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On some new matrix transformations

Abstract

In this paper, we characterize some matrix classes (ω(p,s), V σ λ ), ( ω p (s), V σ λ ) and ( ω p ( s ) , V σ λ ) reg under appropriate conditions.

1 Introduction

Let w denote the set of all real and complex sequences x=( x k ). By l and c, we denote the Banach spaces of bounded and convergent sequences x=( x k ) normed by x= sup k | x k |, respectively. A linear functional L on l is said to be a Banach limit [1] if it has the following properties:

(1) L(x)0 if n0 (i.e., x n 0 for all n),

(2) L(e)=1, where e=(1,1,),

(3) L(Dx)=L(x), where the shift operator D is defined by D( x n )={ x n + 1 }.

Let B be the set of all Banach limits on l . A sequence x is said to be almost convergent if all Banach limits of x coincide. Let c ˆ denote the space of the almost convergent sequences. Lorentz [2] has shown that

c ˆ = { x l : lim m d m , n ( x )  exists, uniformly in  n } ,

where

d m , n (x)= x n + x n + 1 + x n + 2 + + x n + m m + 1 .

The study of regular, conservative, coercive and multiplicative matrices is important in the theory of summability. In [3], King used the concept of the almost convergence of a sequence introduced by Lorentz to define more general classes of matrices than those of regular and conservative ones.

In [4], Schaefer defined the concepts of σ-conservative, σ-regular and σ-coercive matrices and characterized the matrix classes (c, V σ ), ( c , V σ ) reg and ( l , V σ ), where V σ denotes the set of all bound sequences, all of whose invariant means (or σ-means) are equal. In [5], Mursaleen characterized the classes (c(p), V σ ), ( c ( p ) , V σ ) reg and ( l (p), V σ ) of matrices, which generalized the results due to Schaefer [4]. In [6], Mohiuddine and Aiyup defined the space ω(p,s) and obtained necessary and sufficient conditions to characterize the matrices of classes (ω(p,s), V σ ), ( ω p (s), V σ ) and ( ω p ( s ) , V σ ) reg .

Matrix transformations between sequence spaces have also been discussed by Savaş and Mursaleen [7], Basarir and Savaş [8], Mursaleen [5, 916], Vatan and Simsek [17], Savaş [1824], Vatan [25] and many others.

In this paper we characterize the matrix classes from this space to the space V σ λ , i.e., we obtain necessary and sufficient conditions to characterize the matrices of classes (ω(p,s), V σ λ ), ( ω p (s), V σ λ ) and ( ω p ( s ) , V σ λ ) reg .

2 Preliminaries

Let σ be a one-to-one mapping from the set N of natural numbers into itself. A continuous linear functional φ on l is said to be an invariant mean or σ-mean if and only if

(i) φ(x)0 when the sequence x=( x k ) has x k 0 for all k;

(ii) φ(e)=1;

(iii) φ(x)=φ( x σ ( k ) ) for all x l .

Let V σ denote the set of bounded sequences all of whose σ-means are equal. We say that a sequence x=( x k ) is σ-convergent if and only if x V σ . For σ(n)=n+1, the set V σ is reduced to the set c ˆ of almost convergent sequences [2, 26].

If x=( x n ), write Tx=( x σ ( n ) ). It is easy to show that

V σ = { x l : lim m t m n ( x ) = L ,  uniformly in  n ; L = σ - lim x } ,

where

t m n (x)= 1 m + 1 j = 0 m T j x n

and σ m (n) denotes the m th iterate of σ at n.

If p k is real and positive, we define (see Maddox [27])

c 0 (p)= { x : lim k | x k | p k = 0 }

and

c(p)= { x : lim k | x k l | p k = 0  for some  l } .

The classes (ω(p,s), V σ ), ( ω p (s), V σ ) and ( ω p ( s ) , V σ ) reg have been defined by Mohiuddine and Aiyup [6] and, for p=( p k ) with p k >0, the space ω(p,s) is defined for s0 by

ω(p,s)= { x : 1 n k = 1 n k s | x k l | p k 0 , n  for some  l , s 0 } ,

where s=( s k ) is an arbitrary sequence with s k 0 (k=1,2,). If p k =p, for each k, we have ω(p,s)= ω p (s).

The sequence space

ω(p)= { x : 1 n k = 1 n | x k l | p k 0 , n }

for some l, which has been investigated by Maddox is the special case of ω(p,s) which corresponds to s=0. Obviously ω(p)ω(p,s).

We further define the following.

Let λ=( λ m ) be a non-decreasing sequence of positive numbers tending to ∞ such that

λ m + 1 λ m +1, λ 1 =1.

A sequence x=( x k ) of real numbers is said to be (σ,λ)-convergent to a number L if and only if x V σ λ , where

V σ λ = { x l : lim m t m n ( x ) = L ,  uniformly in  n ; L = ( σ , λ ) - lim x } , t m n ( x ) = 1 λ m i I m x σ i ( n ) ,

and I m =[m λ m +1,m]. Note that c V σ λ l . For σ(n)=n+1, V σ λ reduces to the space V ˆ λ of almost λ-convergent sequences [28]; and if we take σ(n)=n+1 and λ m =m, then V σ λ reduces to c ˆ (see [29]). Further, if we take λ m =m, then V σ λ reduces to V σ .

If E is a subset of ω, then we write E + for a generalized Köthe-Toeplitz dual of E; i.e.,

E + = { a : k a k x k  converges for every  x E } .

If 0< p k 1, then ω + (p)=M, where

M= { a : r = 0 max r { ( 2 r N 1 ) 1 p k | a k | } <  for some integer  N > 1 } ,

and max is the maximum taken over 2 r k< 2 r + 1 (see Theorem 4, [30]).

If X is a topological linear space, we denote by X the continuous dual of X; i.e., the set of all continuous linear functionals on X. Obviously,

[ ω ( p , s ) ] = { a : r = 0 max r { ( 2 r N 1 ) 1 p k | a k s k | } <  for some integer  N > 1 } .

3 Main results

Let X and Y be two nonempty subsets of the space w of complex sequences. Let A=( a n k ) (n,k=1,2,) be an infinite matrix of complex numbers. We write Ax=( A n (x)) if A n (x):= k a n k x k converges for each n. (Throughout, k will denote summation over k from k=1 to k=.) If x=( x k )X implies that Ax=( A n (x))Y, we say that A defines a (matrix) transformation from X to Y and we denote it by A:XY. By (X,Y) we mean the class of matrices A such that A:XY.

We now characterize the matrices in the class ( c 0 (p), V σ 0 λ (p)). We write

t m , n (Ax)= k a(n,k,m) x k ,

where

a(n,k,m)= 1 λ m i I m a σ i ( n ) , k .

Theorem 3.1 Let 0< p k 1, then A(ω(p,s), V σ λ ) if and only if

(i) there exists an integer B>1 such that for every n

D n = sup m r = 0 max r ( 2 r B 1 ) 1 p k | a ( n , k , m ) s k | <;

(ii) α ( k ) = { a n k } n = 1 V σ λ for each k;

(iii) α= { k a n k } n = 1 V σ λ .

In this case the σ-lim of Ax is (limx)[u k u k ]+ k u k x k for every xw(p,s), where u=σ-lima and u k =σ-lim a ( k ) , k=1,2, .

Proof Suppose that A(ω(p,s), V σ λ ). Define e k =(0,0,,1,0,) having 1 in the k th coordinate sequence. Since e and e k are in ω(p,s), necessity of (ii) and (iii) is clear. We know that k a(n,k,m) x k converges for each m, n and xω(p,s). Therefore ( a ( n , k , m ) ) k ω + (p,s) and

r = 0 max r ( 2 r B 1 ) 1 p k | a ( n , k , m ) s k | <

for each m, n (see [31]). Furthermore, if f m n (x)= t m n (Ax), then { f m n } m is a sequence of continuous linear functionals on ω(p,s) such that lim m t m n (Ax) exists. Therefore, by using the Banach-Steinhaus theorem, the necessity of (i) follows immediately.

Conversely, suppose that the conditions (i), (ii) and (iii) hold and xω(p,s). We know that ( a ( n , k , m ) ) k and u k are in ω + (p,s) and that the series k a(n,k,m) x k and k u k x k converge for each m, n. Write

c(n,k,m)=a(n,k,m) u k .

Then

k a(n,k,m) x k = k u k x k +l k c(n,k,m)+ k c(n,k,m)( x k l)

by (ii) for some integer k 0 >0, we have

lim m k k 0 c(n,k,m)( x k l)=0,uniformly in n,

where l is the limit of x for xω(p,s). Since

sup m , n r max r ( 2 r B 1 ) 1 p k | c ( n , k , m ) | 2 D n , lim m k k 0 | a ( n , k , m ) u k s k | | s k ( x k l ) | = 0 ,

uniformly in n, whence

lim n k a(n,k,m) x k lu+ k u k ( x k l).

 □

Theorem 3.2 Let 1 p k <, then A( ω p (s), V σ λ ) if and only if

(i) for every n

M(A)= sup m r 2 r p ( r | a ( n , k , m ) s k | q ) 1 q <,

where p 1 + q 1 =1;

(ii) α ( k ) V σ λ for each k;

(iii) α V σ λ .

Proof Assume that the conditions are satisfied and let x ω p (s). Then

| t m n ( A x ) | r = 0 r | a ( n , k , m ) s k x k s k | r = 0 ( r | a ( n , k , m ) s k | q ) 1 q ( r | x k | p ) 1 p ,

and hence t m n (Ax) is absolutely and uniformly convergent for each m, n. Note that (i) and (ii) imply that

r = 0 2 r p ( r | s k u k | ) 1 q M(A)<,

so that by Hölder’s inequality, k u k x k <. Now as in the converse part of Theorem 3.1, it follows that A( ω p (s), V σ λ ).

Conversely, suppose that A( ω p (s), V σ λ ). Since e k and e are in ω p (s), the necessity of (ii) and (iii) is clear. For the necessity of (i), suppose that

t m n (Ax)= k a(n,k,m) x k

exits for each n whenever x ω p (s). Then, for each n and r0, write

f n r (x)= r a(n,k,m) x k .

Then { f n r } m is a sequence of continuous linear functionals on ω p (s). Since

| f n r ( x ) | ( r | a ( n , k , m ) s k | q ) 1 q ( r | s k x k | p ) 1 p 2 r p ( r | a ( n , k , m ) s k | q ) 1 q x,

it follows that for each n,

lim j r = 0 j f m r (x)= t m n (Ax)

is in the dual space ω p . Hence there exists a K m n such that

| a ( n , k , m ) s k | K m n x.
(3.1)

For each n and any integer j>0, define x ω p (s) as in [30] (Theorem 7, p.173), we get

r = 0 j 2 r p ( r | a ( n , k , m ) s k | q ) 1 q K m n .

Hence, for each n,

r = 0 2 r p ( r | a ( n , k , m ) s k | q ) 1 q K m n <.
(3.2)

Since t m n (Ax) is absolutely convergent, we get

| t m n ( A x ) | r = 0 2 r p ( r | a ( n , k , m ) s k | q ) 1 q x,

so that

K m n r = 0 2 r p ( r | a ( n , k , m ) s k | q ) 1 q .
(3.3)

By virtue of (3.2) and (3.3),

K m n = r = 0 2 r p ( r | a ( n , k , m ) s k | q ) 1 q .

Finally, by (Theorem 11, [30], p.114) for every n, the existence of lim m t m n (Ax) on ω p (s) implies that

sup m K m n = sup m r = 0 2 r p ( r | a ( n , k , m ) s k | q ) 1 q <,

which is (i). □

Theorem 3.3 Let 0< p k <, then A ( ω p ( s ) , V σ ) reg if and only if conditions (i), (ii) with σ-lim=0 and (iii) with σ-lim=+1 of Theorem  3.2 hold.

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Acknowledgements

Dedicated to Professor Hari M Srivastava.

We would like to thank the referees for their valuable comments.

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Correspondence to Rahmet Savaş.

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Savaş, R., Savaş, E. On some new matrix transformations. J Inequal Appl 2013, 254 (2013). https://doi.org/10.1186/1029-242X-2013-254

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Keywords

  • Banach Space
  • Dual Space
  • Matrix Transformation
  • Sequence Space
  • Nonempty Subset