- Open Access
On some new matrix transformations
© Savaş and Savaş; licensee Springer. 2013
- Received: 21 November 2012
- Accepted: 18 April 2013
- Published: 20 May 2013
In this paper, we characterize some matrix classes , and under appropriate conditions.
- Banach Space
- Dual Space
- Matrix Transformation
- Sequence Space
- Nonempty Subset
Let w denote the set of all real and complex sequences . By and c, we denote the Banach spaces of bounded and convergent sequences normed by , respectively. A linear functional L on is said to be a Banach limit  if it has the following properties:
(1) if (i.e., for all n),
(2) , where ,
(3) , where the shift operator D is defined by .
The study of regular, conservative, coercive and multiplicative matrices is important in the theory of summability. In , 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 , Schaefer defined the concepts of σ-conservative, σ-regular and σ-coercive matrices and characterized the matrix classes , and , where denotes the set of all bound sequences, all of whose invariant means (or σ-means) are equal. In , Mursaleen characterized the classes , and of matrices, which generalized the results due to Schaefer . In , Mohiuddine and Aiyup defined the space and obtained necessary and sufficient conditions to characterize the matrices of classes , and .
Matrix transformations between sequence spaces have also been discussed by Savaş and Mursaleen , Basarir and Savaş , Mursaleen [5, 9–16], Vatan and Simsek , Savaş [18–24], Vatan  and many others.
In this paper we characterize the matrix classes from this space to the space , i.e., we obtain necessary and sufficient conditions to characterize the matrices of classes , and .
Let σ be a one-to-one mapping from the set N of natural numbers into itself. A continuous linear functional φ on is said to be an invariant mean or σ-mean if and only if
(i) when the sequence has for all k;
(iii) for all .
Let denote the set of bounded sequences all of whose σ-means are equal. We say that a sequence is σ-convergent if and only if . For , the set is reduced to the set of almost convergent sequences [2, 26].
and denotes the m th iterate of σ at n.
where is an arbitrary sequence with (). If , for each k, we have .
for some l, which has been investigated by Maddox is the special case of which corresponds to . Obviously .
We further define the following.
and max is the maximum taken over (see Theorem 4, ).
Let X and Y be two nonempty subsets of the space w of complex sequences. Let () be an infinite matrix of complex numbers. We write if converges for each n. (Throughout, will denote summation over k from to .) If implies that , we say that A defines a (matrix) transformation from X to Y and we denote it by . By we mean the class of matrices A such that .
Theorem 3.1 Let , then if and only if
(ii) for each k;
In this case the of Ax is for every , where and , .
for each m, n (see ). Furthermore, if , then is a sequence of continuous linear functionals on such that exists. Therefore, by using the Banach-Steinhaus theorem, the necessity of (i) follows immediately.
Theorem 3.2 Let , then if and only if
(ii) for each k;
so that by Hölder’s inequality, . Now as in the converse part of Theorem 3.1, it follows that .
which is (i). □
Theorem 3.3 Let , then if and only if conditions (i), (ii) with and (iii) with of Theorem 3.2 hold.
Dedicated to Professor Hari M Srivastava.
We would like to thank the referees for their valuable comments.
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