Pure and Applied Mathematics Journal
Volume 4, Issue 6, December 2015, Pages: 237-241

Weihgted Cesaro Sequence Space and Related Matrix Transformation

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

Department of Mathematics, Eden University College, Dhaka, Bangladesh

Email address:

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

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


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



To begin with, we show that the space  is paranormed by






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


We want to show that .

Let  and  . Then using inequality (1) we get






which implies that the series  convergent.


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  


It follows that




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


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  .



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


for every n and  where  is the closed sphere in  with centre at the origin  and radius .

Now choose an integer , such that



there exists an integer , such that


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



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


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


(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



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.


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