- Email: [email protected]

` THE SPACES OF CESARO ALMOST CONVERGENT SEQUENCES AND CORE THEOREMS∗ Kuddusi Kayaduman Science and Art Faculty, Gaziantep University, Gaziantep/Turkey E-mail: [email protected]

Mehmet S ¸ eng¨ on¨ ul Science and Art Faculty, Nev¸sehir University, Nev¸sehir/Turkey E-mail: [email protected]

Abstract As known, the method to obtain a sequence space by using convergence ﬁeld of an inﬁnite matrix is an old method in the theory of sequence spaces. However, the study of convergence ﬁeld of an inﬁnite matrix in the space of almost convergent sequences is so new (see [15]). The purpose of this paper is to introduce the new spaces f$ and f$0 consisting of all sequences whose Ces` aro transforms of order one are in the spaces f and f0 , respectively. Also, in this paper, we show that f$ and f$0 are linearly isomorphic to the spaces f and f0 , respectively. The β- and γ-duals of the spaces f$ and f$0 are computed. Furthermore, the classes (f$ : μ) and (μ : f$) of inﬁnite matrices are characterized for any given sequence space μ, and determined the necessary and suﬃcient conditions on a matrix A to satisfy BC -core(Ax) ⊆ K-core(x), K-core(Ax) ⊆ BC -core(x), BC -core(Ax) ⊆ BC -core(x), BC -core(Ax) ⊆ st-core(x) for all x ∈ ∞ . Key words almost convergence; matrix domain of a sequence space; β- and γ-duals and matrix transformations; core theorems; isomorphism 2010 MR Subject Classiﬁcation

1

46A45; 40J05; 40C05

Introduction

By w, we shall denote the space of all real or complex valued sequences. Each linear subspace of w is called a sequence space. We shall write ∞ , c, c0 , 1 , cs and bs for the spaces of all bounded, convergent, null sequences, absolutely convergent series, convergent series and bounded series, respectively. Let λ and μ be two sequence spaces and A = (ank ) be an inﬁnite matrix of real or complex numbers ank , where n, k ∈ N = {0, 1, 2, · · ·}. Then, we can say that A deﬁnes a matrix mapping from λ to μ, and we denote it by writing A : λ → μ, if for every sequence x = (xk ) ∈ λ the sequence Ax = {(Ax)n } , the A-transform of x, is in μ where ank xk (n ∈ N). (1.1) (Ax)n = k ∗ Received

April 18, 2011; revised November 14, 2011.

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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 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, we have Ax = {(Ax)n }n∈N ∈ μ for all x ∈ λ. The matrix domain λA of an inﬁnite matrix A in a sequence space λ is deﬁned by λA = {x = (xk ) ∈ w : Ax ∈ λ} .

(1.2)

If we take λ = c, then cA is called convergence domain of A. We write the limit of Ax as limA x = lim ank xk , and A is called regular if limA x = lim x for each convergent sequence n→∞ k x. By using the matrix domain of a particular limitation method so many sequence spaces have been built and published in famous maths journals. By reviewing the literature, one can reach them easily (for instance, see Wang [42], Ng and Lee [37], Malkowsky [31], Altay and Ba¸sar [3–8], Malkowsky and Sava¸s [32], Ba¸sarır [18], Aydın and Ba¸sar [9–13], Ba¸sar, Altay and Mursaleen [17], Kiri¸s¸ci and Ba¸sar [27], S ¸ eng and Ba¸sar [41], Altay [1], Polat and Ba¸sar [39] and, Malkowsky, Mursaleen and Suantai [33]). Finally, the new technique for deducing certain topological properties, such as AB-, KB-, AD-properties, solidity and monotonicity etc., and determining the α-, β- and γ-duals of the domain of a triangle matrix in a sequence space was given by Altay and Ba¸sar [2]. Furthermore, quite recently, Ba¸sar and Kiri¸s¸ci [15] 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). The rest of this paper is structured as follows: Some required deﬁnitions and consequences related with almost convergent sequences space and core of a sequence of real numbers are given in Section 2. In Section 3, we introduce the sequence space f$ and f$0 which are BK-spaces. In Section 4, we state and prove the theorems determining the β- and γ-duals of the sequence spaces f$ and f$0 . In Section 5, the classes (λ : μ) and (μ : λ) of inﬁnite matrices of real numbers are characterized, where λ = {f$, f$0 } and μ is any sequence space. Section 6 is devoted some core theorems related with the space f$.

2

Preliminaries

Lorentz [29] introduced that a sequence x = (xk ) ∈ ∞ is said to be almost convergent n xk+p to the generalized limit α if and only if lim n+1 = α uniformly in p and is denoted by n→∞ k=0

f − lim x = α. The space of all almost convergent and almost null sequences are denoted by f and f0 , respectively. That is, f=

n xk+p = α uniformly in p x = (xk ) ∈ w : lim n→∞ n+1 k=0

and f0 =

n xk+p = 0 uniformly in p . x = (xk ) ∈ w : lim n→∞ n+1 k=0

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Lorentz [29] also obtained the necessary and suﬃcient conditions for an inﬁnite matrix to contain f in its convergence domain. These conditions on an inﬁnite matrix A = (ank ) consist of the standard Silverman Toeplitz conditions for regularity plus the condition lim |ank − n→∞ k

an,k+1 | = 0. Such matrices are called strongly regular. One of the best known strongly regular matrix is C, the Ces`aro matrix of order one which is a lower triangular matrix deﬁned by ⎧ ⎨ 1 , 0 ≤ k ≤ n, cnk = n + 1 ⎩ 0, k>n for all n, k ∈ N. Let K be a subset of N. The natural density δ of K is deﬁned by δ(K) = lim

1 |{k n→∞ n

≤

n : k ∈ K}|, where the vertical bars indicate the number of elements in the enclosed set. The sequence x = (xk ) is said to be statistically convergent to the number l if for every ε, δ({k : |xk − l| ≥ ε}) = 0 (see [25]). In this case, we write st- lim x = l. We shall 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 [20], [25] and others). Let us consider the following functionals deﬁned on ∞ : l(x) = lim inf xk , k→∞

L(x) = lim sup xk , k→∞

1 qσ (x) = lim sup sup xσi (n) , p→∞ n∈N p + 1 i=0 p

1 xn+i . n∈N p + 1 p

L∗ (x) = lim sup sup p→∞

i=0

In [35], the σ-core of a real bounded sequence x is deﬁned as the closed interval [−qσ (−x), qσ (x)], and also the inequalities qσ (Ax) ≤ L(x) (σ-core of Ax ⊆ K-core of x), qσ (Ax) ≤ qσ (x) (σ-core of Ax ⊆ σ-core of x), for all x ∈ ∞ , was studied. Here the Knopp core, in short K-core of x is the interval [l(x), L(x)] (see [21]). When σ(n) = n + 1, since qσ (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)] (see [38]). The concepts of B-core and σ-core were studied by many authors [22–24, 26, 35, 36, 38]. Recently, Fridy and Orhan [25] 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 is the closed interval [st- lim inf x, st- lim sup x] and also determined necessary and suﬃcient conditions for a matrix A to yield K-core(Ax) ⊆ st-core(x) for all x ∈ ∞ . Deﬁnition 2.1 Let x ∈ ∞ . Then, BC -core of x is deﬁned by the closed interval [−T ∗ (−x), T ∗ (x)], where ∗

T (x) = lim sup sup

n

n→∞ p∈N k=0

1 xj+p . n + 1 j=0 k + 1 k

Therefore, it is easy to see that BC -core of x is α if and only if f − lim x = α.

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

3

The Sequence Spaces f$ and f$0

Now, we introduce the new spaces f$ and f$0 , as the sets of all sequences such that their C-transforms are in the spaces f and f0 , respectively, that is f$ = and f$0 =

n x = (xk ) ∈ w : lim n→∞

k=0

n x = (xk ) ∈ w : lim n→∞

k=0

1 xj+p = α uniformly in p n + 1 j=0 k + 1 k

k 1 xj+p = 0 uniformly in p . n + 1 j=0 k + 1

With the notation of (1.2), we can write f$ = fC and f$0 = (f0 )C . Deﬁne the sequence y = (yk ), which will be frequently used, as the C-transform of a sequence x = (xk ), i.e., yn =

n xk n+1

for all n ∈ N.

(3.1)

k=0

Now, we begin with the following theorem. Theorem 3.1 The sequence spaces f$ and f$0 are linearly isomorphic to the spaces f and f0 , respectively. Proof Since the fact “the spaces f$0 and f0 are linearly isomorphic” can also be proved in the similar way, we consider only the spaces f$ and f . In order to prove the fact 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 (3.1), from f$ to f by x → y = T x. The linearity of T is clear. Further, it is trivial that x = θ = (0, 0, · · ·) whenever T x = θ and hence T is injective. Let y = (yk ) ∈ f$ and deﬁne the sequence x = (xk ) by xk = (k + 1)yk − kyk−1 (k ∈ N). Then, we have lim

n→∞

n j=0

j j n 1 xi+k 1 (k + 1)yi+k − kyk−1+i = lim n + 1 i=0 j + 1 n→∞ j=0 n + 1 i=0 j+1

1 yj+k uniformly in k n→∞ n + 1 j=0 n

= lim

= f − lim yk , $ Consequently, we see that T is surjective. Hence, T is linear bijecwhich shows that x ∈ f. tion which therefore shows that the spaces f$ and f are linearly isomorphic, as desired. This completes the proof. 2

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Theorem 3.2 Deﬁne the norm . . n j . 1 xi+k .. . . xf$ = sup . n+1 j + 1. n j=0

(3.2)

i=0

Then the sets f$ and f$0 are linear spaces with the co-ordinatewise addition and scalar multiplication which are the BK-spaces with the norm in (3.2). Proof The ﬁrst part of the theorem is easy. We will prove second part of the theorem. Since (3.1) holds and f and f0 are BK-spaces [43] with the norm .∞ by Example 7.3.2 (b) of Boos [19] also the matrix C is normal, Theorem 4.3.2 of Wilansky [43] gave the fact that the spaces f$ and f$0 are BK-spaces. 2 In Theorem 3.3, we use some techniques due to M´oricz and Rhoades [34]. Theorem 3.3 Deﬁne the sequences (αn ) and (βn ) by αn = inf

p>0

p+n k=p

1 xj , n + 1 j=0 k + 1 k

βn = sup p>0

p+n k=p

1 xj n + 1 j=0 k + 1 k

for all n ∈ N. Then, αn ≤ βn for each n ∈ N and (i) The sequence (α2n ) is non-decreasing. (ii) The sequence (β2n ) is non-increasing. Proof It is trivial that αn = inf

p>0

p+n k=p

1 xj 1 xj ≤ sup = βn n + 1 j=0 k + 1 p>0 n + 1 j=0 k + 1 p+n

k

k

k=p

for each n ∈ N. Since part (ii) can be proved in the similar way, we prove only part (i). n+1 p+2

k xj p>0 2n+1 + 1 j=0 k + 1 k=p ⎡ ⎤ n+1 p+2

k n 1 + 1 x 2 j ⎣ ⎦ = inf n+1 p>0 2 + 1 2n + 1 k + 1 k=p j=0 ⎤ ⎡ n+1 p+2

k 1 x 1 j ⎣ (2n + 1)⎦ = inf n+1 p>0 2 + 1 2n + 1 k + 1 k=p j=0 ⎡ n p+2

k 1 xj 1 ⎣ (2n + 1) = inf n+1 p>0 2 + 1 2n + 1 k + 1 k=p j=0 ⎤ n+1 p+2

k xj 1 (2n + 1)⎦ + n 2 +1 k + 1 n j=0

α2n+1 = inf

1

k=p+2 +1

≥

1 [(2n + 1)α2n + (2n + 1)α2n ] = α2n . 2n+1 + 2

This step completes the proof. Theorem 3.4 lim (β2r − α2r ) = 0 if and only if x ∈ f$. r→∞

2

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Proof Suppose that lim (β2n − α2n ) = 0. For each n choose r to satisfy 2r ≤ n ≤ 2r+1 . n→∞ r We may write n in a dyadic representation of the form n = ni 2i , where each ni is 0 or 1, i=0

i = 0, 1, 2, · · · , r − 1, and nr = 1. Then,

1 n+1

p+n i i=p j=0

r

p+

xj 1 = i+1 n+1 ⎡

ni 2i

i=0

i=p

i xj i +1 j=0

0 p+n i 02

1 ⎣ 1 = n + 1 20 + 1

i=p

j=0

xj (20 + 1) i+1

2 n

i 22 xj 1 (21 + 1) + 1 2 +1 i+1 1 j=0

i=n1 2

+··· 1 + r 2 +1 9

r n r2

⎤

i xj (2r + 1)⎦ i + 1 r−1 j=0

i=nr−1 2

: 1 (2r + 1)α2r + (2r−1 + 1)α2r−1 + · · · + 20 α20 n+1 r 1 nj (2j + 1)αj , ≥ n + 1 j=0 =

since nj ∈ {0, 1} and hence inf

p>0

p+n k=p

1 xj 1 = αn ≥ nj (2j + 1)α2j n + 1 j=0 k + 1 n + 1 j=0 k

r

and sup p>0

p+n k=p

1 xj 1 = βn ≤ nj (2j + 1)β2j . n + 1 j=0 k + 1 n + 1 j=0 k

r

Thus

1 nj (2j + 1)(β2j − α2j ). n + 1 j=0 r

0 ≤ βn − αn ≤

(3.3) k

+1) then, T is regular If T = (tnk ) is the lower triangular matrix with nonzero entries tnk = nk (2 n+1 matrix, so that lim (β2r − α2r ) = 0. From equality (3.3), we see that lim (βn − αn ) = 0. r→∞

n→∞

Conversely, let us take x ∈ f$. Then, since lim

n→∞

p+n k=p

1 xj =α n + 1 j=0 k + 1 k

implies lim αn = inf lim

n→∞

p>0 n→∞

p+n k=p

1 xj =α n + 1 j=0 k + 1 k

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and lim βn = sup lim

n→∞

p>0 n→∞

p+n

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1 xj = α, n + 1 j=0 k + 1 k

k=p

we have lim αn − lim βn = lim (αn − βn ) = 0.

n→∞

n→∞

n→∞

If we take n = 2 , then the proof of suﬃciency is obtained. This step completes the proof. 2 p

4

Some Duals of the Space f$

In this section, by using techniques in [2], we state and prove the theorems determining $ the β- and γ-duals of the spaces f$0 and f. For the sequence spaces λ and μ, deﬁne the set S(λ, μ) by S(λ, μ) = {z = (zk ) ∈ w : xz = (xk zk ) ∈ μ for all x = (xk ) ∈ λ} .

(4.1)

With the notation of (4.1), the α-, β- and γ-duals of a sequence space λ, which are respectively denoted by λα , λβ and λγ are deﬁned by λα = S(λ, 1 ), λβ = S(λ, cs) and λγ = S(λ, bs). We begin with following two lemmas which are needed in proving Theorems 4.3 and 4.4. Lemma 4.1 A ∈ (f : ∞ ) if and only if |ank | < ∞. (4.2) sup n∈N k

Lemma 4.2 A ∈ (f : c) if and only if lim ank = a, n→∞

lim ank = ak ; k ∈ N

n→∞

and lim

n→∞

(4.3)

k

|Δ(ank − ak )| = 0.

(4.4)

(4.5)

k

Theorem 4.3 The γ-dual of the spaces f$ and f$0 is the set d1 ∩ d2 , where |(k + 1)Δak | < ∞ d1 = a = (ak ) ∈ w :

(4.6)

k

and d2 = {a = (ak ) ∈ w : {(k + 1)ak } ∈ ∞ } .

(4.7)

Proof Deﬁne the matrix T = (tnk ) via the sequence a = (ak ) ∈ w by ⎧ ⎪ ⎪ ⎨ (k + 1)Δak , 0 ≤ k ≤ n − 1, tnk =

⎪ ⎪ ⎩

(n + 1)an ,

n = k,

0,

otherwise

(4.8)

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for all n, k ∈ N. Bearing in mind relation (3.1), we immediately derive that n

ak xk =

k=0

n−1

(k + 1)Δak yk + (n + 1)an yn = (T y)n

(n ∈ N).

(4.9)

k=0

From (4.9), we see that ax = (ak xk ) ∈ bs whenever x = (xk ) ∈ f$ if and only if T y ∈ ∞ whenever y = (yk ) ∈ f . Then, we derive by Lemma 4.1 that n

|(k + 1)Δak | < ∞

and {(n + 1)an } ∈ ∞ ,

k=0

which yields the desired result f$γ = f$0γ = d1 ∩ d2 . Theorem 4.4 Deﬁne the set d3 by |Δ [(k + 1)Δak − ak ]| < ∞ . d3 = a = (ak ) ∈ w :

2

k

Then f$β = d3 ∩ cs. Proof Consider equality (4.9), again. Thus, we deduce that ax = (ak xk ) ∈ cs whenever x = (xk ) ∈ f$ if and only if T y ∈ c whenever y = (yk ) ∈ f. It is obvious that the columns of that matrix T , deﬁned by (4.8), are in the space c. Therefore, we derive the consequence from Lemma 4.2 that f$β = d3 ∩ cs. 2

5

Some Matrix Mappings Related to the Space f$

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 Ba¸sar [14]. Some authors, such as Ba¸sar [14], Ba¸sar and C ¸ olak [16], Kuttner [28], Lorentz and Zeller [30] worked on the dual summability methods. Now, following Ba¸sar [14], we give a short survey about dual summability methods of the new type. Let us suppose that the inﬁnite matrices A = (ank ) and B = (bnk ) map the sequences x = (xk ) and y = (yk ) which are connected by relation (3.1) to the sequences z = (zn ) and t = (tn ), respectively, i.e., zn = (Ax)n = ank xk (n ∈ N) (5.1) k

and tn = (By)n =

bnk yk (n ∈ N).

(5.2)

k

It is clear here that the method B is applied to the C-transform of the sequence x = (xk ) while the method A is directly applied to the entries of the sequence x = (xk ). So, the methods A and B are essentially diﬀerent. Let us assume that the matrix product BC 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 (5.1), (5.2) are called dual summability methods of the new type if zn reduces to tn (or tn reduces to zn ) under the application of formal summation by parts. This leads us to the fact that BC exists and is equal to A and (BC)x = B(Cx) formally holds, if one side exists. This

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statement is equivalent to the following relation between the entries of the matrices A = (ank ) and B = (bnk ): ank :=

∞ j=k

1 bnj j+1

or bnk := (k + 1)(ank − an,k+1 ) = (k + 1)Δank

(5.3)

for all n, k ∈ N. Now, we give the following theorem concerning to the dual matrices of the new type. Theorem 5.1 Let A = (ank ) and B = (bnk ) be the dual matrices of the new type, μ be any given sequence space. Then, A ∈ (f$ : μ) if and only if B ∈ (f : μ) and {(n + 1)ank }n∈N ∈ c0

(5.4)

for every ﬁxed k ∈ N. Proof Suppose that A = (ank ) and B = (bnk ) are dual matrices of the new type, that is to say that (5.3) holds, and μ is any given sequence space. Let us keep in mind that the spaces f$ and f are linearly isomorphic. Let A ∈ (f$ : μ) and take any y = (yk ) ∈ f . Then, BC exists and (ank )k∈N ∈ d2 ∩ cs which yields that (bnk )k∈N ∈ 1 for each n ∈ N. Hence By exists for each y ∈ f and thus letting m → ∞ in the equality m

bnk yk =

k=0

m m k=0 j=k

1 bnj xk j+1

(5.5)

for all m, n ∈ N, we have by (5.3) that By = Ax which gives the result B ∈ (f : μ). Conversely, suppose that (5.4) holds for ever ﬁxed k ∈ N and B ∈ (f : μ), and take any x = (xk ) ∈ f$. Then, Ax exists. Therefore, we obtain from the equality m

ank xk =

k=0

m−1

m

k=0

k=0

(k + 1)Δank yk + manm ym =

bnk yk (n ∈ N),

(5.6)

as n → ∞ that Ax = By and this shows that A ∈ (f$ : μ). This completes the proof. 2 Theorem 5.2 Suppose that the entries of the inﬁnite matrices D = (dnk ) and E = (enk ) are connected with the relation enk =

n djk (n, k ∈ N) n +1 j=0

(5.7)

and μ be any given sequence space. Then, D ∈ (μ : f$) if and if only E ∈ (μ : f ). Proof Let x = (xk ) ∈ μ and consider the following equality with (5.7) n j=0

1 djk xk = enk xk (m, n, k ∈ N), n+1 m

m

k=0

k=0

which yields as m → ∞ that Dx ∈ f$ whenever x ∈ μ if and if only Ex ∈ f whenever x ∈ μ. This step completes the proof. 2 Now, right here, we give the following propositions which are obtained from Lemmas 4.2, 4.1 and Theorems 5.1, 5.2:

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Proposition 5.3 Let A = (ank ) be an inﬁnite matrix of real or complex numbers. Then, ⎧ . . .

. ∞ ⎪ ⎪ . . anj ⎪ ⎨ (1) lim .Δ − ak .. = 0, . n→∞ j+1 A = (ank ) ∈ (f$ : ∞ ) ⇔ . k . j=k ⎪ ⎪ ⎪ β ⎩ $ (2) {ank } ∈ f for all n ∈ N. Proposition 5.4 Let A = (ank ) be an inﬁnite matrix real or complex numbers. Then, ⎧ ∞ ⎪ anj ⎪ ⎪ (3) lim = a, ⎪ ⎪ n→∞ j +1 ⎪ ⎪ k j=k ⎪ ⎪ ∞ ⎪ ⎪ anj ⎪ ⎪ = ak for each k ∈ N, (4) lim ⎨ n→∞ j +1 j=k . A = (ank ) ∈ (f$ : c) ⇔ . ⎪ .

. ⎪ ∞ ⎪ . . ⎪ a nj ⎪ .Δ ⎪ − ak .. = 0, lim ⎪ (5) n→∞ . ⎪ j+1 ⎪ . ⎪ k . j=k ⎪ ⎪ ⎪ ⎩ (6) {a } $β for all n ∈ N. nk k∈N ∈ f Proposition 5.5 Let A = (ank ) be an inﬁnite matrix real or complex numbers. Then, . . ⎧ . . ∞ ⎪ . ⎪ anj .. ⎪ . ⎪ (7) sup < ∞, ⎪ . ⎪ j + 1 .. ⎪ n∈N . ⎪ k j=k ⎪ ⎪ ⎪ ∞ ⎨ anj $ (8) f − lim = αk exists for each ﬁxed k ∈ N, A = (ank ) ∈ (∞ : f ) ⇔ n→∞ j +1 ⎪ ⎪ j=k ⎪ . . ⎪ ⎪ .m . ∞ ⎪ . ⎪ . 1 a ⎪ n+i,j ⎪ . ⎪ (9) lim − αk .. = 0 uniformly in n. ⎪ . m→∞ ⎩ j+1 . i=0 m + 1 . k

j=k

Proposition 5.6 Let A = (ank ) be an inﬁnite matrix real or complex numbers. Then, . . ⎧ .∞ . ⎪ . ⎪ a nj .. ⎪ . ⎪ (10) sup ⎪ . . < ∞, ⎪ ⎪ n∈N k .j=k j + 1 . ⎪ ⎪ ⎪ ⎨ ∞ anj $ A = (ank ) ∈ (c : f) ⇔ (11) f − lim = αk exists for each ﬁxed k ∈ N, ⎪ n→∞ j+1 ⎪ ⎪ j=k ⎪ ⎪ ∞ ⎪ ⎪ anj ⎪ ⎪ = α. (12) f − lim ⎪ ⎩ n→∞ j+1 k j=k

6

Core Theorems

In this section, we give some core theorems related to the space f$. We need the following lemma due to Das [23] for the proof of next theorem. Lemma 6.1 Let C = cni (p) < ∞ and lim sup |cni (p)| = 0. Then, there is a n→∞ p∈N

y = (yi ) ∈ ∞ such that y ≤ 1 and lim sup sup n→∞ p∈N

i

cni (p)yi = lim sup sup n→∞ p∈N

i

|cni (p)|.

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Theorem 6.2 BC -core(Ax) ⊆ K-core(x) for all x ∈ ∞ if and only if A ∈ (c : f$)reg and . . . . n k 1 . . 1 . (6.1) lim sup aj+p,i .. = 1. . n→∞ p∈N n + 1 k + 1 . .k=0 i j=0 Proof Suppose ﬁrst that BC -core(Ax) ⊆ K-core(x) for all x ∈ ∞ . If x ∈ f$, then we have T ∗ (Ax) = −T ∗(−Ax). By this hypothesis, we get −L(−x) ≤ −T ∗ (−Ax) ≤ T ∗ (Ax) ≤ L(x). If x ∈ c, then L(x) = −L(−x) = lim x. So, we have f − lim Ax = T ∗ (Ax) = −T ∗ (−Ax) = lim x which implies that A ∈ (c, f$)reg . Now, let us consider the sequence C = (cni (p)) of inﬁnite matrices deﬁned by 1 1 aj+p,i n+1 k + 1 j=0 n

cni (p) =

k

for all n, i, p ∈ N.

k=0

Then, it is easy to see that the conditions of Lemma 6.1 are satisﬁed for the matrix sequence C. Thus, by using the hypothesis, we can write 1 ≤ lim inf sup |cni (p)| ≤ lim sup sup |cni (p)| n→∞ p∈N

= lim sup sup n→∞ p∈N

n→∞ p∈N

i

i

∗

cni (p)yi = T (Ay) ≤ L(y) ≤ y ≤ 1.

i

This gives the necessity of (6.1). Conversely, assume that A ∈ (c : f$)reg and (6.1) holds for all x ∈ ∞ . 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 i0 ∈ N such that xi < L(x) + ε for all i > i0 . Now, we can write cni (p)xi = cni (p)xi + (cni (p))+ xi − (cni (p))− xi i

i

≤ x

i≥i0

|cni (p)| + [L(x) + ε]

i

i

i≥i0

|cni (p)| + x

[|cni (p)| − cni (p)] .

i

Thus, by applying the lim sup sup and using hypothesis, we have T ∗ (Ax) ≤ L(x) + ε. This n→∞ p∈N

completes the proof, since ε is arbitrary and x ∈ ∞ . 2 The proof of the following two theorems are entirely analogous to the proof of Theorem 6.2. So, we omit the detail. Theorem 6.3 K-core(Ax) ⊆ BC -core(x) for all x ∈ ∞ if and only if A ∈ (f$ : c)reg and (6.1) holds. Theorem 6.4 BC -core(Ax) ⊆ BC -core(x) for all x ∈ ∞ if and only if A is f$- regular and (6.1) holds. $ reg if and only if A ∈ (c : f$)reg and Theorem 6.5 A ∈ (S ∩ ∞ : f) . . . n k 1 .. . 1 . (6.2) lim aj+p,i .. = 0 uniformly in p . n→∞ n+1 . k + 1 j=0 . i∈E k=0

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for every E ⊆ N with natural density zero. Proof Let A ∈ (S ∩ ∞ : f$)reg . Then, A ∈ (c : f$)reg immediately ⎧ follows from the fact ⎨ x , i ∈ E, i where E is that c ⊂ S ∩ ∞ . Now, deﬁne a sequence t = (ti ) for x ∈ ∞ as ti = ⎩ 0, i ∈ / E, any subset of N with δ(E) = 0. Then, st- lim ti = 0 and t ∈ S0 , so we have⎧At ∈ f$0 . On the ⎨ a , i ∈ E, ni for other hand, since (At)n = ani ti , the matrix B = (bni ) deﬁned by bni = ⎩ 0, i∈E i∈ /E all n, must belong to the class (∞ : f$0 ). Hence, the necessity of (6.2) follows from part (i) of Proposition 5.5. $ reg and (6.2) holds. Let x ∈ S ∩ ∞ and st- lim x = . Conversely, suppose that A ∈ (c : f) Write E = {i : |xi − | ≥ ε} for any given ε > 0, so that δ(E) = 0. Since A ∈ (c, f$)reg and f$ − lim ani = 1, we have i

f$ − lim(Ax) = f$ − lim

ani (xi − ) +

i

= lim sup n→∞ p∈N

i

ani

= f$ − lim

i

ani (xi − ) +

i

1 1 aj+p,i (xi − ) + . n+1 k + 1 j=0 n

k

k=0

On the other hand, since . . . . . . . n k n k 1 .. . . . 1 1 1 . . . aj+p,i (xi − ). ≤ x∞ aj+p,i .. + εA, . . n+1. k + 1 j=0 . . . i n + 1 k=0 k + 1 j=0 i∈E k=0 condition (6.2) implies that lim

n→∞

i

1 1 aj+p,i (xk − )| = 0 n+1 k + 1 j=0 n

k

uniformly in p.

k=0

Hence, f$ − lim (Ax) = st- lim x; that is, A ∈ (S ∩ m : f$)reg , which completes the proof. Theorem 6.6 BC -core(Ax) ⊆ st-core(x) for all x ∈ ∞ if and only if A ∈ (S ∩ ∞ and (6.1) holds.

2 $ : f )reg

Proof Assume that BC -core(Ax) ⊆ st-core(x) for all x ∈ ∞ . Then T ∗ (Ax) ≤ β(x) for all x ∈ ∞ where β(x) = st- lim sup x. Hence, since β(x) = st- lim sup x ≤ L(x) for all x ∈ ∞ (see [25]), we have (6.1) from Theorem 6.2. Furthermore, one can also easily see that −β(−x) ≤ −T ∗ (−Ax) ≤ T ∗ (Ax) ≤ β(x), i.e., st- lim inf x ≤ −T ∗ (−Ax) ≤ T ∗ (Ax) ≤ st − lim sup x. If x ∈ S ∩ ∞ , then st- lim inf x = st- lim sup x = st- lim x. Thus, the last inequality implies that st- lim x = −T ∗ (−Ax) = T ∗ (Ax) = f$ − lim Ax, that is, A ∈ (S ∩ ∞ : f$)reg . Conversely, assume that A ∈ (S ∩ ∞ : f$)reg and (6.1) holds. If x ∈ ∞ , then β(x) is ﬁnite. Let E be a subset of N deﬁned by E = {i : xi > β(x) + ε} for a given ε > 0. Then it is obvious that δ(E) = 0 and xi ≤ 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− .

No.6

` K. Kayaduman & M. S ¸ eng¨ on¨ ul: SPACES OF CESARO ALMOST CONVERGENT

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Now, we can write

cni (p)xi =

i

cni (p)xi +

i

=

i

≤ x

cni (p)xi

i≥i0

cni (p)xi +

c+ ni (p)xi −

i≥i0

|cni (p)| +

i

+x

c+ ni (p)xi

i≥i0 i∈E

[|cni (p)| − cni (p)]

|cni (p)| + [β(x) + ε]

i

+x

+

i≥i0 i∈E /

c− ni (p)xi

i≥i0

c+ ni (p)xi

i≥i0

≤ x

|cni (p)| + x

i≥i0 i∈E

|cni (p)|

i≥i0 i∈E /

[|cni (p)| − cni (p)] .

i≥i0

Applying the operator lim sup sup and using hypothesis, we obtained that T ∗ (Ax) ≤ β(x) + ε. n→∞ p∈N

This completes the proof since ε is arbitrary.

2

Acknowledgements The authors would like to thank Dr. Feyzi Ba¸sar, Fatih University, Science and Art Faculty, Dept. of Math., for his much encouragement, support and constructive ˙ on¨ criticism. Finally, we thank to Dr. B. Altay, In¨ u University, Education Faculty, Dept. of Math. Edu., for his careful reading and making a useful comment which improved the presentation and the readability of the paper. References [1] Altay B. On the space of p-summable diﬀerence sequences of order m, (1 ≤ p ≤ ∞). Stud Sci Math Hungar, 2006, 43(4): 387–402 [2] Altay B, Ba¸sar F. Certain topological properties and duals of the matrix domain of a triangle matrix in a sequence space. J Math Anal Appl, 2007, 336(1): 632–645 [3] Altay B, Ba¸sar F. Some Euler sequence spaces of non-absolute type. Ukrainian Math J, 2005, 57(1): 1–17 [4] Altay B, Ba¸sar F. Some paranormed Riezs sequence spaces of non-absolute type. Southeast Asian Bull Math, (2006), 30(5) 591–608 [5] Altay B, Ba¸sar F. Some paranormed sequence spaces of non-absolute type derived by weighted mean. J Math Anal Appl, 2006, 319(2): 494–508 [6] Altay B, Ba¸sar F. Generalization of the sequence space (p) derived by weighted mean. J Math Anal Appl, 2007, 330(1): 174–185 [7] Altay B, Ba¸sar F. Certain topological properties and duals of the matrix domain of a triangle matrix in a sequence space. J Math Anal Appl, 2007, 336(1): 632–645 [8] Altay B, Ba¸sar F. The matrix domain and the ﬁne spectrum of the diﬀerence operator on the sequence space p , (0 < p < 1). Commun Math Anal, 2007, 2(2): 1–11 [9] Aydın C, Ba¸sar F. On the new sequence spaces which include the spaces c0 and c. Hokkaido Math J, 2004, 33(2): 383–398 [10] Aydın C, Ba¸sar F. Some new paranormed sequence spaces. Inform Sci, 2004, 160(1–4): 27–40 [11] Aydın C, Ba¸sar F. Some new diﬀerence sequence spaces. Appl Math Comput, 2004, 157(3): 677–693 [12] Aydın C, Ba¸sar F. Some new sequence spaces which include the spaces p and ∞ . Demonstratio Math, 2005, 38(3): 641–656

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