Abstract and Applied Analysis
Volume 2013, Issue 1 783731
Research Article
Open Access

Almost Sequence Spaces Derived by the Domain of the Matrix Ar

Ali Karaisa, 

Corresponding Author

Ali Karaisa

Department of Mathematics-Computer Science, Faculty of Sciences, Necmettin Erbakan University, Meram Yerleşkesi, Meram, 42090 Konya, Turkey konya.edu.tr

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Ümıt Karabıyık, 

Ümıt Karabıyık

Department of Mathematics-Computer Science, Faculty of Sciences, Necmettin Erbakan University, Meram Yerleşkesi, Meram, 42090 Konya, Turkey konya.edu.tr

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First published: 10 November 2013
https://doi.org/10.1155/2013/783731
Citations: 2
Academic Editor: Feyzi Başar

Abstract

By using Ar, we introduce the sequence spaces , , and of normed space and BK-space and prove that , and are linearly isomorphic to the sequence spaces f, f0, and fs , respectively. Further, we give some inclusion relations concerning the spaces , , and the nonexistence of Schauder basis of the spaces fs and is shown. Finally, we determine the β- and γ-duals of the spaces and . 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 ℓp 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 | xk| and , 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 = (ank) an infinite matrix of real or complex numbers ank, 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 = (xk) ∈ μ the sequence Ax = (Ax) n, the A-transform of x is in γ, where
()
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 μA of an infinite matrix A in a sequence space μ is defined by
()

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 in [2], and in [3], μG = Z(u, v; μ) in [4], and cΛ = cλ in [5], and and 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 = (xk) and xk⩾0 for every k, (ii) ϕ(xσ(k)) = ϕ(xk), 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 = (xk) ∈ 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 − lim x = α. In other words, f − lim xk = α uniformly in n if and only if
()
The characterization given above was proved by Lorentz in [11]. We denote the sets of all almost convergent sequences f and series fs by
()
where
()
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|tmn(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 and which are the matrix domains of the Ar matrix in the almost convergent sequence and series spaces fs and f, respectively. In addition, we give some inclusion relations concerning the spaces , , and the non-existence of Schauder basis of the spaces fs and 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 and and characterize the classes , , and , 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 , , and Derived by the Domain of the Matrix Ar

In the present section, we introduce the sequence spaces , , and as the set of all sequences such that Ar-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 , , and . Recently, Aydın and Başar [14] studied the sequence spaces and :
()
where Ar denotes the matrix defined by
()
Now we introduce the sequence spaces , , and as the sets of all sequences such that their Ar-transforms are in the spaces f, f0, and fs, respectively; that is,
()
We can redefine the spaces , , and by the notation of (2):
()
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 = (yk), which will be frequently used, as the Ar-transform of a sequence x = (xk); that is,
()

Theorem 2. The spaces and have no Schauder basis.

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

Theorem 3. The following statements hold.

  • (i)

    The sets and are linear spaces with the coordinatewise addition and scalar multiplication which are BK-spaces with the norm

    ()

  • (ii)

    The set is a linear space with the coordinatewise addition and scalar multiplication which is a BK-space with the norm

    ()

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 is normal, Theorem 4.3.2 of Wilansky [17, p.61] gives the fact that the spaces and 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 , , and are linearly isomorphic to the sequence spaces f, f0, and fs, respectively; that is, , , and .

Proof. To prove the fact that , we should show the existence of a linear bijection between the spaces and f. Consider the transformation T defined with the notation of (2) from to f by x ↦ y = Tx = Arx. The linearity of T is clear. Further, it is clear that x = θ whenever Tx = θ, and hence, T is injective.

Let , and define the sequence x = (xk(r)) by

()
whence
()
which implies that . As a result, T is surjective. Hence, T is a linear bijection which implies that the spaces and f are linearly isomorphic, as desired. Similarly, the isomorphisms and can be proved.

Theorem 5. The inclusion strictly holds.

Proof. Let x = (xk) ∈ c. Since c ⊂ f, x ∈ f. Because Ar is regular for 0 < r < 1, Arx ∈ c. Therefore, since lim Arx = f − lim Arx, we see that . So we have that the inclusion holds. Further, consider the sequence t = (tk(r)) defined by tk(r) = (2k + 1)/(1 + rk)(−1)k ∀ k ∈ ℕ. Then, since Art = (−1) n ∈ f, . One can easily see that t ∉ f. Thus, , and this completes the proof.

Theorem 6. The sequence spaces and ℓ∞ overlap, but neither of them contains the other.

Proof. Let us consider the sequence u = (uk(r)) defined by uk(r) = 1/(1 + rk) for all ℕ. Then, since Aru = e ∈ f, . It is clear that u ∈ ℓ∞. This means that the sequence spaces and ℓ∞ are not disjoint. Now, we show that the sequence space and ℓ∞ do not include each other. Let us consider the sequence t = (tk(r)) defined as in proof of Theorem 5 above and z = (zk(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 + rk)’s are increasing by factors of 10. Then, since Art = (−1) n ∈ f, , but t ∉ ℓ∞. Therefore, . Also, the sequence since Arz = (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 . Hence, the sequence spaces and ℓ∞ overlap, but neither of them contains the other. This completes the proof.

Theorem 7. Let the spaces , , and be given. Then,

  • (i)

    strictly hold;

  • (ii)

    strictly hold.

Proof. (i) Let which means that Arx ∈ f0. Since f0 ⊂ f, Arx ∈ f. This implies that . Thus, we have .

Now, we show that this inclusion is strict. Let us consider the sequence u = (uk(r)) defined as in proof of Theorem 6 for all k ∈ ℕ. Consider the following:

()
which means that ; that is to say, the inclusion is strict.
  • (ii)

    Let which means that Arx ∈ c. Since c ⊂ f, Arx ∈ f. This implies that . Thus, we have . Furthermore, let us consider the sequence t = {tk(r)} defined as in proof of Theorem 5 for all k ∈ ℕ. Then, since Art = (−1) n ∈ f∖c, . This completes the proof.

3. Certain Matrix Mappings on the Sets , and Some Duals

In this section, we will characterize some matrix transformations between the spaces of Ar almost convergent sequence and almost convergent series in addition to paranormed and classical sequence spaces after giving β- and γ-duals of the spaces and . 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−1*Y = {a ∈ ω : a · x ∈ Y} and
()
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 Xα, Xβ, and Xγ, are defined by
()
It is obvious that Xα ⊂ Xβ ⊂ Xγ. Also, it can easily be seen that the inclusions Xα ⊂ Yα, Xβ ⊂ Yβ, and Xγ ⊂ Yγ hold whenever Y ⊂ X.

Lemma 8 (see [18].)A = (ank)∈(f : ℓ∞) if and only if

()

Lemma 9 (see [18].)A = (ank)∈(f : c) if and only if (18) holds and there are α, αk ∈ ℂ such that

()
()
()

Theorem 10. Define the sets and by

()
where Δ(ak/(1 + rk)) = ak/(1 + rk) − ak+1/(1 + rk+1) for all k ∈ ℕ. Then .

Proof. Take any sequence a = (ak) ∈ ω, and consider the following equality:

()
where is
()
for all k, n ∈ ℕ. Thus, we deduce from (23) that ax = (akxk) ∈ bs whenever if and only if Ty ∈ ℓ∞ whenever y = (yk) ∈ f where is defined in (24). Therefore, with the help of Lemma 8,  .

Theorem 11. The β-dual of the space is the intersection of the sets

()
where for all k ∈ ℕ. Then, .

Proof. Let us take any sequence a ∈ ω. By (23), ax = (akxk) ∈ cs whenever if and only if Ty ∈ c whenever y = (yk) ∈ f. It is obvious that the columns of that matrix T in c where defined in (24), we derive the consequence by Lemma 9 that .

Theorem 12. The γ-dual of the space is the intersection of the sets

()
In other words, we have .

Proof. We obtain from (23) that ax = (akxk) ∈ bs whenever if and only if Ty ∈ ℓ∞ whenever y = (yk) ∈ fs, where is defined in (24). Therefore, by Lemma 19(viii), .

Theorem 13. Define the set by

()
Then, .

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:
()
for all k, n ∈ ℕ. Assume that the infinite matrices A = (ank) and B = (bnk) map the sequences x = (xk) and y = (yk) which are connected with relation (10) to the sequences u = (un) and v = (vn), respectively; that is,
()
()
One can easily conclude here that the method A is directly applied to the terms of the sequence x = (xk), while the method B is applied to the Ar-transform of the sequence x = (xk). So the methods A and B are essentially different.
Now, suppose that the matrix product BAr 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 BAr exists and is equal to A and (BAr)x = B(Arx) formally holds if one side exists. This statement is equivalent to the following relation between the entries of the matrices A = (ank) and B = (bnk) which are connected with the relation
()

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 and .

Theorem 14. Suppose that the entries of the infinite matrices A = (ank) and B = (bnk) are connected with relation (31) for all k, n ∈ ℕ, and let λ be any given sequence space. Then, if and only if

()

Proof. Suppose that A = (ank) and B = (bnk) are connected with the relation (31), and let λ be any given sequence space, and keep in mind that the spaces and f are norm isomorphic.

Let , and take any sequence , and keep in mind that y = Arx. Then, ; that is, (32) holds for all n ∈ ℕ and BAr exists which implies that (bnk) k∈ℕ ∈ ℓ1 = fβ for each n ∈ ℕ. Thus, By exists for all y ∈ f, and thus, we have m → ∞ in the equality

()
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 (bnk) k∈ℕ ∈ ℓ1 for all n ∈ ℕ which gives together with for each n ∈ ℕ that Ax exists. Then, we obtain from the equality
()
for all m, n ∈ ℕ, as m → ∞, that Ax = By, and this shows that .

Theorem 15. Suppose that the entries of the infinite matrices E = (enk) and F = (fnk) are connected with the relation

()
for all m, n ∈ ℕ and λ is any given sequence space. Then, if and only if F ∈ (λ : f).

Proof. Let x = (xk) ∈ λ, and consider the following equality:

()
for all k, m, n ∈ ℕ, which yields as m → ∞ that 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 = (ank) and B = (bnk) are connected with the relation (31). Then, if and only if B ∈ (fs : λ) and 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 = (enk) and F = (fnk) are connected with relation (35). Then, 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 , , and may be derived by replacing the entries of A and B by those ofB = A(Ar) −1, and F = ArE, 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 = (ank) be an infinite matrix. Then, the following statements hold:

  • (i)

    A ∈ (c0(p) : f)   if and only if

    ()

  • (ii)

    A ∈ (c(p) : f)   if and only if (37) and

    ()

  • (iii)

    A ∈ (ℓ∞(p) : f)    if and only if (37) and

    ()

Lemma 19. Let A = (ank) be an infinite matrix. Then, the following statements hold:

  • (i)

    (Duran, [19]) A ∈ (ℓ∞ : f) if and only if (18) holds and

    ()
    ()

  • (ii)

    (King, [20]) A ∈ (c : f) if and only if (18), (40) hold and

    ()

  • (iii)

    (Başar and Çolak, [21]) A ∈ (cs : f) if and only if (40) holds and

    ()

  • (iv)

    (Başar and Çolak, [21]) A ∈ (bs : f) if and only if (40), (43) hold and

    ()
    ()

  • (v)

    (Duran, [19]) A ∈ (f : f) if and only if (18), (40), and (42) hold and

    ()

  • (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 and

    ()

  • (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 and

    ()

  • (x)

    (Başar, [22]) A ∈ (fs : fs) if and only if (45), (48) hold and

    ()

  • (xi)

    (Başar and Çolak, [21]) A ∈ (cs : fs) if and only if (48) holds;

  • (xii)

    (Başar, [25]) A ∈ (f : cs) if and only if

    ()
    ()
    ()
    ()

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

Corollary 20. Let A = (ank) be an infinite matrix. Then, the following statements hold.

  • (i)

    if and only if for all n ∈ ℕ and (40), (44) hold with instead of ank, (46) holds with instead of a(n, k, m), and (45) holds with instead of a(n, k).

  • (ii)

    if and only if for all n ∈ ℕ and (19), (43), (44), and (47) hold with instead of ank.

  • (iii)

    if and only if for all n ∈ ℕ and (43) and (44) hold with instead of ank.

  • (iv)

    if and only if for all n ∈ ℕ and (45), (48), and (49) hold with instead of a(n, k).

  • (v)

    if and only if (48) holds with instead of a(n, k).

  • (vi)

    if and only if (44) holds with instead of ank and (45), (48) hold with instead of a(n, k).

  • (vii)

    if and only if (45), (48), and (49) hold with instead of a(n, k).

Corollary 21. Let A = (ank) be an infinite matrix. Then, the following statements hold.

  • (i)

    if and only if (37)and (38) hold with instead of a(n, k, m).

  • (ii)

    if and only if (37) holds with instead of a(n, k, m).

  • (iii)

    if and only if (37) and (39) hold with instead of a(n, k, m).

Corollary 22. Let A = (ank) be an infinite matrix. Then, the following statements hold.

  • (i)

    if and only if for all n ∈ ℕ and (18) holds with instead of ank.

  • (ii)

    if and only if for all n ∈ ℕ and (18), (19), (20), and (21) hold with instead of ank.

  • (iii)

    if and only if for all n ∈ ℕ and (50),(53) hold with instead of a(n, k) and (51),(52) hold with instead of ank.

Corollary 23. Let A = (ank) be an infinite matrix. Then, the following statements hold.

  • (i)

    if and only if (18), (40) hold with instead of ank and (41) holds with instead of a(n, k, m).

  • (ii)

    if and only if (18), (40), and (46) hold with instead of a(n, k, m) and (42) holds with instead of ank.

  • (iii)

    if and only if (18), (40), and (42) hold with instead of ank.

  • (iv)

    if and only if (40), (43), and (44) hold with instead of ank and (45) holds with instead of a(n, k).

  • (v)

    if and only if (40), (44) hold with instead of ank, (46) holds with instead of a(n, k, m), and (45) holds with instead of a(n, k).

  • (vi)

    if and only if (40) and (43) hold with instead of ank.

Remark 24. Characterization of the classes , , , and 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.

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