Matrix Transformations from Cesaro Vector-Valued Sequence Space into Orlicz Sequence Space

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Oravan Arunphalungsanti and Kantita Wijanto Department of Mathematics, Faculty of Science, Mahanakorn University of Technology, Bangkok 10530, Thailand

e-mail : orav_ple@yahoo.com

Abstract. The purpose of this paper, we give matrix characterizations from Cesaro vector-valued sequence space Ces(X, p) into Orlicz sequence space ℓ_{M} and by using this result we obtain matrix characterizations from Ces(X, p) into h_{M} and ℓ_{r}, 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}} < ∞},
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}} < ∞\},

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2000 Mathematics Subject Classification: 46A45.

Key words and phrases: Sequence space, Matrix transformation, Orlicz sequence space.

\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). S. Suantai[6, 7] gave characterization of infinite matrices mapping Nakano vector-valued sequence space ℓ(X, p) into any BK-space, ℓ_{∞} and ℓ_{∞}(q). S. Suantai[9, 11] has given matrix characterizations from ℓ_{∞}(X, p), c_{0}(X) and ℓ(X, p) into the Orlicz sequence space.

In this paper, we interested to find characterizations from Ces(X, p) into Orlicz sequence space.

2. Notation and Definitions

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}}).

A function M : ℝ → [0, ∞) is said to be an Orlicz function if it is even, convex, continuous and vanishing only at 0. We define the Orlicz sequence space by the formula

ℓ_{M} = {x = (x_{k}) ∈ ℓ^{0} : ρ_{M}(cx) = ∑_{k=1}^{∞} M(cx_{k}) < ∞ for some c > 0}

where ℓ^{0} stands for the space of all real sequences. We consider ℓ_{M} equipped with the Luxemburg norm

∥x∥ = inf\{ε > 0 : ρ_{M}(\frac{x}{ε}) ≤ 1\}.

Let h_{M} denote the subspace order continuous elements, i.e

h_{M} = {x = (x_{k}) : ρ_{M}(cx) < ∞ for any c > 0}.

It is know that ℓ_{M} is a BK-space and h_{M} is a closed subspace of ℓ_{M}.

We say that an Orlicz function M satisfiesthe δ_{2} condition (M ∈ δ_{2} for short) if there exist constants k ≥ 2 and u_{0} > 0 such that

M(2u) ≤ KM(u)

whenever |u| ≤ u_{0}.

In C. Sudsukh [8], this Proposition is useful to characterize condition of matrix transformations.

Proposition 2.1. Let E and E_{n}(n ∈ N) be X-valued sequence spaces, and F and F_{n}(n ∈ N) scalar sequence space, and let u and v be sequences of real numbers with u_{k} ≠ 0 for all k ∈ N. Then we have

(i) (⋃_{n=1}^{∞} E_{n}, F) = ⋂_{n=1}^{∞}(E_{n}, F),

(ii) (E, ⋂_{n=1}^{∞} F_{n}) = ⋂_{n=1}^{∞}(E, F_{n}),

(iii) (E_{u}, F_{v}) =_{v} (E, F,)_{u^{−1}},

(iv) (E_{1} + E_{2}, F) = (E_{1}, F) ∩ (E_{2}, F), (v) (E, F_{1}) = (E, F_{2}) ∩ (Φ(X), F_{1}) if E an F K-space with AD, F_{2} is an

F K-space and F_{1} is a closed subspace of F_{2}.

Proposition 2.2. Let M be Orlicz function and x ∈ ℓ_{M}.

(i) If ∥x∥ ≤ 1, then ρ_{M}(x) ≤ ∥x∥.

(ii) If ∥x∥ > 1, then ρ_{M}(x) > ∥x∥.

(iii) If M ∈ δ_{2}, then ∥x∥ = 1 ⇒ ρ_{M}(x) = 1.

Proof. See [1, Theorem 1.38 and Theorem 1.39] □

3. Main Results

We first give characterizations of infinite matrices mapping the Cesaro vectorvalued sequence space Ces(X, p) into the Orlicz sequence space.

Theorem 3.1. Let p = (p_{k}) be a bounded sequence of positive real numbers with p_{k} > 1 for all k ∈ N and A = (f_{k}^{n}) an infinite matrix. Then A ∈ (Ces(X, p), ℓ_{M}) if and only if

(1) for each k ∈ N, (f_{k}^{n}(x))_{n=1}^{∞} ∈ ℓ_{M} for all x ∈ X and

(2) there exists m_{0} ∈ N such that

sup_{k} sup_{∥x∥≤1} ∑_{n=1}^{∞}(M ∘ (m_{0}^{−p_{k}}f_{k}^{n}))(x) ≤ 1.

Proof. Assume that A ∈ (Ces(X, p), ℓ_{M}), we want to show conditions hold. Since e^{k}(x) ∈ Ces(X, p) for all x ∈ X and all k ∈ N, we have Ae^{k}(x) ∈ ℓ_{M}, so (1) is obtained. By Zeller’s theorem, we have that A : Ces(X, p) → ℓ_{M} is continuous. Then there exists m_{0} ∈ N such that

x = (x_{k}) ∈ Ces(X, p), ∥x∥ ≤ \frac{1}{m_{0}} ⇒ ∥Ax∥ ≤ 1. (3.1)

Let x ∈ X with ∥x∥ ≤ 1 and k ∈ N. We have m_{0}^{−p_{k}}e^{k}(x) ∈ Ces(X, p) and ∥m_{0}^{−p_{k}}e^{k}(x)∥ ≤ \frac{1}{m_{0}} . By (3.1) we have ∥(m_{0}^{−p_{k}}f_{k}^{n}(x))_{n=1}^{∞}∥ ≤ ∥A(m_{0}^{−p_{k}}e^{k}(x))∥ ≤ 1. By Proposition 2.2(i) we obtain that ∑_{n=1}^{∞} M(m_{0}^{−p_{k}}f_{k}^{n}(x)) ≤ 1. This implies that

sup_{k} sup_{∥x∥≤1} ∑_{n=1}^{∞}(M ∘ (m_{0}^{−p_{k}}f_{k}^{n}))(x) ≤ 1 ,

thus condition (2) holds.

Conversely, assume that the conditions (1) and (2) hold. By (2), there exists m_{0} ∈ N such that ∑_{n=1}^{∞} M(m_{0}^{−p_{k}}f_{k}^{n}(x)) ≤ 1 for all k ∈ N and all x ∈ X with ∥x∥ ≤ 1. From Proposition 2.2(i) we have that ∥A(m_{0}^{−p_{k}}e^{k}(x))∥ ≤ ∥(m_{0}^{−p_{k}}f_{k}^{n}(x))_{n=1}^{∞}∥ ≤ 1 for all k ∈ N and all x ∈ X with ∥x∥ ≤ 1. Hence, for x ∈ X with x ≠ 0, we have

∥A(m_{0}^{−p_{k}}e^{k}(x))∥ ≤ ∥(m_{0}^{−p_{k}}f_{k}^{n}(x))_{n=1}^{∞}∥ ≤ ∥x∥ (3.2)

Let x = (x_{k}) ∈ Ces(X, p) and k ∈ N. By (3.2) we have

∥Ae^{k}(x_{k})∥ = ∥A(m_{0}^{−p_{k}} · m_{0}^{p_{k}}e^{k}(x_{k}))∥
= m_{0}^{p_{k}}∥A(m_{0}^{−p_{k}} · e^{k}(x_{k}))∥
≤ m_{0}^{p_{k}} · ∥x_{k}∥
≤ m_{0}^{p_{k}} · ∑_{n=1}^{k} ∥x_{n}∥  (3.3)

Since, for all x = (x_{k}) ∈ Ces(X, p) we have that ∑_{k=1}^{∞}(\frac{1}{k} ∑_{n=1}^{k} ∥x_{n}∥)^{p_{k}} < ∞ when p_{k} > 1. Let ∑_{k=1}^{∞}(\frac{1}{k} ∑_{n=1}^{k} ∥x_{n}∥)^{p_{k}} = L. Fixed for each k, so

(∑_{n=1}^{k} ∥x_{n}∥) ≤ L. (3.4)

By (3.3) and (3.4) we have,

∑_{k=1}^{∞} ∥Ae^{k}(x_{k})∥ ≤ ∑_{k=1}^{∞} m_{0}^{p_{k}} · (∑_{n=1}^{k} ∥x_{n}∥)

= ∑_{k=1}^{∞} m_{0}^{p_{k}} · \frac{(\frac{1}{k}∑_{n=1}^{k} ∥x_{n}∥)}{\frac{1}{k}} · \frac{(\frac{1}{k}∑_{n=1}^{k} ∥x_{n}∥)^{p_{k}}}{(\frac{1}{k}∑_{n=1}^{k} ∥x_{n}∥)^{p_{k}}}

= ∑_{k=1}^{∞} m_{0}^{p_{k}} · (\frac{1}{k} ∑_{n=1}^{k} ∥x_{n}∥)^{p_{k}} · \frac{1}{\frac{1}{k}(\frac{1}{k}∑_{n=1}^{k} ∥x_{n}∥)^{p_{k}−1}}

≤ m_{0}^{sup_{k} p_{k}} ∑_{k=1}^{∞} (\frac{1}{k} ∑_{n=1}^{k} ∥x_{n}∥)^{p_{k}} · \frac{1}{((\frac{1}{k})^{\frac{p_{k}}{p_{k}−1}} ∑_{n=1}^{k} ∥x_{n}∥)^{p_{k}−1}}

= m_{0}^{G} ∑_{k=1}^{∞} (\frac{1}{k} ∑_{n=1}^{k} ∥x_{n}∥)^{p_{k}} · ((\frac{1}{k})^{\frac{p_{k}}{p_{k}−1}} ∑_{n=1}^{k} ∥x_{n}∥)^{1−p_{k}} ; G = sup_{k} p_{k}

≤ m_{0}^{G} ∑_{k=1}^{∞} (\frac{1}{k} ∑_{n=1}^{k} ∥x_{n}∥)^{p_{k}}(1 · ∑_{n=1}^{k} ∥x_{n}∥)^{1−p_{k}}

≤ m_{0}^{G} ∑_{k=1}^{∞} (\frac{1}{k} ∑_{n=1}^{k} ∥x_{n}∥)^{p_{k}}(L)^{1−p_{k}}  ; L = max(1, inf_{k} L)

≤ m_{0}^{G} · L^{1−inf_{k} p_{k}} ∑_{k=1}^{∞} (\frac{1}{k} ∑_{n=1}^{k} ∥x_{n}∥)^{p_{k}}

= V ∑_{k=1}^{∞} (\frac{1}{k} ∑_{n=1}^{k} ∥x_{n}∥)^{p_{k}}  ; V = m_{0}^{G}L^{1−M}, M = inf_{k} p_{k}
< ∞.

Therefore ∑_{k=1}^{∞} Ae^{k}(x_{k}) converges absolutely in ℓ_{M}. Since ℓ_{M} is Banach, ∑_{k=1}^{∞} Ae^{k}(x_{k}) converges in ℓ_{M}. Let y = (y_{k}) ∈ ℓ_{M} be the sum of the series ∑_{k=1}^{∞} Ae^{k}(x_{k}). By continuity of p_{m}, we have for each m ∈ N,

y_{m} = p_{m}(y) = lim_{n→∞} ∑_{k=1}^{n} p_{m}(Ae^{k}(x_{k})) = lim_{n→∞} ∑_{k=1}^{n} f_{k}^{m}(x_{k})

This implies that Ax exists and (Ax)_{m} = ∑_{k=1}^{∞} f_{k}^{m}(x_{k}) = y_{m}, so that Ax ∈ ℓ_{M}.

Theorem 3.2. Let p = (p_{k}) be a bounded sequence of positive real numbers with p_{k} > 1 for all k ∈ N and A = (f_{k}^{n}) an infinite matrix. Then A ∈ (Ces(X, p), h_{M}) if and only if

(1) for each k ∈ N, (f_{k}^{n}(x))_{n=1}^{∞} ∈ h_{M} for all x ∈ X and

(2) there exists m_{0} ∈ N such that

sup_{k} sup_{∥x∥≤1} ∑_{n=1}^{∞}(M ∘ (m_{0}^{−p_{k}}f_{k}^{n}))(x) ≤ 1.

Proof. Since h_{M} is a closed subspace of ℓ_{M}, the theorem is obtained by applying Theorem 3.1 and Proposition 2.1(v). □

Theorem 3.3. For an infinite matrix A = (f_{k}^{n}), A ∈ (Ces(X), ℓ_{M}) if and only if

(1) for each k ∈ N, (f_{k}^{n}(x))_{n=1}^{∞} ∈ ℓ_{M} for all x ∈ X and

(2) there exists m_{0} ∈ N such that

sup_{k} sup_{∥x∥≤1} ∑_{n=1}^{∞}(M ∘ (m_{0}^{−1}f_{k}^{n}))(x) ≤ 1.

By Theorem 3.1 and M(t) = |t|^{r}, r ≥ 1, we have ℓ_{M} = ℓ_{r}. We have this result.

Corollary 3.4. Let p = (p_{k}) be a bounded sequence of positive real numbers with p_{k} > 1 for all k ∈ N and r ≥ 1. Then for an infinite matrix A = (f_{k}^{n}), A ∈ (Ces(X, p), ℓ_{r}) if and only if

(1) for each k ∈ N, ∑_{n=1}^{∞} |f_{k}^{n}(x)|^{r} < ∞ for all x ∈ X and

(2) there exists m_{0} ∈ N such that

sup_{k} sup_{∥x∥≤1} ∑_{n=1}^{∞} |m_{0}^{−p_{k}}f_{k}^{n}(x)|^{r} ≤ 1.

Corollary 3.5. For r ≥ 1 and for an infinite matrix A = (f_{k}^{n}), A ∈ (Ces(X), ℓ_{r}) if and only if

(1) for each k ∈ N, ∑_{n=1}^{∞} |f_{k}^{n}(x)|^{r} < ∞ for all x ∈ X and

(2) sup_{k} sup_{∥x∥≤1} ∑_{n=1}^{∞} |f_{k}^{n}(x)|^{r} < ∞.

Acknowledgement. The author would like to thank Dr.Chanan Sudsukh for worth guidance and Mahanakorn University of Technology for financial support to do this paper.

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