Pure and Applied Mathematics Journal
Volume 4, Issue 3, June 2015, Pages: 90-95

Generalized Riesz Sequence Space of Non-Absolute Type and Some Matrix Mapping

Md. Fazlur Rahman, A. B. M. Rezaul Karim

Department of Mathematics, Eden University College, Dhaka, Bangladesh

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(Md. F. Rahman)
(A. B. M. R. Karim)

To cite this article:

Md. Fazlur Rahman, A. B. M. Rezaul Karim. Generalized Riesz Sequence Space of Non-Absolute Type and Some Matrix Mapping. Pure and Applied Mathematics Journal. Vol. 4, No. 3, 2015, pp. 90-95. doi: 10.11648/j.pamj.20150403.15

Abstract: Recently several authors defined and studied Riesz sequence space (u, p) of non-absolute type. In this paper for some weight s 0, we define the generalized Risez sequence space (u, p, s) of non-absolute type and determine its Kothe-Toeplitz dual. We also consider the matrix mapping (u, p, s) to  and (u, p, s) to c, where  is the space of all bounded sequences and c is the space of all convergent sequences.

Keywords: Sequence Space, Kothe-Toeplitz Dual, Matrix Mappin

1. Introduction

Throughout the paper N, R denote the set of positive integers and the set of all real numbers. We also denote the collection of all finite subsets of N by F. Let  be the space of all sequences, real or complex; ,  and are respectively the space of all bounded sequences, convergent sequences and null sequences. Let  be a bounded sequence of strictly positive real numbers with and .

Then the sequence spaces  and  were defined by Maddox [7] (see also [5, 13]) as follows:


with ,


which are complete spaces paranormed by and  if and only if inf .

We shall assume throughout that  provided .

In [15] Stieglitz and Tietz defined


Let  be a sequence of positive real numbers and let us write

for  N . Then the matrix  of the Riesz mean  is given by

The Riesz mean (R,) is regular if and only if  as  see (Peterson [4, p.10], [11], [12], [14], [17], [18]).

In a recent paper Sheikh and Ganie [16] defined and studied the Riesz sequence space of non-absolute type by




The main purpose of this paper is to define the generalized Riesz sequence space .We determine the Kothe-Toeplitz dual of  and then consider the matrix mapping to  and  to c.

In [2] Bulut and Cakar defined and studied the sequence space  and in [3] Khan and Khan defined and investigated the Cesaro sequence space ces(p,s). In the same vein we define the generalized Riesz sequence space in the following way.

Definition. For  we define


If s = 0 then  reduces to ,which is defined and studied in [16].

Define the sequence  by


Let X and Y be two subsets of . Let  be an infinite matrix of real or complex numbers , where N. Then the matrix A defines the A- transformation from X into Y, if for every sequence, the sequence , the A-transform of x exists and is in Y, where

For simplicity in notation, here and what follows, the summation without limits runs from 0 to . By , we denote the class of all such matrices. A sequence x is to be A-summable to  if Ax converges to , which is called as the A-limit of x.

We mention the following inequality (see[6,9]) which will be used later. For any integer  and any two complex numbers a and b have

|a b|                   (2)

Where  and.

Theorem 1.1.  is a complete linear metric space paranormed by g defined by


With 0and .

Proof. The linearity of  with respect to the co-ordinate wise addition and scalar multiplication follows from the inequalities which are satisfied for  (see [6, p.30])


and for any R (see [8])


It is clear that , where and  for all . The inequality (4) and (5) together gives the subadditivity of g and .

Consider any sequence  of points of  such that  and a sequence  of scalars such that . Then  is bounded, since by subadditivity the inequality

holds. Thus we have,

 as .

Thus the scalar multiplication is continuous. Hence g is a paranorm on the space .

It is quite routine to show that  is a metric space with the metric   provided that , where g is defined by (3); and using a similar method to that in [9] one can show that  is complete under the metric mentioned above .

2. Kothe-Toeplitz Duals

If X is a sequence space we define [13]

, and  are called the - (or Kothe-Toeplitz), - (or generalized Kothe-Toeplitz) and - dual spaces of X, respectively. Note that.

In this section we shall obtain the - , - and - dual of . For our purpose we need the following lemma.

Lemma 2.1 ([10, Theorem 5.10 ]). (i) Let 1 for every N. Then  if and only if there exists an integer  such that


(ii) Let  for every N. Then  if and only if


Lemma 2.2 ([1, Theorem 6]). (i) Let  for every N. Then  if and only if there exists an integer  such that


(ii) Let  for every N. Then  if and only if


Lemma 2.3 ([1, Theorem 1]). Let  for every N. Then  if and only if (6) and (7) hold and  for N also holds.

Theorem 2.1. Let 1for every N. Define the sets  and as follows:



Proof. Let . Then by (1) one can easily derive that


where N and

Let . Then by combining (8) with (i) of Lemma 2.1 we see that  whenever  if and only if  wheneve . This shows that.

Again, by Abel’s transformation, we have

=, for  N                               (9)

where  is define as

where N. Thus from Lemma 2.3 with (9) we have  whenever  if and only if  whenever . Hence from (6) we derive that


which shows that .

Also, from Lemma 2.2 together with (9) we have  whenever  if and only if  whenever . This again gives the condition (10) which means that.

Theorem 2.2. Let  for every N. Define  and  as




Proof. The proof is similar as that of above theorem 2.1 by using second parts of Lemma 2.1, 2.2, and 2.3 instead of first parts, and so we omit the details.

3. Matrix Mapping on the Set

In this section we characterize the class of matrices  and .

Theorem 3.1. (i) Let 1for every N. Then  if and only if there exists an integer E >1 such that


 N             (12)

(ii) Let 0  for every N. Then  if and only if


Proof. (i). Necessity. Let . Then  exists for  and this implies that  for every fixed N. So by theorem 2.1 the necessities of (11) and (12) hold.

Sufficiency. Suppose the conditions (11) and (12) hold.

For N, consider the equation


When  then from (14) we have


Using inequality (2) we have from (15)


This shows that .

(ii) The proof of second part is similar as that of part (i) and so omitted.

Theorem 3.2. (i) Let 1for every N. Then  if and only if (11), (12) and (13) hold and there is a sequence  of scalars such that

 for all N.              (16)

Proof. Necessity. Suppose that  and 1. Since , so by above theorem the necessities of (11) and (12) hold. For the necessity of condition (16), we take for each fixed k, a sequence with

Then for each N we have , which shows that . This proves the necessity of the condition (16).

Sufficiency. Suppose that the conditions (11), (12), (14) and (16) hold. Then for , we have  for each n and so  exists.

For every N, we have

Letting  together with (11) and (16) gives


Also by letting  we have from (12) that

which leads together with (17) that . Thus the series  converges for every . Writing  for  we have from (15).

 N.   (18)

Comparing this with Lemma 2.3 with  for allN , we have the matrix  belongs to the class .

Thus by (18) we have


Now by combining (19) with the above results one can see that .

Thus the proof is complete.

If  for each N , then we have the following corollary.

Corollary 3.1. Let  for eachN. Then if and only if the conditions (11), (12) and (13) hold, and (16) also holds with for each  N.


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