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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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
,
where
.
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
(1)
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
(3)
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])
(4)
and for any R (see [8])
(5)
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
(6)
(ii) Let for every N. Then if and only if
(7)
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:
and
Then
Proof. Let . Then by (1) one can easily derive that
(8)
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
(10)
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
and
Then
.
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
(11)
N (12)
(ii) Let 0 for every N. Then if and only if
(13)
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
(14)
When then from (14) we have
(15)
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
(17)
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
(19)
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.
References