶𝑐) if and only if (18) holds and there are 𝛼,𝛼_𝑘∈ℂ such that lim_𝑛→∞𝑎_𝑛𝑘=𝛼_𝑘∀k∈ℕ,lim_{𝑛→∞𝑘}𝑎_𝑛𝑘=𝛼,lim_{𝑛→∞𝑘}Δ𝑎_𝑛𝑘−𝛼_𝑘=0. (19)(20)(21)

Theorem 10. Define the sets 𝑡^𝑟₁ and 𝑡^𝑟₂ by 𝑡^𝑟₁=𝑎=𝑎_𝑘∈𝜔∶_𝑘Δ𝑎_𝑘1+𝑟^𝑘(,𝑡𝑘+1)<∞_𝑟2=𝑎=𝑎_𝑘∈𝜔∶sup_𝑘𝑎_𝑘(𝑘+1)1+𝑟^𝑘<∞, (22) where Δ(𝑎_𝑘/(1+𝑟^𝑘))=𝑎_𝑘/(1+𝑟^𝑘)−𝑎_𝑘+1/(1+𝑟^𝑘+1) for all 𝑘∈ℕ. Then (𝑎^𝑟_𝑓)^𝛾=𝑡^𝑟₁∩𝑡^𝑟₂.

Proof. Take any sequence 𝑎=(𝑎_𝑘)∈𝜔, and consider the following equality: ^𝑛_𝑘=0𝑎_𝑘𝑥_𝑘=^𝑛_𝑘=0𝑎_𝑘^𝑘_{𝑖=𝑘−1}(−1)^𝑘−𝑗𝑖+11+𝑟^𝑖𝑦_𝑖=^𝑛−1_𝑘=0Δ𝑎_𝑘1+𝑟^𝑘(𝑘+1)𝑦_𝑘+𝑛+11+𝑟^𝑛𝑎_𝑛𝑦_𝑛=𝑇𝑦_𝑛, (23) where 𝑇={𝑡^𝑟_𝑛𝑘} is 𝑡^𝑟_𝑛𝑘=Δ𝑎_𝑘1+𝑟^𝑘(𝑘+1)(0⩽𝑘⩽𝑛−1)𝑛+11+𝑟^𝑛𝑎_𝑛()0()𝑘=𝑛𝑘>𝑛 (24) for all 𝑘,𝑛∈ℕ. Thus, we deduce from (23) that 𝑎𝑥=(𝑎_𝑘𝑥_𝑘)∈𝑏𝑠 whenever 𝑥=(𝑥_𝑘)∈𝑎^𝑟_𝑓 if and only if 𝑇𝑦∈ℓ_∞ whenever 𝑦=(𝑦_𝑘)∈𝑓 where 𝑇={𝑡^𝑟_𝑛𝑘} is defined in (24). Therefore, with the help of Lemma 8,  (𝑎^𝑟_𝑓)^𝛾=𝑡^𝑟₁∩𝑡^𝑟₂.

Theorem 11. The 𝛽-dual of the space 𝑎^𝑟_𝑓 is the intersection of the sets 𝑡^𝑟₃=𝑎=𝑎_𝑘∈𝜔∶lim_{𝑛→∞𝑘}Δ𝑡^𝑟_𝑛𝑘−𝛼_𝑘,𝑡=0_𝑟4=𝑎=𝑎_𝑘∈𝜔∶𝑘+11+𝑟^𝑘𝑎_𝑘∈𝑐𝑠, (25) where lim_𝑛→∞𝑡^𝑟_𝑛𝑘=𝛼_𝑘 for all 𝑘∈ℕ. Then, (𝑎^𝑟_𝑓)^𝛽=𝑡^𝑟₃∩𝑡^𝑟₄.

Proof. Let us take any sequence 𝑎∈𝜔. By (23), 𝑎𝑥=(𝑎_𝑘𝑥_𝑘)∈𝑐𝑠 whenever 𝑥=(𝑥_𝑘)∈𝑎^𝑟_𝑓 if and only if 𝑇𝑦∈𝑐 whenever 𝑦=(𝑦_𝑘)∈𝑓. It is obvious that the columns of that matrix 𝑇 in 𝑐 where 𝑇={𝑡^𝑟_𝑛𝑘} defined in (24), we derive the consequence by Lemma 9 that (𝑎^𝑟_𝑓)^𝛽=𝑡^𝑟₃∩𝑡^𝑟₄.

Theorem 12. The 𝛾-dual of the space 𝑎^𝑟_𝑓𝑠 is the intersection of the sets 𝑐^𝑟₁=𝑎=𝑎_𝑘∈𝜔∶_𝑘ΔΔ𝑎_𝑘1+𝑟^𝑘()+𝑎𝑘+1_𝑘1+𝑟^𝑘(,𝑐𝑘+1)<∞_𝑟2=𝑎=𝑎_𝑘𝑎∈𝜔∶^𝑘(𝑘+1)1+𝑟^𝑘∈𝑐₀. (26) In other words, we have (𝑎^𝑟_𝑓𝑠)^𝛾=𝑐^𝑟₁∩𝑐^𝑟₂.

Proof. We obtain from (23) that 𝑎𝑥=(𝑎_𝑘𝑥_𝑘)∈𝑏𝑠 whenever 𝑥=(𝑥_𝑘)∈𝑎^𝑟_𝑓𝑠 if and only if 𝑇𝑦∈ℓ_∞ whenever 𝑦=(𝑦_𝑘)∈𝑓𝑠, where 𝑇={𝑡^𝑟_𝑛𝑘} is defined in (24). Therefore, by Lemma 19(viii), (𝑎^𝑟_𝑓𝑠)^𝛾=𝑐^𝑟₁∩𝑐^𝑟₂.

Theorem 13. Define the set 𝑐^𝑟₃ by 𝑐^𝑟₃=𝑎=𝑎_𝑘∈𝜔∶lim_{𝑛→∞𝑘}Δ²𝑡^𝑟_𝑛𝑘𝑒𝑥𝑖𝑠𝑡𝑠. (27) Then, (𝑎^𝑟_𝑓𝑠)^𝛽=𝑐^𝑟₁∩𝑐^𝑟₂∩𝑐^𝑟₃.

Proof. This may be obtained in the same way as mentioned in the proof of Theorem 12 with Lemma 19(viii) instead of Lemma 19(vii). So we omit details.

For the sake of brevity, the following notations will be used: 𝑎()=1𝑛,𝑘,𝑚𝑚+1^𝑚_𝑖=0𝑎_{𝑛+𝑖,𝑘},𝑎(𝑛,𝑘)=^𝑛_𝑖=0𝑎_𝑖𝑘,𝑎_𝑛𝑘𝑎=Δ^𝑛𝑘1+𝑟^𝑘()=𝑎𝑘+1^𝑛𝑘1+𝑟^𝑘−𝑎_𝑛,𝑘+11+𝑟^𝑘+1(𝑘+1),Δ𝑎_𝑛𝑘=𝑎_𝑛,𝑘−𝑎_𝑛,𝑘+1,𝑎_𝑛𝑘=^𝑛_𝑗=01+𝑟^𝑗𝑒_𝑗𝑘,𝑛+1 (28) for all 𝑘,𝑛∈ℕ. Assume that the infinite matrices 𝐴=(𝑎_𝑛𝑘) and 𝐵=(𝑏_𝑛𝑘) map the sequences 𝑥=(𝑥_𝑘) and 𝑦=(𝑦_𝑘) which are connected with relation (10) to the sequences 𝑢=(𝑢_𝑛) and 𝑣=(𝑣_𝑛), respectively; that is, 𝑢_𝑛=(𝐴𝑥)_𝑛=_𝑘𝑎_𝑛𝑘𝑥_𝑘∀𝑛∈ℕ,𝑣_𝑛=𝐵𝑦_𝑛=_𝑘𝑏_𝑛𝑘𝑦_𝑘∀𝑛∈ℕ. (29)(30) One can easily conclude here that the method 𝐴 is directly applied to the terms of the sequence 𝑥=(𝑥_𝑘), while the method 𝐵 is applied to the 𝐴^𝑟-transform of the sequence 𝑥=(𝑥_𝑘). So the methods 𝐴 and 𝐵 are essentially different.

Now, suppose that the matrix product 𝐵𝐴^𝑟 exists which is a much weaker assumption than the conditions on the matrix 𝐵 belonging to any matrix class, in general. It is not difficult to see that the sequence in (30) reduces to the sequence in (29) under the application of formal summation by parts. This leads us to the fact that 𝐵𝐴^𝑟 exists and is equal to 𝐴 and (𝐵𝐴^𝑟)𝑥=𝐵(𝐴^𝑟𝑥) formally holds if one side exists. This statement is equivalent to the following relation between the entries of the matrices 𝐴=(𝑎_𝑛𝑘) and 𝐵=(𝑏_𝑛𝑘) which are connected with the relation 𝑎_𝑛𝑘=𝑏_𝑛𝑘𝑎=Δ^𝑛𝑘1+𝑟^𝑘(𝑘+1)or𝑎_𝑛𝑘=1+𝑟^𝑘∞_𝑗=𝑘𝑏_𝑛𝑗1+𝑗∀𝑘,𝑛∈ℕ. (31)

Note that the methods 𝐴 and 𝐵 are not necessarily equivalent since the order of summation may not be reversed. We now give the following fundamental theorem connected with the matrix mappings on/into the almost convergent spaces 𝑎^𝑟_𝑓 and 𝑎^𝑟_𝑓s.

Theorem 14. Suppose that the entries of the infinite matrices 𝐴=(𝑎_𝑛𝑘) and 𝐵=(𝑏_𝑛𝑘) are connected with relation (31) for all 𝑘,𝑛∈ℕ, and let 𝜆 be any given sequence space. Then, 𝐴∈(𝑎^𝑟_𝑓∶𝜆) if and only if 𝐵∈𝑓∶𝜆,𝑛+11+𝑟^𝑘𝑎_{𝑛𝑘𝑘∈ℕ}∈𝑐₀. (32)

Proof. Suppose that 𝐴=(𝑎_𝑛𝑘) and 𝐵=(𝑏_𝑛𝑘) are connected with the relation (31), and let 𝜆 be any given sequence space, and keep in mind that the spaces 𝑎^𝑟_𝑓 and 𝑓 are norm isomorphic.
Let 𝐴∈(𝑎^𝑟_𝑓∶𝜆), and take any sequence 𝑥∈𝑎^𝑟_𝑓, and keep in mind that 𝑦=𝐴^𝑟𝑥. Then, (𝑎_𝑛𝑘)_𝑘∈ℕ∈(𝑎^𝑟_𝑓)^𝛽; that is, (32) holds for all 𝑛∈ℕ and 𝐵𝐴^𝑟 exists which implies that (𝑏_𝑛𝑘)_𝑘∈ℕ∈ℓ₁=𝑓^𝛽 for each 𝑛∈ℕ. Thus, 𝐵𝑦 exists for all 𝑦∈𝑓, and thus, we have 𝑚→∞ in the equality ^𝑚_𝑘=0𝑏_𝑛𝑘𝑦_𝑘=^𝑚𝑚_{𝑘=0𝑗=𝑘}1+𝑟^𝑘𝑏_𝑛𝑗𝑥1+𝑗^𝑘 (33) for all 𝑚,𝑛∈ℕ, and we have (31) 𝐵𝑦=𝐴𝑥 which means that 𝐵∈(𝑓∶𝜆). On the other hand, assume that (32) holds and 𝐵∈(𝑓∶𝜆). Then, we have (𝑏_𝑛𝑘)_𝑘∈ℕ∈ℓ₁ for all 𝑛∈ℕ which gives together with (𝜐_𝑛𝑘)_𝑘∈ℕ∈(𝑎^𝑟_𝑓)^𝛽 for each 𝑛∈ℕ that 𝐴𝑥 exists. Then, we obtain from the equality ^𝑚_𝑘=0𝑎_𝑛𝑘𝑥_𝑘=^𝑚−1_𝑘=0Δ𝑎_𝑛𝑘1+𝑟^𝑘(𝑘+1)𝑦_𝑘+𝑚+11+𝑟^𝑚𝑎_𝑛𝑚𝑦_𝑚=^𝑚_𝑘=0𝑏_𝑛𝑘𝑦_𝑘 (34) for all 𝑚,𝑛∈ℕ, as 𝑚→∞, that 𝐴𝑥=𝐵𝑦, and this shows that 𝐴∈(𝑎^𝑟_𝑓∶𝜆).

Theorem 15. Suppose that the entries of the infinite matrices 𝐸=(𝑒_𝑛𝑘) and 𝐹=(𝑓_𝑛𝑘) are connected with the relation 𝑓_𝑛𝑘=𝑎_𝑛𝑘 (35) for all 𝑚,𝑛∈ℕ and 𝜆 is any given sequence space. Then, 𝐸∈(𝜆∶𝑎^𝑟_𝑓) if and only if 𝐹∈(𝜆∶𝑓).

Proof. Let 𝑥=(𝑥_𝑘)∈𝜆, and consider the following equality: ^𝑛𝑚_𝑗=0𝑘=01+𝑟^𝑗𝑒𝑛+1^𝑗𝑘𝑥_𝑘=^𝑚_𝑘=0𝑓_𝑛𝑘𝑥_𝑘 (36) for all 𝑘,𝑚,𝑛∈ℕ, which yields as 𝑚→∞ that 𝐸𝑥∈𝑎^𝑟_𝑓 whenever 𝑥∈𝜆 if and only if 𝐹𝑥∈𝑓 whenever 𝑥∈𝜆. This step completes the proof.

Theorem 16. Let 𝜆 be any given sequence space, and the matrices 𝐴=(𝑎_𝑛𝑘) and 𝐵=(𝑏_𝑛𝑘) are connected with the relation (31). Then, 𝐴∈(𝑎^𝑟_𝑓𝑠∶𝜆) if and only if 𝐵∈(𝑓𝑠∶𝜆) and (𝑎_𝑛𝑘)_𝑘∈ℕ∈(𝑎^𝑟_𝑓𝑠)^𝛽 for all 𝑛∈ℕ.

Proof. The proof is based on the proof of Theorem 14.

Theorem 17. Let 𝜆 be any given sequence space, and the elements of the infinite matrices 𝐸=(𝑒_𝑛𝑘) and 𝐹=(𝑓_𝑛𝑘) are connected with relation (35). Then, 𝐸=(𝑒_𝑛𝑘)∈(𝜆∶𝑎^𝑟_𝑓𝑠) if and only if 𝐹∈(𝜆∶𝑓𝑠).

Proof. The proof is based on the proof of Theorem 15.

By Theorems 14, 15, 16, and 17, we have quite a few outcomes depending on the choice of the space 𝜆 to characterize certain matrix mappings. Hence, by the help of these theorems, the necessary and sufficient conditions for the classes (𝑎^𝑟_𝑓∶𝜆), (𝜆∶𝑎^𝑟_𝑓), (𝑎^𝑟_𝑓𝑠∶𝜆) and (𝜆∶𝑎^𝑟_𝑓𝑠) may be derived by replacing the entries of 𝐴 and 𝐵 by those of𝐵=𝐴(𝐴^𝑟)⁻¹, and 𝐹=𝐴^𝑟𝐸, respectively, where the necessary and sufficient conditions on the matrices 𝐸 and 𝐹 are read from the concerning results in the existing literature

Lemma 18. Let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix. Then, the following statements hold:(i)𝐴∈(𝑐₀(𝑝)∶𝑓)   if and only if∃𝑁>1∋sup_{𝑚∈ℕ𝑘}|𝑎(𝑛,𝑘,𝑚)|𝑁^{−1/𝑝𝑘}<∞,∀𝑛∈ℕ,∃𝛼_𝑘∈ℂ∀𝑘∈ℕ∋lim_𝑚→∞𝑎(𝑛,𝑘,𝑚)=𝛼_𝑘𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦𝑖𝑛𝑛; (37)(ii)𝐴∈(𝑐(𝑝)∶𝑓)   if and only if (37) and∃𝛼∈ℂ∋lim_{𝑚→∞𝑘}𝑎(𝑛,𝑘,𝑚)=𝛼𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦𝑖𝑛𝑛; (38)(iii)𝐴∈(ℓ_∞(𝑝)∶𝑓)    if and only if (37) and∃𝑁>1∋lim_{𝑚→∞𝑘}𝑎(𝑛,𝑘,𝑚)−𝛼_𝑘𝑁^1/𝑝𝑘=0𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦𝑖𝑛𝑛. (39)

Lemma 19. Let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix. Then, the following statements hold:(i)(Duran, [19]) 𝐴∈(ℓ_∞∶𝑓) if and only if (18) holds and𝑓−lim𝑎_𝑛𝑘=𝛼_𝑘𝑒𝑥𝑖𝑠𝑡𝑠𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑓𝑖𝑥𝑒𝑑𝑘,lim_{𝑚→∞𝑘}𝑎(𝑛,𝑘,𝑚)−𝛼_𝑘=0𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦𝑖𝑛𝑛; (40)(41)(ii)(King, [20]) 𝐴∈(𝑐∶𝑓) if and only if (18), (40) hold and𝑓−lim_𝑘𝑎_𝑛𝑘=𝛼; (42)(iii)(Başar and Çolak, [21]) 𝐴∈(𝑐𝑠∶𝑓) if and only if (40) holds andsup_{𝑛∈ℕ𝑘}Δ𝑎_𝑛𝑘<∞; (43)(iv)(Başar and Çolak, [21]) 𝐴∈(𝑏𝑠∶𝑓) if and only if (40), (43) hold andlim_𝑘𝑎_𝑛𝑘=0𝑒𝑥𝑖𝑠𝑡𝑠𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑓𝑖𝑥𝑒𝑑𝑛,lim_{𝑞→∞𝑘}1𝑞+1^𝑞_𝑖=0Δ𝑎(𝑛+𝑖,𝑘)−𝛼_𝑘=0𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦𝑖𝑛𝑛; (44)(45)(v)(Duran, [19]) 𝐴∈(𝑓∶𝑓) if and only if (18), (40), and (42) hold andlim_{𝑚→∞𝑘}Δ𝑎(𝑛,𝑘,𝑚)−𝛼_𝑘=0𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦𝑖𝑛𝑛; (46)(vi)(Başar, [22]) 𝐴∈(𝑓𝑠∶𝑓) if and only if (40), (44), (46), and (45) hold;(vii)(Öztürk, [23]) 𝐴∈(𝑓𝑠∶𝑐) if and only if (19), (43), and (44) hold andlim_{𝑛→∞𝑘}Δ²𝑎_𝑛𝑘=𝛼; (47)(viii)𝐴∈(𝑓𝑠∶ℓ_∞) if and only if (43) and (44) hold;(ix)(Başar and Solak, [24]) 𝐴∈(𝑏𝑠∶𝑓𝑠) if and only if (44), (45) hold andsup_{𝑛∈ℕ𝑘}|(()Δ𝑎𝑛,𝑘)|<∞,𝑓−lim𝑎𝑛,𝑘=𝛼_𝑘𝑒𝑥𝑖𝑠𝑡𝑠𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑓𝑖𝑥𝑒𝑑𝑘; (48)(x)(Başar, [22]) 𝐴∈(𝑓𝑠∶𝑓𝑠) if and only if (45), (48) hold andlim_{𝑞→∞𝑘}1𝑞+1^𝑞_𝑖=0Δ²𝑎(𝑛+𝑖,𝑘)−𝛼_𝑘=0𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦𝑖𝑛𝑛; (49)(xi)(Başar and Çolak, [21]) 𝐴∈(𝑐𝑠∶𝑓𝑠) if and only if (48) holds;(xii)(Başar, [25]) 𝐴∈(𝑓∶𝑐𝑠) if and only ifsup_{𝑛∈ℕ𝑘}|𝑎(𝑛,𝑘)|<∞,_𝑛𝑎_𝑛𝑘=𝛼_𝑘𝑒𝑥𝑖𝑠𝑡𝑠𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑓𝑖𝑥𝑒𝑑𝑘,_𝑛𝑘𝑎_𝑛𝑘=𝛼,lim_{𝑚→∞𝑘}Δ𝑎(𝑛,𝑘)−𝛼_𝑘=0. (50)(51)(52)(53)

Now we give our main results which are related to matrix mappings on/into the spaces of almost convergent series 𝑎^𝑟_𝑓𝑠 and sequences 𝑎^𝑟_𝑓.

Corollary 20. Let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix. Then, the following statements hold.(i)𝐴∈(𝑎^𝑟_𝑓𝑠∶𝑓) if and only if {𝑎_𝑛𝑘}_𝑘∈ℕ∈(𝑎^𝑟_𝑓)^𝛽 for all 𝑛∈ℕ and (40), (44) hold with 𝑎_𝑛𝑘 instead of 𝑎_𝑛𝑘, (46) holds with 𝑎(𝑛,𝑘,𝑚) instead of 𝑎(𝑛,𝑘,𝑚), and (45) holds with 𝑎(𝑛,𝑘) instead of 𝑎(𝑛,𝑘).(ii)𝐴∈(𝑎^𝑟_𝑓𝑠∶𝑐) if and only if {𝑎_𝑛𝑘}_𝑘∈ℕ∈(𝑎^𝑟_𝑓)^𝛽 for all 𝑛∈ℕ and (19), (43), (44), and (47) hold with 𝑎_𝑛𝑘 instead of 𝑎_𝑛𝑘.(iii)𝐴∈(𝑎^𝑟_𝑓𝑠∶ℓ_∞) if and only if {𝑎_𝑛𝑘}_𝑘∈ℕ∈(𝑎^𝑟_𝑓)^𝛽 for all 𝑛∈ℕ and (43) and (44) hold with 𝑎_𝑛𝑘 instead of 𝑎_𝑛𝑘.(iv)𝐴∈(𝑎^𝑟_𝑓𝑠∶𝑓𝑠) if and only if {𝑎_𝑛𝑘}_𝑘∈ℕ∈(𝑎^𝑟_𝑓𝑠)^𝛽 for all 𝑛∈ℕ and (45), (48), and (49) hold with 𝑎(𝑛,𝑘) instead of 𝑎(𝑛,𝑘).(v)𝐴∈(𝑐𝑠∶𝑎^𝑟_𝑓𝑠) if and only if (48) holds with 𝑎(𝑛,𝑘) instead of 𝑎(𝑛,𝑘).(vi)𝐴∈(𝑏𝑠∶𝑎^𝑟_𝑓𝑠) if and only if (44) holds with 𝑎_𝑛𝑘 instead of 𝑎_𝑛𝑘 and (45), (48) hold with 𝑎(𝑛,𝑘) instead of 𝑎(𝑛,𝑘).(vii)𝐴∈(𝑓𝑠∶𝑎^𝑟_𝑓𝑠) if and only if (45), (48), and (49) hold with 𝑎(𝑛,𝑘) instead of 𝑎(𝑛,𝑘).

Corollary 21. Let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix. Then, the following statements hold.(i)𝐴∈(𝑐(𝑝)∶𝑎^𝑟_𝑓𝑠) if and only if (37)and (38) hold with 𝑎(𝑛,𝑘,𝑚) instead of 𝑎(𝑛,𝑘,𝑚).(ii)𝐴∈(𝑐₀(𝑝)∶𝑎^𝑟_𝑓𝑠) if and only if (37) holds with 𝑎(𝑛,𝑘,𝑚) instead of 𝑎(𝑛,𝑘,𝑚).(iii)𝐴∈(ℓ_∞(𝑝)∶𝑎^𝑟_𝑓𝑠) if and only if (37) and (39) hold with 𝑎(𝑛,𝑘,𝑚) instead of 𝑎(𝑛,𝑘,𝑚).

Corollary 22. Let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix. Then, the following statements hold.(i)𝐴∈(𝑎^𝑟_𝑓∶ℓ_∞) if and only if {𝑎_𝑛𝑘}_𝑘∈ℕ∈(𝑎^𝑟_𝑓)^𝛽 for all 𝑛∈ℕ and (18) holds with 𝑎_𝑛𝑘 instead of 𝑎_𝑛𝑘.(ii)𝐴∈(𝑎^𝑟_𝑓∶𝑐) if and only if {𝑎_𝑛𝑘}_𝑘∈ℕ∈(𝑎^𝑟_𝑓)^𝛽 for all 𝑛∈ℕ and (18), (19), (20), and (21) hold with 𝑎_𝑛𝑘 instead of 𝑎_𝑛𝑘. (iii)𝐴∈(𝑎^𝑟_𝑓∶𝑐𝑠) if and only if {𝑎_𝑛𝑘}_𝑘∈ℕ∈(𝑎^𝑟_𝑓)^𝛽 for all 𝑛∈ℕ and (50),(53) hold with 𝑎(𝑛,𝑘) instead of 𝑎(𝑛,𝑘) and (51),(52) hold with 𝑎_𝑛𝑘 instead of 𝑎_𝑛𝑘.

Corollary 23. Let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix. Then, the following statements hold.(i)𝐴∈(ℓ_∞∶𝑎^𝑟_𝑓) if and only if (18), (40) hold with 𝑎_𝑛𝑘 instead of 𝑎_𝑛𝑘 and (41) holds with 𝑎(𝑛,𝑘,𝑚) instead of 𝑎(𝑛,𝑘,𝑚).(ii)𝐴∈(𝑓∶𝑎^𝑟_𝑓) if and only if (18), (40), and (46) hold with 𝑎(𝑛,𝑘,𝑚) instead of 𝑎(𝑛,𝑘,𝑚) and (42) holds with 𝑎_𝑛𝑘 instead of 𝑎_𝑛𝑘.(iii)𝐴∈(𝑐∶𝑎^𝑟_𝑓) if and only if (18), (40), and (42) hold with 𝑎_𝑛𝑘 instead of 𝑎_𝑛𝑘.(iv)𝐴∈(𝑏𝑠∶𝑎^𝑟_𝑓) if and only if (40), (43), and (44) hold with 𝑎_𝑛𝑘 instead of 𝑎_𝑛𝑘 and (45) holds with 𝑎(𝑛,𝑘) instead of 𝑎(𝑛,𝑘).(v)𝐴∈(𝑓𝑠∶𝑎^𝑟_𝑓) if and only if (40), (44) hold with 𝑎_𝑛𝑘 instead of 𝑎_𝑛𝑘, (46) holds with 𝑎(𝑛,𝑘,𝑚) instead of 𝑎(𝑛,𝑘,𝑚), and (45) holds with 𝑎(𝑛,𝑘) instead of 𝑎(𝑛,𝑘).(vi)𝐴∈(𝑐𝑠∶𝑎^𝑟_𝑓) if and only if (40) and (43) hold with 𝑎_𝑛𝑘 instead of 𝑎_𝑛𝑘.

Remark 24. Characterization of the classes (𝑎^𝑟_𝑓∶𝑓_∞), (𝑓_∞∶𝑎^𝑟_𝑓), (𝑎^𝑟_𝑓𝑠∶𝑓_∞), and (𝑓_∞∶𝑎^𝑟_𝑓𝑠) is redundant since the spaces of almost bounded sequences 𝑓_∞ and ℓ_∞ are equal.

Acknowledgment

The authors thank the referees for their careful reading of the original paper and for the valuable comments.