Open Access

On some new matrix transformations

Journal of Inequalities and Applications20132013:254

https://doi.org/10.1186/1029-242X-2013-254

Received: 21 November 2012

Accepted: 18 April 2013

Published: 20 May 2013

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 n 0 (i.e., x n 0 for all n),

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

(3) L ( D x ) = 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 T x = ( 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 s 0 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 A x = ( 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 A x = ( A n ( x ) ) Y , we say that A defines a (matrix) transformation from X to Y and we denote it by A : X Y . By ( X , Y ) we mean the class of matrices A such that A : X Y .

We now characterize the matrices in the class ( c 0 ( p ) , V σ 0 λ ( p ) ) . We write
t m , n ( A x ) = 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 ( lim x ) [ u k u k ] + k u k x k for every x w ( p , s ) , where u = σ - lim a 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 x ω ( 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 ( A x ) , then { f m n } m is a sequence of continuous linear functionals on ω ( p , s ) such that lim m t m n ( A x ) 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 x ω ( 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 x ω ( 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 l u + 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 ( A x ) 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 ( A x ) = k a ( n , k , m ) x k
exits for each n whenever x ω p ( s ) . Then, for each n and r 0 , 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 ( A x )
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 ( A x ) 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 ( A x ) 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.

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

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

Authors’ Affiliations

(1)
Department of Mathematics, Medeniyet University
(2)
Department of Mathematics, Istanbul Ticaret University

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© Savaş and Savaş; licensee Springer. 2013

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