On the Riesz Almost Convergent Sequences Space

Abstract

The purpose of this paper is to introduce new spaces 𝑓 and 𝑓₀ that consist of all sequences whose Riesz transforms of order one are in the spaces 𝑓 and 𝑓₀, respectively. We also show that 𝑓 and 𝑓₀ are linearly isomorphic to the spaces 𝑓 and 𝑓₀, respectively. The 𝛽- and 𝛾-duals of the spaces 𝑓 and 𝑓₀ are computed. Furthermore, the classes (𝑓∶𝜇) and (𝜇∶𝑓) of infinite matrices are characterized for any given sequence space 𝜇 and determine the necessary and sufficient conditions on a matrix 𝐴 to satisfy 𝐵_𝑅−core(𝐴𝑥)⊆𝐾−core(𝑥), 𝐵_𝑅−core(𝐴𝑥)⊆𝑠𝑡−core(𝑥) for all 𝑥∈ℓ_∞.

1. Introduction and Preliminaries

Let 𝑤 be the space of all real or complex valued sequences. Then, each linear subspace of 𝑤 is called a sequence space. For example, the notations ℓ_∞, 𝑐, 𝑐₀, ℓ₁, 𝑐𝑠, and 𝑏𝑠 are used for the sequence spaces of all bounded, convergent, and null sequences, absolutely convergent series, convergent series, and bounded series, respectively. Let 𝜆 and 𝜇 be two sequence spaces and 𝐴=(𝑎_𝑛𝑘) an infinite matrix of real or complex numbers 𝑎_𝑛𝑘, where 𝑛,𝑘∈ℕ={0,1,2,…}. Then, 𝐴 defines a matrix mapping from 𝜆 to 𝜇 and is denoted by 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) By (𝜆∶𝜇), we denote the class of 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. The matrix domain 𝜆_𝐴 of an infinite matrix 𝐴 in a sequence space 𝜆 is defined by𝜆_𝐴=𝑥=𝑥_𝑘∈𝑤∶𝐴𝑥∈𝜆. (1.2) If we take 𝜆=𝑐, then 𝑐_𝐴 is called, convergence domain of 𝐴, and we write the limit of 𝐴𝑥 as lim_𝐴𝑥=lim_𝑛→∞∑_𝑘𝑎_𝑛𝑘𝑥_𝑘. Further 𝐴 is called regular if lim_𝐴𝑥=lim𝑥 for each convergent sequence 𝑥.

Let 𝜆 be a sequence space. Then 𝜆 is called solid if and only if ℓ_∞𝜆⊂𝜆, [1]. The approach constructing a new sequence space by means of the matrix domain of a particular limitation method has recently been employed by Altay and Başar [2], Başarir [3], Aydın and Başar [4], Kirişçi and Başar [5], Şengönül and Başar [6], Polat and Başar [7], and Malkowsky et al. [8]. Finally, the new technique for deducing certain topological properties, such as AB-, KB-, and AD-properties, and solidity and monotonicity, and determining the 𝛼-, 𝛽- and 𝛾-duals of the domain of a triangle matrix in a sequence space is given by Altay and Başar [9].

Furthermore, quite recently, Kirişçi and Başar [10] introduced the new sequence space 𝑓 derived from the space 𝑓 of almost convergent sequences by means of the domain of the generalized difference matrix 𝐵(𝑟,𝑠).

Define the sets 𝑓 and 𝑓₀ by𝑓=𝑥=𝑥_𝑘∈𝑤∶∃𝛼∈ℂ∋lim_𝑛→∞^𝑛_𝑘=0𝑥_𝑘+𝑝,𝑓𝑛+1=𝛼uniformlyin𝑝₀=𝑥=𝑥_𝑘∈𝑤∶lim_𝑛→∞^𝑛_𝑘=0𝑥_𝑘+𝑝.𝑛+1=0uniformlyin𝑝 (1.3)

If 𝑥∈𝑓, then 𝑥 is said to be almost convergent to the generalized limit 𝛼. When 𝑥∈𝑓, we write 𝑓−lim𝑥=𝛼.

Lorentz [11] introduced this concept and obtained the necessary and sufficient conditions for an infinite matrix to contain 𝑓 in its convergence domain. These conditions on an infinite matrix 𝐴=(𝑎_𝑛𝑘) consist of the standard Silverman Toeplitz conditions for regularity plus the condition lim_𝑛→∞∑_𝑘|𝑎_𝑛𝑘−𝑎_𝑛,𝑘+1|=0. Such matrices are called strongly regular. One of the best known strongly regular matrices is 𝐶, the Cesàro matrix of order one which is a lower triangular matrix defined by𝑐_𝑛𝑘=1𝑛+1,0≤𝑘≤𝑛,0,𝑘>𝑛, (1.4)

for all 𝑛,𝑘∈ℕ.

A matrix 𝑈 is called the generalized Cesàro matrix if it is obtained from 𝐶 by shifting rows. Let 𝑝∶ℕ→ℕ. Then, 𝑈=(𝑢_𝑛𝑘) is defined by𝑢_𝑛𝑘=1(𝑛)(𝑛)𝑛+1,𝑝≤𝑘≤𝑝+𝑛,0,otherwise, (1.5)

for all 𝑛,𝑘∈ℕ.

Let us suppose that 𝐺 is the set of all such matrices obtained by using all possible functions 𝑝. Now, right here, let us give a new definition for the set of almost convergent sequences that was introduced by Butković et al. [12]:

Lemma 1.1. The set 𝑓 of all almost convergent sequences is equal to the set ⋂_U∈𝐺𝑐_𝑈.

Other one of the best known regular matrices is 𝑅=(𝑟_𝑛𝑘), the Riesz matrix which is a lower triangular matrix defined by𝑟_𝑛𝑘=𝑟_𝑘𝑅_𝑛,0≤𝑘≤𝑛,0,𝑘>𝑛, (1.6)

for all 𝑛,𝑘∈ℕ, where (𝑟_𝑘) is real sequence with 𝑟_𝑘≥0(𝑘∈ℕ) and 𝑅_𝑛=𝑟₀+𝑟₁+⋯+𝑟_𝑛.

Let 𝐾 be a subset of ℕ. The natural density 𝛿 of 𝐾 is defined by𝛿(𝐾)=lim_𝑛→∞1𝑛|{𝑘≤𝑛∶𝑘∈𝐾}|, (1.7)

where the vertical bars indicate the number of elements in the enclosed set. The sequence 𝑥=(𝑥_𝑘) is said to be statistically convergent to the number 𝑙 if, for every 𝜀, 𝛿({𝑘∶|𝑥_𝑘−𝑙|≥𝜀})=0 (see [13]). In this case, we write 𝑠𝑡−lim𝑥=𝑙. We will also write 𝒮 and 𝒮₀ to denote the sets of all statistically convergent sequences and statistically null sequences. The statistically convergent sequences were studied by several authors (see [13, 14] and others).

Let us consider the following functionals defined on ℓ_∞:𝑙(𝑥)=liminf_𝑘→∞𝑥_𝑘,𝐿(𝑥)=limsup_𝑘→∞𝑥_𝑘,𝑞_𝜎(𝑥)=limsup_𝑝→∞sup_𝑛∈ℕ1𝑝+1^𝑝_𝑖=0𝑥_{𝜎𝑖(𝑛)},𝐿^∗(𝑥)=limsup_𝑝→∞sup_𝑛∈ℕ1𝑝+1^𝑝_𝑖=0𝑥_𝑛+𝑖. (1.8)

In [15], the 𝜎-core of a real bounded sequence 𝑥 is defined as the closed interval [−𝑞_𝜎(−𝑥),𝑞_𝜎(𝑥)] and also the inequalities 𝑞_𝜎(𝐴𝑥)≤𝐿(𝑥) (𝜎-core of 𝐴𝑥⊆𝐾-core of 𝑥) 𝑞_𝜎(𝐴𝑥)≤𝑞_𝜎(𝑥) (𝜎-core of 𝐴𝑥⊆𝜎-core of 𝑥), for all 𝑥∈ℓ_∞, have been studied. Here the Knopp core, in short 𝐾-core, of 𝑥 is the interval [𝑙(𝑥),𝐿(𝑥)]. In particular, when 𝜎(𝑛)=𝑛+1, since 𝑞_𝜎(𝑥)=𝐿^∗(𝑥), 𝜎-core of 𝑥 is reduced to the Banach core, in short 𝐵-core, of 𝑥 defined by the interval [−𝐿^∗(−𝑥),𝐿^∗(𝑥)].

The concepts of 𝐵-core and 𝜎-core have been studied by many authors [16, 17].

Recently, Fridy and Orhan [13] have introduced the notions of statistical boundedness, statistical limit superior (or briefly 𝑠𝑡−limsup), and statistical limit inferior (or briefly 𝑠𝑡−liminf), defined the statistical core (or briefly 𝑠𝑡-core) of a statistically bounded sequence as the closed interval [𝑠𝑡−liminf𝑥,𝑠𝑡−limsup𝑥], and also determined the necessary and sufficient conditions for a matrix 𝐴 to yield 𝐾-core(𝐴𝑥)⊆𝑠𝑡-core(𝑥) for all 𝑥∈ℓ_∞.

Let us write𝑇^∗(𝑥)=limsup_𝑛→∞sup^𝑛_{𝑝∈ℕ𝑘=0}1𝑛+1^𝑘_𝑗=0𝑥_𝑗+𝑝𝑘+1. (1.9)

Quite recently, 𝐵_𝐶-core of a sequence has been introduced by the closed intervals [−𝑇^∗(−𝑥),𝑇^∗(𝑥)] and also the inequalities𝑇^∗(𝐴𝑥)≤𝐿(𝑥),𝐿(𝐴𝑥)≤𝑇^∗(𝑥),𝑇^∗(𝐴𝑥)≤𝑇^∗(𝑥),𝑇^∗(𝐴𝑥)≤𝑠𝑡−limsup𝑥 (1.10)

have been studied for all 𝑥∈ℓ_∞ in [18].

Definition 1.2. Let 𝑥∈ℓ_∞. Then, 𝐵_𝑅-core of 𝑥 is defined by the closed interval [−𝜏^∗(−𝑥),𝜏^∗(𝑥)], where 𝜏^∗(𝑥)=limsup_𝑛→∞sup^𝑛_{𝑝∈ℕ𝑗=0}1𝑛+1^𝑗_𝑖=0𝑟_𝑖𝑥_𝑖+𝑝𝑅_𝑗,−𝜏^∗(−𝑥)=liminf_𝑛→∞sup^𝑛_{𝑝∈ℕ𝑗=0}1𝑛+1^𝑗_𝑖=0𝑟_𝑖𝑥_𝑖+𝑝𝑅_𝑗. (1.11) Therefore, it is easy to see that 𝐵_𝑅-core of 𝑥 is ℓ if and only if 𝑓−lim𝑥=ℓ.

As known, the method to obtain a new sequence space by using the convergence field of an infinite matrix is an old method in the theory of sequence spaces. However, the study of the convergence field of an infinite matrix in the space of almost convergent sequences is new.

2. The Sequence Spaces 𝑓 and 𝑓₀

In this section we introduce the new spaces 𝑓 and 𝑓₀ as the sets of all sequences such that their 𝑅-transforms are in the spaces 𝑓 and 𝑓₀, respectively, that is𝑓=𝑥=𝑥_𝑘∈𝑤∶∃𝛼∈ℂ∋lim_𝑛→∞^𝑛_𝑗=01𝑛+1^𝑗𝑖=0𝑟_𝑖𝑥_𝑖+𝑝𝑅_𝑗,𝑓=𝛼uniformlyin𝑝₀=𝑥=𝑥_𝑘∈𝑤∶lim_𝑛→∞^𝑛_𝑗=01𝑛+1^𝑗𝑖=0𝑟_𝑖𝑥_𝑖+𝑝𝑅_𝑗=𝛼uniformlyin𝑝. (2.1)

With the notation of (1.2), we can write 𝑓=𝑓_𝑅 and 𝑓₀=(𝑓₀)_𝑅. Define the sequence 𝑦=(𝑦_𝑘), which will be frequently used, as the 𝑅-transform of a sequence 𝑥=(𝑥_𝑘), that is,𝑦_𝑛=^𝑛_𝑘=0𝑟_𝑘𝑥_𝑘𝑅_𝑛∀𝑛∈ℕ. (2.2) If 𝑅=𝐶, which is Cesàro matrix, order 1, then the space 𝑓 and 𝑓₀ correspond to the spaces 𝑓 and 𝑓₀ (see [18]).

Suppose that 𝐺′={𝒢∶𝒢=𝑈∘𝑅,𝑈∈𝐺and𝑅isRieszmatrix}. Then we have the following proposition.

Proposition 2.1. 𝑓=⋂_{𝑇∈𝐺′}𝑐_𝑇 dir.

Proof. The proof is similar to the proof of Lemma 1.1 so we omit the details, (see [12]).

Consider the function 𝑓‖⋅‖∶𝑓→ℝ, and define‖𝑥‖𝑓=sup_𝑛^𝑛_𝑗=01𝑛+1^𝑗𝑖=0𝑟_𝑖𝑥_𝑖+𝑝𝑅_𝑗. (2.3) The function 𝑓‖⋅‖ is a norm and (𝑓𝑓,‖⋅‖) is BK-space. The proof of this is as follows.

Theorem 2.2. The sets 𝑓 and 𝑓₀ are linear spaces with the coordinate wise addition and scalar multiplication that is the BK- space with the 𝑓‖𝑥‖=‖𝑅𝑥‖_𝑓.

Proof. The first part of the theorem can be easily proved. We prove the second part of the theorem. Since (1.2) holds and 𝑓 and 𝑓₀ are the BK-spaces [1] with respect to their natural norm, also the matrix 𝑅 is normal and gives the fact that the spaces 𝑓 and 𝑓₀ are BK-spaces.

Theorem 2.3. The sequence spaces 𝑓 and 𝑓₀ are linearly isomorphic to the spaces 𝑓 and 𝑓₀, respectively.

Proof. Since the fact “the spaces 𝑓₀ and 𝑓₀ are linearly isomorphic” can also be proved in a similar way, we consider only the spaces 𝑓 and 𝑓. In order to prove the fact that 𝑓≅𝑓, we should show the existence of a linear bijection between the spaces 𝑓 and 𝑓. Consider the transformation 𝑇 defined, with the notation of (2.2), from 𝑓 to 𝑓 by 𝑥↦𝑦=𝑇𝑥. The linearity of 𝑇 is clear. Further, it is trivial that 𝑥=𝜃=(0,0,…) whenever 𝑇𝑥=𝜃 and hence 𝑇 is injective.
Let 𝑦=(𝑦_𝑘)∈𝑓, and define the sequence 𝑥=(𝑥_𝑘) by 𝑥_𝑘=1𝑟_𝑘𝑅_𝑘𝑦_𝑘−𝑅_𝑘−1𝑦_𝑘−1,(𝑘∈ℕ). (2.4) Then, we have 𝑓_𝑅−lim𝑥=lim_𝑛→∞^𝑛_𝑗=01𝑛+1^𝑗𝑖=0𝑟_𝑖𝑥_𝑖+𝑝𝑅_𝑗uniformlyin𝑝=lim_𝑛→∞^𝑛_𝑗=01𝑛+1^𝑗𝑖=0𝑟_𝑖1/𝑟_𝑖𝑅_𝑖𝑦_𝑖+𝑝−𝑅_𝑖−1𝑦_{𝑖+𝑝−1}𝑅_𝑗uniformlyin𝑝=lim_𝑛→∞1𝑛+1^𝑛𝑗=0𝑦_𝑗+𝑝uniformlyin𝑝=𝑓−lim𝑦uniformlyin𝑝, (2.5) which shows that 𝑥∈𝑓. Consequently, we see that 𝑇 is surjective. Hence, 𝑇 is a linear bijection that therefore shows that the spaces 𝑓 and 𝑓 are linearly isomorphic, as desired. This completes the proof.

Theorem 2.4. The spaces 𝑓 and 𝑓₀ are not solid sequence spaces.

Proof. If we take 𝑢=(𝑢_𝑘)=(𝑅₀/𝑟₀,−(𝑅₀/𝑟₀+𝑅₁/𝑟₁),(𝑅₁/𝑟₁+𝑅₂/𝑟₂),…,(−1)^𝑘(𝑅_𝑘/𝑟_𝑘+𝑅_𝑘+1/𝑟_𝑘+1),…) and 𝑣=(𝑣_𝑘)=(1,−1,1,…,(−1)^𝑘,…), then we see that 𝑢∈𝑓 and 𝑣∈ℓ_∞. Let 𝑢𝑣 be 𝑡, that is, (𝑢𝑣)_𝑘=𝑡_𝑘. Then, since (𝑡_𝑘)=(𝑅₀/𝑟₀,(𝑅₀/𝑟₀+𝑅₁/𝑟₁),(𝑅₁/𝑟₁+𝑅₂/𝑟₂),…,(𝑅_𝑘/𝑟_𝑘+𝑅_𝑘+1/𝑟_𝑘+1),…) and it is not hard to see by taking into account the definition Riesz matrix that 𝑓_𝑅−lim𝑡=lim_𝑛→∞∑^𝑛_𝑗=01/(𝑛+1)∑^𝑗_𝑖=0(𝑟_𝑖𝑡_𝑖+𝑝)/𝑅_𝑗=∞. This shows that the multiplication ℓ_∞𝑓 of the spaces ℓ_∞ and 𝑓 is not a subset of 𝑓 and therefore the space 𝑓 is not solid. The proof for the space 𝑓₀ is similar to the proof of the space 𝑓, so we omit it.

Theorem 2.5. Let the spaces 𝑓 and 𝑓₀ be given. Then,(1)the inclusion 𝑓₀⊂𝑓 holds and the space 𝑓 is not a subset of the space ℓ_∞,(2) if (1/𝑅_𝑛)∈𝑐 and (𝑟_𝑘)∈ℓ₁, then ℓ_∞⊂𝑓 strictly holds.

Proof. (1) Clearly, the inclusion 𝑓₀⊂𝑓 holds. Let us consider the sequence given by 𝑢_𝑘=−𝑅_𝑘𝑟_𝑘+𝑅_𝑘+1𝑟_𝑘+1𝑅,𝑘isodd,_𝑘𝑟_𝑘+𝑅_𝑘+1𝑟_𝑘+1,𝑘iseven, (2.6) Since (𝑅_𝑘)→∞(𝑘→∞), the sequence (𝑢_𝑘) is not a bounded sequence. But clearly 𝑢∈𝑓. This shows that to us, the space 𝑓 is not a subset of the space ℓ_∞.
(2) If (1/𝑅_𝑛)∈𝑐 and (𝑟_𝑘)∈ℓ₁, then for all 𝑥∈ℓ_∞ we have 𝑅𝑥∈𝑐. Therefore, since lim(𝑅𝑥)=𝑓−lim(𝑅𝑥), we see that 𝑥∈𝑓.

In Theorem 2.6, we will use some similar techniques that are due to Móricz and Rhoades [19].

Theorem 2.6. Define the sequences (𝛼_𝑛) and (𝛽_𝑛) by 𝛼_𝑛=inf_𝑝>0^𝑛_𝑗=01𝑛+1^𝑗𝑖=0𝑟_𝑖𝑥_𝑖+𝑝𝑅_𝑗,𝛽_𝑛=sup_𝑝>0^𝑛_𝑗=01𝑛+1^𝑗𝑖=0𝑟_𝑖𝑥_𝑖+𝑝𝑅_𝑗 (2.7) for all 𝑛∈ℕ. Then, 𝛼_𝑛≤𝛽_𝑛 for each 𝑛∈ℕ and(i)the sequence (𝛼_2𝑛) is nondecreasing,(ii)the sequence (𝛽_2𝑛) is nonincreasing.

Proof. It is trivial that 𝛼_𝑛=inf_𝑝>0^𝑛_𝑗=01𝑛+1^𝑗𝑖=0𝑟_𝑖𝑥_𝑖+𝑝𝑅_𝑗≤sup_𝑝>0^𝑛_𝑗=01𝑛+1^𝑗𝑖=0𝑟_𝑖𝑥_𝑖+𝑝𝑅_𝑗=𝛽_𝑛 (2.8) for each 𝑛∈ℕ.
Since the part (ii) can be proved in a similar way, we prove only part (i) 𝛼_2𝑛+1=inf_𝑝>0^𝑝+2𝑛+1_𝑗=𝑝12^𝑛+1+1^𝑗𝑖=0𝑟_𝑖𝑥_𝑖𝑅_𝑗=inf_𝑝>012^𝑛+12+1^𝑛+12^𝑛+1^{𝑝+2𝑛+1𝑗=𝑝𝑗𝑖=0}𝑟_𝑖𝑥_𝑖𝑅_𝑗=inf_𝑝>012^𝑛+11+12^𝑛+1^{𝑝+2𝑛+1𝑗=𝑝𝑗𝑖=0}𝑟_𝑖𝑥_𝑖𝑅_𝑗2^𝑛+1=inf_𝑝>012^𝑛+11+12^𝑛+1^{𝑝+2𝑛𝑗=𝑝𝑗𝑖=0}𝑟_𝑖𝑥_𝑖𝑅_𝑗2^𝑛+1+12^𝑛+1^{𝑝+2𝑛+1𝑗=𝑝+2𝑛+1𝑗𝑖=0}𝑟_𝑖𝑥_𝑖𝑅_𝑗2^𝑛≥1+12^𝑛+12+1^𝑛+1𝛼_2𝑛+2^𝑛+1𝛼_2𝑛=𝛼_2𝑛. (2.9) This step completes the proof.

Theorem 2.7. lim_𝑝→∞(𝛽_2𝑝−𝛼_2𝑝)=0 if and only if 𝑥∈𝑓.

Proof. Suppose that lim_𝑛→∞(𝛽_2𝑛−𝛼_2𝑛)=0. For each 𝑛, choose 𝑟 to satisfy 2^𝑟≤𝑛≤2^𝑟+1. We may write 𝑛 in a dyadic representation of the form 𝑛=∑^𝑟_𝑖=0𝑛_𝑖2^𝑖, where each 𝑛_𝑖 is 0 or 1, 𝑖=0,1,2,…,𝑟−1, and 𝑛_𝑟=1. Then, 1𝑛+1^{𝑝+𝑛𝑗=𝑝𝑗𝑖=0}𝑟_𝑖𝑥_𝑖𝑅_𝑗=1𝑛+1^{𝑝+∑𝑟𝑖=0𝑛𝑖2𝑖𝑗=𝑝𝑗𝑖=0}𝑟_𝑖𝑥_𝑖𝑅_𝑗=11𝑛+12⁰+1^{𝑝+𝑛020𝑗=𝑝𝑗𝑖=0}𝑟_𝑖𝑥_𝑖𝑅_𝑗2⁰+1+12¹+1^{𝑛222𝑗=𝑛121𝑗𝑖=0}𝑟_𝑖𝑥_𝑖𝑅_𝑗2¹1+1+⋯+2^𝑟+1^{𝑛𝑟2𝑟𝑗=𝑛𝑟−12𝑟−1𝑗𝑖=0}𝑟_𝑖𝑥_𝑖𝑅_𝑗2^𝑟=1+12𝑛+1^𝑟+1𝛼_2𝑟+2^𝑟−1+1𝛼_2𝑟−1+⋯+2⁰𝛼₂₀≥1𝑛+1^𝑟𝑗=0𝑛_𝑗2^𝑗+1𝛼_𝑗 (2.10) since 𝑛_𝑗∈{0,1}, and hence inf_𝑝>0^𝑝+𝑛_𝑗=𝑝1𝑛+1^𝑗𝑖=0𝑟_𝑖𝑥_𝑖𝑅_𝑗=𝛼_𝑛≥1𝑛+1^𝑟𝑗=0𝑛_𝑗2^𝑗+1𝛼_2𝑗,sup∑_𝑝>0^𝑝+𝑛_𝑗=𝑝1𝑛+1^𝑗_𝑖∑₌₀𝑟_𝑖𝑥_𝑖𝑅_𝑗=𝛽_𝑛≤1𝑛+1^𝑟_𝑗∑₌₀𝑛_𝑗2^𝑗+1𝛼_2𝑗. (2.11) Thus, 0≤𝛽_𝑛−𝛼_𝑛≤1𝑛+1^𝑟𝑗=0𝑛_𝑗2^𝑗+1𝛽_2𝑗−𝛼_2𝑗. (2.12) If 𝑇 is the lower triangular matrix with nonzero entries 𝑡_𝑛𝑘=𝑛_𝑘(2^𝑘+1)/(𝑛+1), then, 𝑇 is a regular matrix so that lim_𝑟→∞(𝛽_2𝑟−𝛼_2𝑟)=0. From the equality (2.12), we see that lim_𝑛→∞(𝛽_𝑛−𝛼_𝑛)=0.
Conversely, assume that 𝑥∈𝑓. Then, since lim_𝑛→∞^𝑝+𝑛_𝑗=𝑝1𝑛+1^𝑗𝑖=0𝑟_𝑖𝑥_𝑖𝑅_𝑗=𝛼 (2.13) implies lim_𝑛→∞𝛼_𝑛=inf_𝑝>0lim_𝑛→∞^𝑝+𝑛_𝑗=𝑝1𝑛+1^𝑗𝑖=0𝑟_𝑖𝑥_𝑖𝑅_𝑗=𝛼,lim_𝑛→∞𝛽_𝑛=sup_𝑝>0lim_𝑛→∞^𝑝+𝑛_𝑗=𝑝1𝑛+1^𝑗𝑖=0𝑟_𝑖𝑥_𝑖𝑅_𝑗=𝛼, (2.14) we have lim_𝑛→∞𝛼_𝑛−lim_𝑛→∞𝛽_𝑛=lim_𝑛→∞𝛼_𝑛−𝛽_𝑛=0. (2.15) If we take 𝑛=2^𝑝, then the proof of sufficiency is obtained. This step completes the proof.

3. Some Duals of the Spaces 𝑓 and 𝑓₀

In this section, by using techniques in [9], we have stated and proved the theorems determining the 𝛽- and 𝛾-duals of the spaces 𝑓₀ and 𝑓. For the sequence spaces 𝜆 and 𝜇, define the set 𝑆(𝜆,𝜇) by𝑆𝜆,𝜇=𝑧=𝑧_𝑘∈𝑤∶𝑥𝑧=𝑥_𝑘𝑧_𝑘∈𝜇∀𝑥=𝑥_𝑘∈𝜆. (3.1) With the notation of (3.1), the 𝛼-, 𝛽-, and 𝛾-duals of a sequence space 𝜆, which are, respectively, denoted by 𝜆^𝛼, 𝜆^𝛽, and 𝜆^𝛾, are defined by𝜆^𝛼=𝑆𝜆,ℓ₁,𝜆^𝛽=𝑆(𝜆,𝑐𝑠),𝜆^𝛾=𝑆(𝜆,𝑏𝑠). (3.2)

The following two lemmas are introduced in [20] which we need in proving Theorems 3.3 and 3.4.

Lemma 3.1. 𝐴∈(𝑓∶ℓ_∞) if and only if sup_{𝑛∈ℕ𝑘}𝑎_𝑛𝑘<∞. (3.3)

Lemma 3.2. 𝐴∈(𝑓∶𝑐) if and only if lim_{𝑛→∞𝑘}𝑎_𝑛𝑘=𝑎,lim_𝑛→∞𝑎_𝑛𝑘=𝑎_𝑘;𝑘∈ℕ,lim_{𝑛→∞𝑘}Δa_𝑛𝑘−𝑎_𝑘=0. (3.4)

Theorem 3.3. The 𝛾-duals of the spaces 𝑓 and 𝑓₀ are the set 𝑑₁∩𝑑₂, where 𝑑₁=𝑎=𝑎_𝑘∈𝑤∶_𝑘𝑅_𝑘Δ𝑎_𝑘𝑟_𝑘,𝑑<∞₂=𝑎=𝑎_𝑘𝑎∈𝑤∶^𝑛𝑟_𝑛𝑅_𝑛∈ℓ_∞. (3.5)

Proof. Define the matrix 𝑇=(𝑡_𝑛𝑘) via the sequence 𝑎=(𝑎_𝑘)∈𝑤 by 𝑡_𝑛𝑘=𝑅_𝑘Δ𝑎_𝑘𝑟_𝑘𝑎,0≤𝑘≤𝑛−1,_𝑛𝑟_𝑛𝑅_𝑛,𝑛=𝑘,0,otherwise, (3.6) for all 𝑛,𝑘∈ℕ. Here, Δ(𝑎_𝑘/𝑟_𝑘)=(𝑎_𝑘/𝑟_𝑘−𝑎_𝑘+1/𝑟_𝑘+1). By using (2.2), we derive that ^𝑛_𝑘=0𝑎_𝑘𝑥_𝑘=^𝑛−1_𝑘=0𝑅_𝑘Δ𝑎_𝑘𝑟_𝑘𝑦_𝑘+𝑎_𝑛𝑟_𝑛𝑅_𝑛𝑦_𝑛=𝑇𝑦_𝑛,(𝑛∈ℕ). (3.7) From (3.7), we see that 𝑎𝑥=(𝑎_𝑘𝑥_𝑘)∈𝑏𝑠 whenever 𝑥=(𝑥_𝑘)∈𝑓 if and only if 𝑇𝑦∈ℓ_∞ whenever 𝑦=(𝑦_𝑘)∈𝑓. Then, we derive by Lemma 3.1 that ^𝑛_𝑘=0𝑅_𝑘Δ𝑎_𝑘𝑟_𝑘𝑎<∞,^𝑛𝑟_𝑛𝑅_𝑛∈ℓ_∞, (3.8) which yields the desired result 𝑓^𝛾=𝑓^𝛾₀=𝑑₁∩𝑑₂.

Theorem 3.4. Define the set 𝑑₃ by 𝑑₃=𝑎=𝑎_𝑘∈𝑤∶_𝑘Δ𝑅_𝑘Δ𝑎_𝑘𝑟_𝑘−𝑎_𝑘<∞. (3.9) Then, 𝑓^𝛽=𝑑₃∩𝑐𝑠.

Proof. Consider equality (3.7), again. Thus, we deduce that 𝑎𝑥=(𝑎_𝑘𝑥_𝑘)∈𝑐𝑠 whenever 𝑥=(𝑥_𝑘)∈𝑓 if and only if 𝑇𝑦∈𝑐 whenever 𝑦=(𝑦_𝑘)∈𝑓. It is obvious that the columns of that matrix 𝑇, defined by (3.6), are in the space 𝑐. Therefore, we derive the consequence from Lemma 3.2 that 𝑓^𝛽=𝑑₃∩𝑐𝑠.

4. Some Matrix Mappings Related to the Spaces 𝑓 and 𝑓₀

In this section, we characterize the matrix mappings from 𝑓 into any given sequence space via the concept of the dual summability methods of the new type introduced by Başar [21].

Note that some researchers, such as, Başar [21], Başar and Çolak [22], Kuttner [23], and Lorentz and Zeller [24], worked on the dual summability methods. Now, following Başar [21], we give a short survey about dual summability methods of the new type.

Let us suppose that the infinite matrices A=(𝑎_𝑛𝑘) and 𝐵=(𝑏_𝑛𝑘) map the sequences 𝑥=(𝑥_𝑘) and 𝑦=(𝑦_𝑘), which are connected by the relation (2.2) to the sequences 𝑧=(𝑧_𝑛) and 𝑡=(𝑡_𝑛), respectively, that is,𝑧_𝑛=(𝐴𝑥)_𝑛=_𝑘𝑎_𝑛𝑘𝑥_𝑘,(),𝑡𝑛∈ℕ_𝑛=𝐵𝑦_𝑛=_𝑘𝑏_𝑛𝑘𝑦_𝑘,(𝑛∈ℕ). (4.1) It is clear here that the method 𝐵 is applied to the 𝑅-transform of the sequence 𝑥=(𝑥_𝑘) while the method 𝐴 is directly applied to the entries of the sequence 𝑥=(𝑥_𝑘). So, the methods 𝐴 and 𝐵 are essentially different.

Let us assume that the matrix product 𝐵𝑅 exists, which is a much weaker assumption than the conditions on the matrix 𝐵 belonging to any matrix class, in general. The methods 𝐴 and 𝐵 in (4.1), (4.2) are called dual summability methods of the new type if 𝑧_𝑛 reduces to 𝑡_𝑛 (or 𝑡_𝑛 reduces to 𝑧_𝑛) under the application of formal summation by parts. This leads us to the fact that 𝐵𝑅 exists and is equal to 𝐴 and (𝐵𝑅)𝑥=𝐵(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):𝑎_𝑛𝑘∶=^∞_𝑗=𝑘𝑟_𝑗𝑅_𝑗𝑏_𝑛𝑗or𝑏_𝑛𝑘∶=𝑅_𝑘𝑎_𝑛𝑘𝑟_𝑘−𝑎_𝑛,𝑘+1𝑟_𝑘+1=𝑅_𝑘Δ𝑎_𝑛𝑘𝑟_𝑘 (4.2) for all 𝑛,𝑘∈ℕ.

Now, we give the following theorem concerning the dual matrices of the new type.

Theorem 4.1. Let 𝐴=(𝑎_𝑛𝑘) and 𝐵=(𝑏_𝑛𝑘) be the dual matrices of the new type and 𝜇 any given sequence space. Then, 𝐴∈(𝑓∶𝜇) if and only if 𝐵∈(𝑓∶𝜇) and 𝑅_𝑛𝑟_𝑛𝑎_{𝑛𝑘𝑛∈ℕ}∈𝑐₀ (4.3) for every fixed 𝑘∈ℕ.

Proof. Suppose that 𝐴=(𝑎_𝑛𝑘) and 𝐵=(𝑏_𝑛𝑘) are dual matrices of the new type, that is to say (4.2) holds and 𝜇 is an any given sequence space. Since the spaces 𝑓 and 𝑓 are linearly isomorphic, now let 𝐴∈(𝑓∶𝜇) and 𝑦=(𝑦_𝑘)∈𝑓. Then, 𝐵𝑅 exists and (𝑎_𝑛𝑘)_𝑘∈ℕ∈𝑑₂∩𝑐𝑠, which yields that (𝑏_𝑛𝑘)_𝑘∈ℕ∈ℓ₁ for each 𝑛∈ℕ. Hence, 𝐵𝑦 exists for each 𝑦∈𝑓, and thus letting 𝑚→∞ in the equality ^𝑚_𝑘=0𝑏_𝑛𝑘𝑦_𝑘=^𝑚_𝑘=0^𝑚_𝑗=𝑘𝑟_𝑗𝑅_𝑗𝑏_𝑛𝑗𝑥_𝑘 (4.4) for all 𝑚,𝑛∈ℕ, we have by (4.2) that 𝐵𝑦=𝐴𝑥, which gives the result 𝐵∈(𝑓∶𝜇).
Conversely, let {𝑎_𝑛𝑘}_𝑘∈ℕ∈𝑓^𝛽 for each 𝑛∈ℕ and 𝐵∈(𝑓∶𝜇) hold, and take any 𝑥=(𝑥_𝑘)∈𝑓. Then, 𝐴𝑥 exists. Therefore, we obtain from the equality ^𝑚_𝑘=0𝑎_𝑛𝑘𝑥_𝑘=^𝑚−1_𝑘=0𝑅_𝑘Δ𝑎_𝑛𝑘𝑟_𝑘𝑦_𝑘+𝑎_𝑛𝑚𝑟_𝑛𝑦_𝑚=^𝑚_𝑘=0𝑏_𝑛𝑘𝑦_𝑘;(𝑛∈ℕ), (4.5) as 𝑛→∞ that 𝐴𝑥=𝐵𝑦, and this shows that 𝐴∈(𝑓∶𝜇). This completes the proof.

Theorem 4.2. Suppose that the entries of the infinite matrices 𝐷=(𝑑_𝑛𝑘) and 𝐸=(𝑒_𝑛𝑘) are connected with the relation 𝑒_𝑛𝑘=^𝑛_𝑗=0𝑟_𝑗𝑑_𝑗𝑘𝑅_𝑗,(𝑛,𝑘∈ℕ) (4.6) and 𝜇 is any given sequence space. Then, 𝐷∈(𝜇∶𝑓) if and if only 𝐸∈(𝜇∶𝑓).

Proof. Let 𝑥=(𝑥_𝑘)∈𝜇, and consider the following equality with (4.6): _𝑛𝑗=0𝑟_𝑗𝑅_𝑗^𝑚𝑘=0𝑑_𝑗𝑘𝑥_𝑘=^𝑚_𝑘=0𝑒_𝑛𝑘𝑥_𝑘;(𝑚,𝑛,𝑘∈ℕ), (4.7) which yields as 𝑚→∞ that 𝐷𝑥∈𝑓 whenever 𝑥∈𝜇 if and if only 𝐸𝑥∈𝑓 whenever 𝑥∈𝜇. This step completes the proof.

Now, right here, we give the following propositions that are obtained from Lemmas 3.2 and 3.1 and Theorems 4.1 and 4.2.

Proposition 4.3. Let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix of real or complex numbers. Then, 𝐴=𝑎_𝑛𝑘∈𝑓∶ℓ_∞⟺(1)lim_{𝑛→∞𝑘}Δ^∞_𝑗=𝑘𝑟_𝑗𝑎_𝑛𝑗𝑅_𝑗−𝑎_𝑘(2𝑎=0,)_𝑛𝑘∈𝑓^𝛽∀𝑛∈ℕ. (4.8)

Proposition 4.4. Let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix of real or complex numbers. Then, 𝐴=𝑎_𝑛𝑘∈⟺(3)𝑓∶𝑐lim_{𝑛→∞𝑘}^∞_𝑗=𝑘𝑟_𝑗𝑎_𝑛𝑗𝑅_𝑗(4)=𝑎,lim^∞_{𝑛→∞𝑗=𝑘}𝑟_𝑗𝑎_𝑛𝑗𝑅_𝑗=𝑎_𝑘(5)foreach𝑘∈ℕ,lim_{𝑛→∞𝑘}Δ^∞_𝑗=𝑘𝑟_𝑗𝑎_𝑛𝑗𝑅_𝑗−𝑎_𝑘(6𝑎=0,)_{𝑛𝑘𝑘∈ℕ}∈𝑓^𝛽∀𝑛∈ℕ. (4.9)

Proposition 4.5. Let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix of real or complex numbers. Then, 𝐴=𝑎_𝑛𝑘∈ℓ_∞∶𝑓⟺(7)sup_{𝑛∈ℕ𝑘}^∞_𝑗=𝑘𝑟_𝑗𝑎_𝑛𝑗𝑅_𝑗(8)<∞,𝑓−lim^∞_{𝑛→∞𝑗=𝑘}𝑟_𝑗𝑎_𝑛𝑗𝑅_𝑗=𝛼_𝑘(9)existsforeachfixed𝑘∈ℕ,lim_{𝑚→∞𝑘}^𝑚_𝑖=01𝑚+1^∞_𝑗=𝑘𝑟_𝑗𝑎_{𝑛+𝑖,𝑗}𝑅_𝑗−𝛼_𝑘=0uniformlyin𝑛. (4.10)

Proposition 4.6. Let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix of real or complex numbers. Then, 𝐴=𝑎_𝑛𝑘∈𝑓⟺()𝑐∶10sup_{𝑛∈ℕ𝑘}^∞_𝑗=𝑘𝑟_𝑗𝑎_𝑛𝑗𝑅_𝑗()<∞,11𝑓−lim^∞_{𝑛→∞𝑗=𝑘}𝑟_𝑗𝑎_𝑛𝑗𝑅_j=𝛼_𝑘()existsforeachfixed𝑘∈ℕ,12𝑓−lim_{𝑛→∞𝑘}^∞_𝑗=𝑘𝑟_𝑗𝑎_𝑛𝑗𝑅_𝑗=𝛼. (4.11)

5. Core Theorems

In this section, we give some core theorems related to the space 𝑓. We need the following lemma due to Das [25] for the proof of next theorem.

Lemma 5.1. Let ‖𝑐‖=‖𝑐_𝑛𝑗(𝑝)‖<∞ and lim_𝑛→∞sup_𝑝∈ℕ|𝑐_𝑛𝑗(𝑝)|=0. Then, there is a 𝑦=(𝑦_𝑗)∈ℓ_∞ such that ‖𝑦‖≤1 and limsup_𝑛→∞sup_{𝑝∈ℕ𝑗}𝑐_𝑛𝑗𝑝𝑦_𝑗=limsup_𝑛→∞sup_{𝑝∈ℕ𝑗}𝑐_𝑛𝑗𝑝. (5.1)

Theorem 5.2. 𝐵_𝑅−core(𝐴𝑥)⊆𝐾−core(𝑥) for all 𝑥∈ℓ_∞ if and only if 𝐴∈(𝑐∶𝑓)_reg and lim_𝑛→∞sup_{𝑝∈ℕ𝑘}1𝑛+1^𝑛𝑗=01𝑅_𝑗^𝑗𝑖=0𝑟_𝑖𝑎_{𝑖+𝑝,𝑘}=1. (5.2)

Proof. Necessity: Suppose first that 𝐵_𝑅−core(𝐴𝑥)⊆𝐾−core(𝑥) for all 𝑥∈ℓ_∞. If 𝑥∈𝑓, then we have 𝜏^∗(𝐴𝑥)=−𝜏^∗(−𝐴𝑥). By this hypothesis, we get −𝐿(−𝑥)≤−𝜏^∗(−𝐴𝑥)≤𝜏^∗(𝐴𝑥)≤𝐿(𝑥). (5.3) If 𝑥∈𝑐, then 𝐿(𝑥)=−𝐿(−𝑥)=lim𝑥. So, we have 𝑓−lim𝐴𝑥=𝜏^∗(𝐴𝑥)=−𝜏^∗(−𝐴𝑥)=lim𝑥, which implies that 𝐴∈(𝑐,𝑓)_reg.
Now, let us consider the sequence 𝐶=(𝑐_𝑛𝑗(𝑝)) of infinite matrices defined by 𝑐_𝑛𝑗𝑝=1𝑛+1^𝑛𝑗=01𝑅_𝑗^𝑗𝑖=0𝑟_𝑖𝑎_{𝑖+𝑝,𝑘}∀𝑛,𝑖,𝑝∈ℕ. (5.4) Then, it is easy to see that the conditions of Lemma 5.1 are satisfied for the matrix sequence 𝐶. Thus, by using the hypothesis, we can write 1≤liminf_𝑛→∞sup_{𝑝∈ℕ𝑗}𝑐_𝑛𝑗𝑝≤limsup_𝑛→∞sup_{𝑝∈ℕ𝑗}𝑐_𝑛𝑗𝑝=limsup_𝑛→∞sup_{𝑝∈ℕ𝑗}𝑐_𝑛𝑗𝑝𝑦_𝑗=𝜏^∗𝐴𝑦≤𝐿𝑦≤𝑦≤1. (5.5) This gives the necessity of (5.2).
Sufficiency: Conversely, let 𝐴∈(𝑐∶𝑓)_reg and (5.2) hold for all 𝑥∈ℓ_∞. For any real number 𝜆 we write 𝜆⁺=max{𝜆,0} and 𝜆⁻=max{−𝜆,0}; then |𝜆|=𝜆⁺+𝜆⁻ and 𝜆=𝜆⁺−𝜆⁻. Therefore, for any given 𝜀>0, there is a 𝑗₀∈ℕ such that 𝑥_𝑗<𝐿(𝑥)+𝜀 for all 𝑗>𝑗₀. Now, we can write _𝑗𝑐_𝑛𝑗𝑝𝑥_𝑗=_𝑗<𝑗0𝑐_𝑛𝑗𝑝𝑥_𝑗+_𝑗≥𝑗0𝑐_𝑛𝑗𝑝⁺𝑥_𝑗−_𝑗≥𝑗0𝑐_𝑛𝑗𝑝⁻𝑥_𝑗≤‖𝑥‖_𝑗<𝑗0𝑐_𝑛𝑗𝑝+[𝐿(𝑥)+𝜀]_𝑗𝑐_𝑛𝑗𝑝+‖𝑥‖_𝑗𝑐_𝑛𝑗𝑝−𝑐_𝑛𝑗𝑝. (5.6) Thus, by applying limsup_𝑛→∞sup_𝑝∈ℕ and using the hypothesis, we have 𝜏^∗(𝐴𝑥)≤𝐿(𝑥)+𝜀. This completes the proof since 𝜀 is arbitrary and 𝑥∈ℓ_∞.

In particular 𝑟_𝑖=1 for all 𝑖 since 𝑅 is reduced to Cesàro matrix, see [18].

Theorem 5.3. 𝐵_𝐶−core(𝐴𝑥)⊆𝐾−core(𝑥) for all 𝑥∈ℓ_∞ if and only if 𝐴∈(𝑐∶𝑓)_reg and lim_𝑛→∞sup_{𝑝∈ℕ𝑖}1𝑛+1^𝑛𝑘=01𝑘+1^𝑘𝑗=0𝑎_{𝑗+𝑝,𝑖}=1. (5.7)

Theorem 5.4. 𝐴∈(𝒮∩ℓ_∞∶𝑓)_reg if and only if 𝐴∈(𝑐∶𝑓)_reg and lim_{𝑛→∞𝑘∈𝐸}1𝑛+1^𝑛𝑗=01𝑅_𝑗^𝑗𝑖=0𝑟_𝑖𝑎_{𝑖+𝑝,𝑘}=0uniformlyin𝑝 (5.8) for every 𝐸⊆ℕ with natural density zero.

Proof. Necessity: Let 𝐴∈(𝑆∩ℓ_∞,𝑓)_reg. Then, 𝐴∈(𝑐,𝑓)_reg immediately follows from the fact that 𝑐⊂𝑆∩ℓ_∞. Now, define a sequence 𝑡=(𝑡_𝑘) for 𝑥∈ℓ_∞ as 𝑡_𝑘=𝑥_𝑘,𝑘∈𝐸,0,𝑘∉𝐸, (5.9) where 𝐸 is any subset of ℕ with 𝛿(𝐸)=0. Then, 𝑠𝑡−lim𝑡_𝑘=0 and 𝑡∈𝑆₀, and so we have 𝐴𝑡∈𝑓₀. On the other hand, since 𝐴t=∑_𝑘∈𝐸𝑎_𝑛𝑘𝑡_𝑘, the matrix 𝐵=(𝑏_𝑛𝑘) defined by 𝑏_𝑛𝑘=𝑎_𝑛𝑘,𝑘∈𝐸,0,𝑘∉𝐸, (5.10) for all 𝑛, must belong to the class (ℓ_∞,𝑓₀). Hence, the necessity of (5.8) follows from Proposition 4.5.
Sufficiency: Conversely, suppose that 𝐴∈(𝑐,𝑓)_reg and (5.8) holds. Let 𝑥∈𝑆∩ℓ_∞ and 𝑠𝑡−lim𝑥=ℓ. Write 𝐸={𝑘∶|𝑥_𝑘−ℓ|≥𝜀} for any given 𝜀>0 so that 𝛿(𝐸)=0. Since 𝐴∈(𝑐,𝑓)_reg and 𝑓−lim∑_𝑘𝑎_𝑛𝑘=1, we have 𝑓−lim(𝐴𝑥)=𝑓−lim_𝑘𝑎_𝑛𝑘𝑥_𝑘−ℓ+ℓ_𝑘𝑎_𝑛𝑘=𝑓−lim_𝑘𝑎_𝑛𝑘𝑥_𝑘−ℓ+ℓ=lim_𝑛→∞sup_{𝑝∈ℕ𝑘}1𝑛+1^𝑛𝑗=01𝑅_𝑗^𝑗𝑖=0𝑟_𝑖𝑎_{𝑖+𝑝,𝑘}𝑥_𝑘−ℓ+ℓ. (5.11) On the other hand, since _𝑘1𝑛+1^𝑛𝑗=01𝑅_𝑗^𝑗𝑖=0𝑟_𝑖𝑎_{𝑖+𝑝,𝑘}𝑥_𝑘−ℓ≤‖𝑥‖_𝑘∈𝐸1𝑛+1^𝑛𝑗=01𝑅_𝑗^𝑗𝑖=0𝑟_𝑖𝑎_{𝑖+𝑝,𝑘}+𝜀‖𝐴‖, (5.12) condition (5.8) implies that lim_{𝑛→∞𝑘}1𝑛+1^𝑛𝑗=01𝑅_𝑗^𝑗𝑖=0𝑟_𝑖𝑎_{𝑖+𝑝,𝑘}𝑥_𝑘−ℓ=0uniformlyin𝑝. (5.13) Hence, 𝑓−lim(𝐴𝑥)=𝑠𝑡−lim𝑥; that is, 𝐴∈(S∩𝑚,𝑓)_reg, which completes the proof.

Similarly, 𝑟_𝑖=1 for all 𝑖 since 𝑅 is reduced to Cesàro matrix, see [18].

Theorem 5.5. 𝐴∈(𝑆∩ℓ_∞∶𝑓)_reg if and only if 𝐴∈(𝑐∶𝑓)_reg and lim_{𝑛→∞𝑖∈𝐸}1𝑛+1^𝑛𝑘=01𝑘+1^𝑘𝑗=0𝑎_{𝑗+𝑝,𝑖}=0uniformlyin𝑝 (5.14) for every 𝐸⊆ℕ with natural density zero.

Theorem 5.6. 𝐵_𝑅−core(𝐴𝑥)⊆𝑠𝑡−core(𝑥) for all 𝑥∈ℓ_∞ if and only if 𝐴∈(𝒮∩ℓ_∞∶𝑓)_reg and (5.2) holds.

Proof. Necessity: Let 𝐵_𝑅−core(𝐴𝑥)⊆𝑠𝑡−core(𝑥) for all 𝑥∈ℓ_∞. Then, 𝜏^∗(𝐴𝑥)≤𝛽(𝑥) for all 𝑥∈ℓ_∞, where 𝛽(𝑥)=𝑠𝑡−limsup𝑥. Hence, since 𝛽(𝑥)=𝑠𝑡−limsup𝑥≤𝐿(𝑥) for all 𝑥∈ℓ_∞ (see [13]), we have (5.2) from Theorem 5.2. Furthermore, one can also easily see that −𝛽(−𝑥)≤−𝜏^∗(−𝐴𝑥)≤𝜏^∗(𝐴𝑥)≤𝛽(𝑥), that is, 𝑠𝑡−liminf𝑥≤−𝜏^∗(−𝐴𝑥)≤𝜏^∗(𝐴𝑥)≤𝑠𝑡−limsup𝑥. (5.15) If 𝑥∈𝒮∩ℓ_∞, then 𝑠𝑡−liminf𝑥=𝑠𝑡−limsup𝑥=𝑠𝑡−lim𝑥. Thus, the last inequality implies that 𝑠𝑡−lim𝑥=−𝜏^∗(−𝐴𝑥)=𝜏^∗(𝐴𝑥)=𝑓−lim𝐴𝑥, that is, 𝐴∈(𝒮∩ℓ_∞∶𝑓)_reg.
Sufficiency: Conversely, assume that 𝐴∈(𝒮∩ℓ_∞∶𝑓)_reg and (5.2) hold. If 𝑥∈ℓ_∞, then 𝛽(𝑥) is finite. Let 𝐸 be a subset of ℕ defined by 𝐸={𝑙∶𝑥_𝑖>𝛽(𝑥)+𝜀} for a given 𝜀>0. Then it is obvious that 𝛿(𝐸)=0 and 𝑥_𝑖≤𝛽(𝑥)+𝜀 if 𝑙∉𝐸.
For any real number 𝜆 we write 𝜆⁺=max{𝜆,0} and 𝜆⁻=max{−𝜆,0} whence |𝜆|=𝜆⁺+𝜆⁻, 𝜆=𝜆⁺−𝜆⁻ and |𝜆|−𝜆=2𝜆⁻. Now, we can write _𝑗𝑐_𝑛𝑗𝑝𝑥_𝑗=_𝑗<𝑗0𝑐_𝑛𝑗𝑝𝑥_𝑗+_𝑗≥𝑗0𝑐_𝑛𝑗𝑝𝑥_𝑗=_𝑗<𝑗0𝑐_𝑛𝑗𝑝𝑥_𝑗+_𝑗≥𝑗0𝑐⁺_𝑛𝑗𝑝𝑥_𝑗−_𝑗≥𝑗0𝑐⁻_𝑛𝑗𝑝𝑥_𝑗≤‖𝑥‖_𝑗<𝑗0𝑐_𝑛𝑗𝑝+_{𝑗≥𝑗0𝑗∉𝐸}𝑐⁺_𝑛𝑗𝑝𝑥_𝑗+_{𝑗≥𝑗0𝑗∈𝐸}𝑐⁺_𝑛𝑗𝑝𝑥_𝑗+‖𝑥‖_𝑗≥𝑗0𝑐_𝑛𝑗𝑝−𝑐_𝑛𝑗𝑝≤‖𝑥‖_𝑗<𝑗0𝑐_𝑛𝑗𝑝+𝛽(𝑥)+𝜀_{𝑗≥𝑗0𝑗∉𝐸}𝑐_𝑛𝑗𝑝+‖𝑥‖_{𝑗≥𝑗0𝑗∈𝐸}𝑐_𝑛𝑗𝑝+‖𝑥‖_𝑗≥𝑗0𝑐_𝑛𝑗𝑝−𝑐_𝑛𝑗𝑝. (5.16) By applying the operator limsup_𝑛→∞sup_𝑝∈ℕ and using the hypothesis, we obtained that 𝜏^∗(𝐴𝑥)≤𝛽(𝑥)+𝜀. Since 𝜀 is arbitrary, we conclude that 𝜏^∗(𝐴𝑥)≤𝛽(𝑥) for all 𝑥∈ℓ_∞, that is, 𝐵_𝑅−core(𝐴𝑥)⊆𝑠𝑡−core(𝑥) for all 𝑥∈ℓ_∞ and the proof is complete. Now if 𝑟_𝑖=1 for all 𝑖, then 𝑅 is reduced to Cesàro matrix and we have 𝐵_𝐶−core(𝐴𝑥)⊆𝑠𝑡−core(𝑥),∀𝑥∈ℓ_∞ifandonlyif𝐴∈𝑆∩ℓ_∞∶𝑓_reg (5.17) and (5.7) holds, see [18].

Acknowledgment

The authors would like to thank the referees for much constructive criticism and attention for details.