Weihgted Cesaro Sequence Space and Related Matrix Transformation
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. Weihgted Cesaro Sequence Space and Related Matrix Transformation.Pure and Applied Mathematics Journal.Vol.4, No. 6, 2015, pp. 237-241. doi: 10.11648/j.pamj.20150406.12
Abstract: In this paper we define the weighted Cesaro sequence spaces ces (p, q).We prove the space ces(p, q) is a complete paranorm space. In section-2 we determine its Kothe-Toeplitz dual and continuous dual. In section-3 we establish necessary and sufficient condition for a matrix A to map ces (p, q) to and ces(p, q) to , where is the space of all bounded sequences and is the space of all convergent sequences. We also get some known and unknown interesting results as corollaries.
Keywords: Sequence Space, Kothe-Toeplitz Dual, Matrix Transformation
1. Introduction
Let be the space of all (real or complex) sequences and let and respectively the Banach spaces of bounded and convergent sequence endowed with the norm
In [8] Shiue introduce the Cesaro sequence space as
Leibowitz studied some properties of this space and showed that it is a Banach space. Lim [9] defined this space in a different norm as
and
where a sum over the ranges [, determined its dual spaces and characterize some matrix classes. Later in [10] Lim extended this space for the sequence with and defined as
For positive sequence of real numbers
Johnson and Mohapatra [11] defined the Cesaro sequence space
and studied some inclusion relations.
What amounts to the same thing defined by Khan and Rahman [3] as
For with , and a sum over the ranges [). They determined it’s Kothe -Toeplitz dual and characterized some matrix classes.
In this paper we define the Cesaro weighted sequence space in the following way.
Definition. If is a bounded sequence of positive real numbers, then for with , the Cesaro weighted sequence space is defined by
where and a sum over the range .
If for all , then reduces to studied by Lim[10]. Also, if for all n and for all n, then reduces to studied also Lim[9]. Obviously, .
In their paper [1] Maji and Srivastava defined the weighted Cesaro sequence space with a different norms and studied on some operators and inclusion results.
The main purpose of this note is to define and investigate the weighted Cesaro sequence space determine its Kothe-Toeplitz dual and characterize the class of matrices and where and are respectively the spaces of bounded and convergent complex sequences. By specializing sequences and , we get the results of Lim([9], [10]) as corollaries. Meanwhile, we also determine all continuous linear functional on for all
With regard to notation, the dual space of , i.e, the space of all continuous linear functionals on , will be denoted by We write
where for each n the maximum with respect to k in .
Throughout the paper the following well-known inequality (see [6] or [7]) will be frequently used. For any and any two complex numbers a and b we have
(1)
Where
To begin with, we show that the space is paranormed by
(2)
Provided
Clearly
,
where
Since have by Minkowski’s inequality
which shows that g is subadditive.
Finally we have to check the continuity of scalar multiplication. From the definition of , we have So, we may assume that . Now for any complex with , we have
It is quite routine to show that is a metric space with the metric provided that where g is defined by (2). And using a similar method to that in [3] one can show that is complete under the metric mentioned above.
2. Kothe-Toeplitz Duals
If X is a sequence space we define ([2], [5])
Now we are going to give the following theorem by which the generalized Kothe-Toeplitz dual will be determined.
Theorem 1: If then
.
Proof: Let Define
(3)
We want to show that .
Let and . Then using inequality (1) we get
=
=
which implies that the series convergent.
Therefore,
This shows,
Conversely, suppose that is convergent for all but . Then
, for every integer
So we can define a sequence ,
such that , we have
Now we define a sequence in the following way:
for and =0 for where is such that , the maximum is taken with respect to k in
Therefore,
It follows that
diverges.
Moreover
Therefore
That is, which is a contradiction to our assumption. Hence
. That is,
Then combining the two results, we get
.
The continuous dual of is determined by the following theorem.
Theorem 2: Let . Then continuous dual is isomorphic to , which is defined by (3)
Proof: It is easy to check that each can be written in the form
and the 1 appears at the k-th place. Then for any we have
(4)
Where, . By theorem 1, the convergence of for every x in implies that .
If and if we take , then by theorem 1, converges and clearly defines a linear functional on . Using the same kind of argument as in theorem 1, it is easy to check that
whenever , where is defined by (2).
Hence defines an element of
Furthermore, it is easy to see that representation (4) is unique. Hence we can define a mapping
by where the appears in representation (4). It is evident that is linear and bijective. Hence is isomorphic to
3. Matrix Transformations
In the following theorems we shall characterize the matrix classes and Let be an infinite matrix of complex numbers and X, Y two subsets of the space of complex sequences. We say that the matrix A defines a matrix transformation from X into Y and denote it by if for every sequence the sequence is in Y, where
provided the series on the right is convergent.
Theorem 3: Let . Then if and only if there exists an integer , such that .
Where
and
Proof: Sufficiency: Suppose there exists an integer such that Then by inequality (1), we have
Therefore, .
Necessity: Suppose that , but
Then converges for every whence for every n.
By theorem 1, it follows that each defined by
is an element of . Since is complete and since on , by the uniform boundedness principle there exists a number L independent of n and a number , such that
(5)
for every n and where is the closed sphere in with centre at the origin and radius .
Now choose an integer , such that
.
Since
there exists an integer , such that
(6)
Define a sequence as follows:
and for where is the smallest integer such that
Then one can easily show that but , which contradicts (5). This complete the proof of the theorem.
Corollary 3.1 (see [10]). Let . Then if and only if there exists an integer E >1 such that , where
Proof: If for every n in the above theorem, then we obtain the result.
Corollary 3.2 (see [9]). Let . Then if and only if
Proof: If and for all n in the above theorem, then we get the result.
Theorem 4. Let . Then if and only if
(i) and
(ii) there exists an integer , such that , where
and
Proof: Necessity. Suppose . Then exists for each exists for every . Therefore by an argument similar to that in theorem 3 we have condition (ii). Condition (i) is obtained by taking , where is a sequence with 1 at the k-th place and zeros elsewhere.
Sufficiency. The conditions of the theorem imply that
(7)
By (7) it is easy to check that is absolutely convergent for each For each and , we can choose an integer , such that
Then by inequality (1), we have
,
where and
It follows immediately that
This shows that which proved the theorem.
Corollary 4.1 (see [10]). Let . Then if and only if
(i)
(ii) there exists an integer E >1 such that , where
Proof: If for all n in the above theorem, then statements (i) and (ii) follow.
Corollary 4.2 (see [9]). Let . Then if and only if
(i)
(ii)
Proof: If and for all n in the above theorem, then we get the results.
Corollary 4.3. Let . Then if and only if
• the condition of Theorem 3 holds, and
• where is the space of all null sequences.
References