KYUNGPOOK Math. J. 43(2003), 000-000

Matrix Transformations on Cesaro Vector-Valued Sequence Space

Chanan Sudsukh Department of Mathematics Statistics and Computer Science, Faculty of Liberal Arts and Science, Kasetsart University Kamphaeng Saen Campus, Nakhorn Prathom 73140, Thailand

e-mail : faachs@nontri.ku.ac.th

Praiboon Pantaragphong and Oravan Arunphalungsanti Department of Mathematics and Computer Science, Faculty of Science, King Mongkut’s Institute of Technology Ladgrabang, Bangkok 10520, Thailand

e-mail : orav_ple@yahoo.com

Abstract. The purpose of this paper is to find β-dual of Cesaro vector-valued sequence space and give matrix characterizations from Ces(X, p) into sequence spaces ℓ(q), ℓ_{∞}(q), M_{∞}(q) and \underline{ℓ_{∞}}(q) where p = (p_{k}) is a bounded sequence of positive real numbers such that p_{k} > 1 for all k ∈ N.

1. Introduction

Let (X,∥.∥) be a Banach space with a scalar field K, the space of all sequences in X is denote by W (X) and let Φ(X) denote the space of all finite sequences in X. When X = ℝ or C, the corresponding spaces are written as W and Φ. Let N be the set of all natural numbers, we write x = (x_{k}) with x_{k} ∈ X for all k ∈ N. A sequence space in X is linear subspace of W (X). Let p = (p_{k}) be a bounded sequence of positive real numbers, the X-valued sequence space c_{0}(X, p), c(X, p), ℓ_{∞}(X, p), ℓ(X, p), Ces(X, p), \underline{ℓ_{∞}}(X, p), E_{r}(X, p) and F_{r}(X, p) are define by:

c_{0}(X, p) = {x = (x_{k}) : lim_{k→∞} ∥x_{k}∥^{p_{k}} = 0},
c(X, p) = {x = (x_{k}) : lim_{k→∞} ∥x_{k} − a∥^{p_{k}} = 0 for some a ∈ X},
ℓ_{∞}(X, p) = {x = (x_{k}) : sup_{k} ∥x_{k}∥^{p_{k}} < ∞},

Received December 30, 2002, and, in revised form, March 3, 2003.

2000 Mathematics Subject Classification: 46A45.

Key words and phrases: sequence space, matrix transformation and Maddox vectorvalued sequence space.

ℓ(X, p) = {x = (x_{k}) : ∑_{k=1}^{∞} ∥x_{k}∥^{p_{k}} < ∞},

Ces(X, p) = \{x = (x_{k}) : ∑_{k=1}^{∞}(\frac{1}{k} ∑_{n=1}^{k} ∥x_{n}∥)^{p_{k}} < ∞\},
\underline{ℓ_{∞}}(X, p) = \{x = (x_{k}) : lim_{k→∞} ∥δ_{k}x_{k}∥ = 0 for each(δ_{k}) ∈ c_{0}\},
E_{r}(X, p) = {x = (x_{k}) : sup_{k} k^{−r}∥x_{k}∥^{p_{k}} < ∞}, and
F_{r}(X, p) = {x = (x_{k}) : ∑_{k=1}^{∞} k^{r}∥x_{k}∥^{p_{k}} < ∞}.

When X = K, the scalar field of X, the corresponding spaces are written as c_{0}(p), c(p), ℓ_{∞}(p), Ces(p), \underline{ℓ_{∞}}(p), E_{r}(p) and F_{r}(p) respectively.

Grosse and Erdmann [2-3] investigated and gave characterization for infinite matrices to transform between sequence spaces of Maddox. In 1993, F. M. Khan and M. A. Khan[4] gave characterization of infinite matrices of Cesàro sequence space (Ces(p, s)) into the space of convergent series (cs) and the space of bounded series (bs). Wu and Liu [9] gave the matrix transformations from X-valued sequence spaces c_{0}(X, p), ℓ_{∞}(X, p) and ℓ(X, p) into scalar-valued sequence spaces c_{0}(q) and ℓ_{∞}(q). S. Suantai[6, 7, 8] gave characterization of infinite matrices mapping Nakano vector-valued sequence space ℓ(X, p) into any BK-space, ℓ_{∞} and ℓ_{∞}(q). In [1] C. Sudsukh characterized an infinite matrix that transform Maddox vector-valued sequence space into Nakano sequence space and Nakano vector-valued sequence space into Maddox sequence space. In [3] S. Kongnual characterized the matrix transformation of bounded variation vector-valued sequence space into Maddox sequence space.

However, there are many open problems about matrix transformations from vector-valued sequence spaces into scalar-valued sequence spaces. In this paper we study matrix transformations of Cesaro vector-valued sequence space Ces(X, p) into sequence spaces ℓ(q), ℓ_{∞}(q), M_{∞}(q) and \underline{ℓ_{∞}}(q).

2. Definitions and lemmas

For z ∈ X and k ∈ N, we let e^{(k)}(z) be the sequence (0, 0, 0, ..., 0, z, 0, ...) with z in the k^{th} position. For a fixed scalar sequence u = (u_{k}) the sequence space E_{u} is defined by

E_{u} = {x = (x_{k}) ∈ W (X) : (u_{k}x_{k}) ∈ E}.

Suppose that the X-valued sequence space E is endowed with some linear topology τ. Then E is called a K-space if for each n ∈ N the n^{th} coordinate mapping p_{n} : E → X, defined by p_{n}(x) = x_{n}, is continuous on E. If, in addition, (E, τ) is a Fréchet(Banach, LF-, LB-) space, then E is called an F K −(BK−, LF K−, LBK−) space. Now, suppose that E contains Φ(X). Then E is said to have property AB if the set {Σ_{k=1}^{n}e^{k}(x_{k}) : n ∈ N} is bounded in E for every x = (x_{k}) ∈ E. It said to have property AK if Σ_{k=1}^{n}e^{k}(x_{k}) → x ∈ E as n → ∞ for every x = (x_{k}) ∈ E. It has property AD if Φ(X) is dense in E. Let A = (f_{k}^{n}) with f_{k}^{n} in X′, the topological dual of X. Suppose that E is a space of X-valued sequences and F a space of scalar-valued sequences. Then A is said to map E into F, written A : E → F if for each x = (x_{k}) ∈ E, A_{n}(x) = Σ_{k=1}^{∞}f_{k}^{n}(x_{k}) converges for each n ∈ N and if the sequence Ax = (A_{n}(x)) ∈ F. We denote by (E, F) the set of all infinite matrices mapping E into F. If u = (u_{k}) and v = (v_{k}) are scalar sequences, let

u(E, F)_{v} = {A = (f_{k}^{n}) : (u_{n}v_{k}f_{k}^{n})_{n,k} ∈ (E, F)}.

If u_{k} ≠ 0 for all k ∈ N, we write u^{−1} = (\frac{1}{u_{k}}).

In C. Sudsukh [1], this lemma is useful to characterize condition of matrix transformations.

Lemma 2.1. Let E ⊆ W (X) be an F K-space with AK property and F an F Kspace of scalar sequences. Then, for an infinite matrix A = (f_{k}^{n}), A : E → F if and only if

(1) for each n ∈ N, ∑_{k=1}^{∞} f_{k}^{n}(x_{k}) converges for all x = (x_{k}) ∈ E,

(2) for each k ∈ N, (f_{k}^{n}(z))_{n=1}^{∞} ∈ F for all z ∈ X, and

(3) A : Φ(X) → F is continuous when Φ(X) is considered as a subspace of E.

3. Some auxiliary results

In this part we first give useful results that concern with β − dual of Ces(X, p).

Proposition 3.1. Let (f_{k}) be a sequence of continuous linear functional on X and p = (p_{k}) of positive real numbers with p_{k} > 1 and \frac{1}{p_{k}} + \frac{1}{t_{k}} = 1 for all k ∈ N. Then ∑_{k=1}^{∞} f_{k}(x_{k}) converges for all x = (x_{k}) ∈ Ces(X, p) if and only if ∑_{k=1}^{∞}(sup_{n} ∥f_{n}∥.k)^{t_{k}}B^{−t_{k}} < ∞ for some B ∈ N.

Proof. Suppose that ∑_{k=1}^{∞}(sup_{n} ∥f_{n}∥.k)^{t_{k}}B^{−t_{k}} < ∞ for some B ∈ N then for each x = (x_{k}) ∈ Ces(X, p)

∑_{k=1}^{∞} |f_{k}(x_{k})| ≤ ∑_{k=1}^{∞} ∥f_{k}∥.k.B^{−1}B.\frac{1}{k} ∥x_{k}∥

≤ ∑_{k=1}^{∞}[(∥f_{k}∥.k)^{t_{k}}B^{−t_{k}} + B^{p_{k}}(\frac{1}{k} ∥x_{k}∥)^{p_{k}}]

≤ ∑_{k=1}^{∞}[(sup_{k} ∥f_{k}∥.k)^{t_{k}}B^{−t_{k}} + B^{p_{k}}(\frac{1}{k} ∥x_{k}∥)^{p_{k}}]

= ∑_{k=1}^{∞}(sup_{n} ∥f_{n}∥.k)^{t_{k}}B^{−t_{k}} + ∑_{k=1}^{∞} B^{p_{k}}(\frac{1}{k} ∥x_{k}∥)^{p_{k}}

≤ ∑_{k=1}^{∞}(sup_{n} ∥f_{n}∥.k)^{t_{k}}B^{−t_{k}} + B^{sup_{k} p_{k}} ∑_{k=1}^{∞}(\frac{1}{k} ∥x_{k}∥)^{p_{k}}

≤ ∑_{k=1}^{∞}(sup_{n} ∥f_{n}∥.k)^{t_{k}}B^{−t_{k}} + B^{G} ∑_{k=1}^{∞}(\frac{1}{k} ∑_{k=1}^{k} ∥x_{k}∥)^{p_{k}}
< ∞.

Thus ∑_{k=1}^{∞} f_{k}(x_{k}) converges.

Assume that ∑_{k=1}^{∞} f_{k}(x_{k}) converges for each x = (x_{k}) ∈ Ces(X, p). Choose scalar sequence (t_{k}) with |t_{k}| = 1 such that f_{k}(t_{k}x_{k}) = |f_{k}(x_{k})|, ∀k ∈ N. Since (t_{k}x_{k}) ∈ Ces(X, p) by assumption we have ∑_{k=1}^{∞} f_{k}(t_{k}x_{k}) converges then

(3.1)  ∑_{k=1}^{∞} |f_{k}(x_{k})| < ∞ for all x = (x_{k}) ∈ Ces(X, p).

We want to show that ∃B ∈ N such that ∑_{k=1}^{∞}(sup_{n} ∥f_{n}∥.k)^{t_{k}}B^{−t_{k}} < ∞. On contrary, suppose that

(3.2)  ∑_{k=1}^{∞}(sup_{n} ∥f_{n}∥.k)^{t_{k}}b^{−t_{k}} = ∞, ∀b ∈ N.

By (3.2) implies that for each k_{0} ∈ N

(3.3)  ∑_{k>k_{0}}(sup_{n} ∥f_{n}∥.k)^{t_{k}}b_{1}^{−t_{k}} = ∞, ∀b_{1} ∈ N.

From (3.3) we can choose b_{2} > b_{1} and b_{2} > 2^{2} and k_{2} > k_{1} such that

(3.4)  ∑_{k_{1}<k≤k_{2}}(sup_{n} ∥f_{n}∥.k)^{t_{k}}b_{2}^{−t_{k}} > k^{2}.

Doing in this way go on, we have sequence 1 = k_{0} < k_{1} < k_{2} < ... and b_{1} < b_{2} < , ..., b_{i} > 2^{i} such that

∑_{k_{i−1}<k≤k_{i}}(sup_{n} ∥f_{n}∥.k)^{t_{k}}b_{i}^{−t_{k}} > k^{2}.

Choose x_{k} in X with ∥x_{k}∥ = 1 such that ∑_{k_{i−1}<k≤k_{i}}(sup_{n} |f_{n}(x_{n})|.k)^{t_{k}}b_{i}^{−t_{k}} > k^{2}, ∀i ∈ N. Let a_{i} = ∑_{k_{i−1}<k≤k_{i}}(sup_{n} |f_{n}(x_{n})|.k)^{t_{k}}b_{i}^{−t_{k}} and y = (y_{k}), y_{k} = a_{i}^{−1}(sup_{n} |f_{n}(x_{n})|.k)^{t_{k}}|f_{k}(x_{k})|^{−1}x_{k}. Then y ∈ Ces(X, p). Let α = (sup_{n} |f_{n}(x_{n})|) and G = sup_{k} p_{k}, we can separate two cases.

Case α < 1;

[ ∑_{k_{i−1}<k≤k_{i}}(\frac{1}{k} ∑_{j=1}^{k} ∥y_{j}∥)^{p_{k}}]^{\frac{1}{G}} = [ ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{p_{k}}} (∑_{j=1}^{k} ∥a_{i}^{−1}α^{t_{j}}j^{t_{j}}|f_{j}(x_{j})|^{−1}.x_{j}∥)^{p_{k}}]^{\frac{1}{G}}

= [ ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{p_{k}}} (∑_{j=1}^{k} a_{i}^{−1}α^{t_{j}}j^{t_{j}}|f_{j}(x_{j})|^{−1})^{p_{k}}]^{\frac{1}{G}} ; ∥x_{j}∥ = 1

≤ [ ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{p_{k}}} (∑_{j=1}^{k} a_{i}^{−1}1.j^{t_{j}}|f_{j}(x_{j})|^{−1})^{p_{k}}]^{\frac{1}{G}} ; α^{t_{j}} ≤ α < 1, t_{j} > 1

≤ [ ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{p_{k}}} (∑_{j=1}^{k} a_{i}^{−1}1.k^{sup^{k} t_{k}}|f_{j}(x_{j})|^{−1})^{p_{k}}]^{\frac{1}{G}} ; j^{t_{j}} ≤ k^{t_{j}} ≤ k^{sup^{k} t_{k}}, ∀k ∈ N.

Let K_{1,k} and K_{2,k} are partitions of {1, 2, ..., k}, if j ∈ K_{1,k} then |f_{j}(x_{j})| < 1 and if j ∈ K_{2,k} then |f_{j}(x_{j})| ≥ 1 for all j = 1, 2, ..., k. So we have

= [ ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{p_{k}}} ( ∑_{j∈K_{1,k}} a_{i}^{−1}k^{L}|f_{j}(x_{j})|^{−1} + ∑_{j∈K_{2,k}} a_{i}^{−1}k^{L}|f_{j}(x_{j})|^{−1})^{p_{k}}]^{\frac{1}{G}}; L = sup_{k} t_{k}

≤ [ ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{p_{k}}} ( ∑_{j∈K_{1,k}} a_{i}^{−1}k^{L}|f_{j}(x_{j})|^{−1})^{p_{k}}]^{\frac{1}{G}}

+[ ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{p_{k}}} ( ∑_{j∈K_{2,k}} a_{i}^{−1}k^{L}|f_{j}(x_{j})|^{−1})^{p_{k}}]^{\frac{1}{G}}

≤ [ ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{p_{k}}} ( ∑_{j∈K_{1,k}} a_{i}^{−1}k^{L}C)^{p_{k}}]^{\frac{1}{G}} + [ ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{p_{k}}} ( ∑_{j∈K_{2,k}} a_{i}^{−1}k^{L}|f_{j}(x_{j})|^{−1})^{p_{k}}]^{\frac{1}{G}}

; |f_{j}(x_{j})|^{−1} ≤ max_{j} |f_{j}(x_{j})|^{−1} = C, where C > 1

≤ [ ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{p_{k}}} ( ∑_{j∈K_{1,k}} a_{i}^{−1}k^{L}C)^{p_{k}}]^{\frac{1}{G}} + [ ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{p_{k}}} ( ∑_{j∈K_{2,k}} a_{i}^{−1}k^{L}.1)^{p_{k}}]^{\frac{1}{G}}

; |f_{j}(x_{j})|^{−1} ≤ 1

≤ [ ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{p_{k}}} (∑_{j=1}^{k} a_{i}^{−1}k^{L}C)^{p_{k}}]^{\frac{1}{G}} + [ ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{p_{k}}} (∑_{j=1}^{k} a_{i}^{−1}k^{L}.1)^{p_{k}}]^{\frac{1}{G}}

; ∑_{j∈K_{1,k}} + ∑_{j∈K_{2,k}} ≤ ∑_{j=1}^{k} + ∑_{j=1}^{k}

≤ [C^{G}k_{i}^{L.G} ∑_{k_{i−1}<k≤k_{i}} a_{i}^{−1}]^{\frac{1}{G}} + [k_{i}^{L.G} ∑_{k_{i−1}<k≤k_{i}} a_{i}^{−1}]^{\frac{1}{G}} ; k^{L.G} ≤ k_{i}^{L.G}

< [C^{G}k_{i}^{L.G} ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{2}} ]^{\frac{1}{G}} + [k_{i}^{L.G} ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{2}} ]^{\frac{1}{G}} ; a_{i} > k^{2} ⇒ a_{i}^{−1} < k^{−2}

= (C^{G}k_{i}^{L.G})^{\frac{1}{G}} ( ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{2}} )^{\frac{1}{G}} + (k_{i}^{L.G})^{\frac{1}{G}} ( ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{2}} )^{\frac{1}{G}}
= [(C.k_{i}^{L}) + k_{i}^{L}]( ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{2}} )^{\frac{1}{G}}
= [k_{i}^{L}(C + 1)]( ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{2}} )^{\frac{1}{G}} .

Hence, [∑_{k_{i−1}<k≤k_{i}}(\frac{1}{k} ∑_{j=1}^{k} ∥y_{j}∥)^{p_{k}}]^{\frac{1}{G}} ≤ T_{i,1}(∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{2}} )^{\frac{1}{G}} then ∑_{k_{i−1}<k≤k_{i}}(\frac{1}{k} ∑_{j=1}^{k} ∥y_{j}∥)^{p_{k}} ≤ T_{i,1}^{G} ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{2}} . Therefore, ∑_{k=1}^{∞}(\frac{1}{k} ∑_{j=1}^{k} ∥y_{j}∥)^{p_{k}} ≤ T_{i,1}^{G} ∑_{k=1}^{∞} \frac{1}{k^{2}} < ∞, where T_{i,1} = [k_{i}^{L}(C + 1)].

Case α ≥ 1;

[ ∑_{k_{i−1}<k≤k_{i}}(\frac{1}{k} ∑_{j=1}^{k} ∥y_{j}∥)^{p_{k}}]^{\frac{1}{G}}

= [ ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{p_{k}}} (∑_{j=1}^{k} ∥a_{i}^{−1}α^{t_{j}}j^{t_{j}}|f_{j}(x_{j})|^{−1}.x_{j}∥)^{p_{k}}]^{\frac{1}{G}}
= [ ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{p_{k}}} (∑_{j=1}^{k} a_{i}^{−1}α^{t_{j}}j^{t_{j}}|f_{j}(x_{j})|^{−1})^{p_{k}}]^{\frac{1}{G}} ; ∥x_{j}∥ = 1
≤ [ ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{p_{k}}} (∑_{j=1}^{k} a_{i}^{−1}α^{sup^{k} t_{k}}.j^{t_{j}}|f_{j}(x_{j})|^{−1})^{p_{k}}]^{\frac{1}{G}} ; α^{t_{j}} ≤ α^{sup^{k} t_{k}}, α ≥ 1
≤ [ ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{p_{k}}} (∑_{j=1}^{k} a_{i}^{−1}α^{sup^{k} t_{k}}.k^{sup^{k} t_{k}}|f_{j}(x_{j})|^{−1})^{p_{k}}]^{\frac{1}{G}} ; j^{t_{j}} ≤ k^{t_{j}} ≤ k^{sup^{k} t_{k}}.

Now doing same as case α < 1, which separate by partition. We have

= [ ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{p_{k}}} ( ∑_{j∈K_{1,k}} a_{i}^{−1}α^{sup^{k} t_{k}}k^{sup^{k} t_{k}}|f_{j}(x_{j})|^{−1}

+ ∑_{j∈K_{2,k}} a_{i}^{−1}α^{sup^{k} t_{k}}k^{sup^{k} t_{k}}|f_{j}(x_{j})|^{−1})^{p_{k}}]^{\frac{1}{G}}

≤ [C^{G}(α.k_{i})^{L.G} ∑_{k_{i−1}<k≤k_{i}} a_{i}^{−1}]^{\frac{1}{G}} + [(α.k_{i})^{L.G} ∑_{k_{i−1}<k≤k_{i}} a_{i}^{−1}]^{\frac{1}{G}} ; k ≤ k_{i}
< [C^{G}(α.k_{i})^{L.G} ∑_{k_{i−1}<k≤k_{i}} k^{−2}]^{\frac{1}{G}} + [(α.k_{i})^{L.G} ∑_{k_{i−1}<k≤k_{i}} k^{−2}]^{\frac{1}{G}}

; a_{i} > k^{2} ⇒ a_{i}^{−1} < k^{−2}

= [(α.k_{i})^{L}(C + 1)]( ∑_{k_{i−1}<k≤k_{i}} k^{−2})^{\frac{1}{G}}

Thus, [∑_{k_{i−1}<k≤k_{i}}(\frac{1}{k} ∑_{j=1}^{k} ∥y_{j}∥)^{p_{k}}]^{\frac{1}{G}} ≤ T_{i,2}(∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{2}} )^{\frac{1}{G}} then

∑_{k_{i−1}<k≤k_{i}}(\frac{1}{k} ∑_{j=1}^{k} ∥y_{j}∥)^{p_{k}} ≤ T_{i,2}^{G} ∑_{k_{i−1}<k≤k_{i}} \frac{1}{k^{2}} . Therefore,

∑_{k=1}^{∞}(\frac{1}{k} ∑_{j=1}^{k} ∥y_{j}∥)^{p_{k}} ≤ T_{i,2}^{G} ∑_{k=1}^{∞} \frac{1}{k^{2}} < ∞, where T_{i,2} = [(α.k_{i})^{L}(C + 1)]. We obtained y ∈ Ces(X, p). For each i ∈ N, we have

∑_{k_{i−1}<k≤k_{i}} |f_{k}(y_{k})| = ∑_{k_{i−1}<k≤k_{i}} |f_{k}(a_{i}^{−1}(sup_{n} |f_{n}(x_{n}).k|)^{t_{k}}.|f_{k}(x_{k})|^{−1}x_{k})|
= a_{i}^{−1} ∑_{k_{i−1}<k≤k_{i}}(sup_{n} |f_{n}(x_{n}).k|)^{t_{k}}.b_{i}^{−t_{k}}.b_{i}^{t_{k}} ; b_{i}^{−t_{k}}.b_{i}^{t_{k}} = 1
≥ a_{i}^{−1} ∑_{k_{i−1}<k≤k_{i}}(sup_{n} |f_{n}(x_{n}).k|)^{t_{k}}.b_{i}^{−t_{k}} ; b_{i}^{t_{k}} ≥ 1, b_{i} > 2^{i}
= 1.

Then, ∑_{k=1}^{∞} |f_{k}(y_{k})| = ∞ which is contradiction. The proof is complete. □

Remark 3.2. For each T_{i,1} and T_{i,2} in Proposition 3.1 that are bounded.

T_{i,1} = [k_{i}^{L}(C + 1)] = [k_{i}^{sup_{k} t_{k}}(max_{j} |f_{j}(x_{j})|^{−1} + 1)]
T_{i,2} = [(α.k_{i})^{L}(C + 1)] = [(sup_{n} |f_{n}(x_{n})|)^{sup^{k} t_{k}}k_{i}^{sup_{k} t_{k}}(max_{j} |f_{j}(x_{j})|^{−1} + 1)].

Proof. Since (f_{k}) be a sequence of continuous linear functional and in the proof we choose sequence x_{k} in X with ∥x_{k}∥ = 1, so |f_{k}(x_{k})| ≤ ∥f_{k}∥.∥x_{k}∥ = ∥f_{k}∥.1 < ∞. We have |f_{k}(x_{k})| bounded, thus max_{j} |f_{j}(x_{j})|^{−1} and (sup_{n} |f_{n}(x_{n})|) are bounded for all j = 1, 2, ...k. Therefore, T_{i,1} and T_{i,2} are bounded for all i ∈ N. □

4. Main results

In this section, we characterize matrix transformations from Ces(X, p) into Maddox sequence spaces ℓ(q) and ℓ_{∞}(q). By using Lemma 2.1 and Proposition 3.1, we have following theorems.

Theorem 4.1. Let p = (p_{k}) and q = (q_{k}) be bounded sequences of positive real numbers such that p_{k} > 1 and \frac{1}{p_{k}} + \frac{1}{t_{k}} = 1 for all k ∈ N, and A = (f_{k}^{n}) an infinite matrix. Then A : Ces(X, p) → ℓ(q) if and only if

(1) for each n ∈ N there exists B_{n} ∈ N such that

∑_{k=1}^{∞}(sup_{j} ∥f_{j}^{n}∥.k)^{t_{k}}B_{n}^{−t_{k}} < ∞,

(2) for each k ∈ N, ∑_{n=1}^{∞} |f_{k}^{n}(x)|^{q_{n}} < ∞ for every x ∈ X and (3) for each r ∈ N there exists B_{r} ∈ N such that

∑_{k∈K}(\frac{1}{k} ∑_{j=1}^{k} ∥x_{j}∥)^{p_{k}} < \frac{1}{B_{r}} ⇒ ∑_{n=1}^{∞} | ∑_{k∈K} f_{k}^{n}(x_{k})|^{q_{n}} < \frac{1}{r}

for all x = (x_{k}) ∈ Φ(X) and all finite subsets K of N.

Theorem 4.2. Let p = (p_{k}) and q = (q_{k}) be bounded sequences of positive real numbers with p_{k} > 1 and \frac{1}{p_{k}} + \frac{1}{t_{k}} = 1 for all k ∈ N and A = (f_{k}^{n}) an infinite matrix. Then A : Ces(X, p) → ℓ_{∞}(q) if and only if there exists B ∈ N such that

sup_{n}(∑_{k=1}^{∞}(sup_{j} ∥f_{j}^{n}∥.k)^{t_{k}}B^{−t_{k}})^{q_{n}} < ∞.

Proof. By Proposition 3.1, we have ∑_{k=1}^{∞}(sup_{j} ∥f_{j}∥.k)^{t_{k}}B^{−t_{k}} < ∞ for all x = (x_{k}) ∈ Ces(X, p). Since A = (f_{k}^{n}) : Ces(X, p) → ℓ_{∞}(q) and definition of ℓ_{∞}(q), we have that sup_{n}(∑_{k=1}^{∞}(sup_{j} ∥f_{j}^{n}∥.k)^{t_{k}}B^{−t_{k}})^{q_{n}} < ∞. □

Theorem 4.3. Let p = (p_{k}) and q = (q_{k}) be bounded sequences of positive real numbers such that p_{k} > 1 and \frac{1}{p_{k}} + \frac{1}{t_{k}} = 1 for all k ∈ N, and A = (f_{k}^{n}) an infinite matrix. Then A : Ces(X, p) → M_{∞}(q) if and only if

(1) for each m, n ∈ N there exists B ∈ N such that

∑_{k=1}^{∞} m^{\frac{t_{k}}{qn}} (sup_{j} ∥f_{j}^{n}∥)^{t_{k}}.k^{t_{k}}B^{−t_{k}} < ∞

(2) for all m, k ∈ N, ∑_{n=1}^{∞} m^{\frac{1}{qn}} |f_{k}^{n}(x)| < ∞ for every x ∈ X and

(3) for each m, r ∈ N there exists S ∈ N such that

∑_{k∈K}(\frac{1}{k} ∑_{j=1}^{k} ∥x_{j}∥)^{p_{k}} < \frac{1}{S} ⇒ ∑_{n=1}^{∞} m^{\frac{1}{qn}} | ∑_{k∈K} f_{k}^{n}(x_{k})| < \frac{1}{r} ,

for all x = (x_{k}) ∈ Φ(X) and all finite subsets K of N.

Proof. By [1, Proposition 2.3(vii)], we have M_{∞}(q) = ⋂_{m=1}^{∞} ℓ_{(m^{\frac{1}{qn}} )}. By [1, Propo-

sition 2.2(ii) and (iv)] and Theorem 4.1, we have

A : Ces(X, p) → M_{∞}(q) ⇔ A : Ces(X, p) → ⋂_{m=1}^{∞} ℓ_{(m^{\frac{1}{qn}} )}
⇔ A : Ces(X, p) → ℓ_{(m^{\frac{1}{qn}} )} , for all m ∈ N
⇔ (m^{\frac{1}{qn}} f_{k}^{n})_{n,k} : Ces(X, p) → ℓ , for all m ∈ N
⇔ the conditions (1), (2) and (3) hold.

□

Theorem 4.4. Let p = (p_{k}) and q = (q_{k}) be bounded sequences of positive real numbers such that p_{k} > 1 and \frac{1}{p_{k}} + \frac{1}{t_{k}} = 1 for all k ∈ N, and A = (f_{k}^{n}) an infinite matrix. Then A : Ces(X, p) → \underline{ℓ_{∞}}(q) if and only if for each m ∈ N there exists B_{m} ∈ N such that

sup_{n}(∑_{k=1}^{∞} s^{\frac{t_{k}}{qn}} (sup_{j} ∥f_{j}^{n}∥)^{t_{k}}.k^{t_{k}}B_{m}^{−t_{k}}) < ∞.

Proof. By [1, Proposition 2.3(vi)], we have \underline{ℓ_{∞}}(q) = ⋂_{s=1}^{∞} ℓ_{∞_{(s \frac{1}{qn} )}} and [1, Proposi-

tion 2.2(ii) and (iv)], we have

A : Ces(X, p) → \underline{ℓ_{∞}}(q) ⇔ A : Ces(X, p) → ⋂_{s=1}^{∞} ℓ_{∞_{(s \frac{1}{qn} )}}
⇔ A : Ces(X, p) → ℓ_{∞_{(s \frac{1}{qn} )}} , for all s ∈ N
⇔ (s^{\frac{1}{qn}} f_{k}^{n})_{n,k} : Ces(X, p) → ℓ_{∞} , for all s ∈ N
⇔ the condition holds.
□

Acknowledgement. The author would like to thank to my advisor, co-advisor and friend for guidance to do this paper.

References

[1] Sudsukh, C., Matrix Transformations of Vector-Valued Sequence Spaces, Ph. D. thesis of Chaingmai University, (2000).

[2] Grosse, K. and Erdmann. G., The structure of Sequence Spaces of Maddox, Canad. J. Math., 44(1992), 47-54.

[3] Grosse, K. and Erdmann. G., Matrix Transformations between the Sequence Spaces of Maddox, J. of Math. Anal. Appl., 180(1993), 223-238.

[4] Khan, F. M. and Khan, M. A., Matrix Transformations between Cesaro Sequence Spaces, Indian J. Pure Appl. Math., 25(6)(1994), 641-645.

[5] Kongnual, S., Matrix Transformations of Bounded Variation Vector-Valued Sequence Space into Maddox Sequence Space, Master degree Thesis of King Mongkut’s Institute of Technology Ladkrabang, (2001).

[6] Suantai, S., Matrix Transformations between some Vector-Valued Sequence Spaces, Seam., 24(2)(2000).

[7] Suantai, S. and Sudsukh, C., Matrix Transformations of Nakano Vector-Valued Sequence Spaces, Kyungpook Mathematical Journal, 40(1)(2000), 93-97.

[8] Suantai, S., Matrix Transformations from Nakano Vector-Valued Sequence Space into the Orlicz Sequence Space, to appear in the proceeding of functions, Adam Michiscwitz University. Poznan, Poland, (1998).

[9] Wu, C. X. and Lui, L., Matrix Transformations between some Vector-Valued Sequence Spaces, SEA bull, 17(1)(1993), 83-96.