Journal of Function Spaces and Applications

VolumeΒ 2012Β (2012), Article IDΒ 612671, 10 pages

http://dx.doi.org/10.1155/2012/612671

## Some Matrix Transformations of Convex and Paranormed Sequence Spaces into the Spaces of Invariant Means

^{1}Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India^{2}Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 9 February 2012; Accepted 13 April 2012

Academic Editor: AlbertoΒ Fiorenza

Copyright Β© 2012 M. Mursaleen and S. A. Mohiuddine. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We determine the necessary and sufficient conditions to characterize the matrices which transform convex sequences and Maddox sequences into and .

#### 1. Introduction and Preliminaries

By , we denote the space of all real-valued sequences . Any vector subspace of is called a *sequence space*. We write that , , and denote the sets of all bounded, convergent, and null sequences, respectively, and note that ; also and are the set of all convergent and -absolutely convergent series, respectively, where for . In the theory of sequence spaces, a beautiful application of the well-known Hahn-Banach extension theorem gave rise to the concept of the Banach limit. That is, the lim functional defined on can be extended to the whole , and this extended functional is known as the Banach limit [1]. In 1948, Lorentz [2] used this notion of a weak limit to define a new type of convergence, known as the almost convergence. Later on, Raimi [3] gave a slight generalization of almost convergence and named it the -convergence.

A sequence space with a linear topology is called a - if each of the maps defined by is continuous for all . A -space is called an - if is a complete linear metric space. An -space whose topology is normable is called a -space. An -space is said to have property if and is a basis for , where is a sequence whose only nonzero term is a 1 in th place for each and , the set of all finitely nonzero sequences. If is dense in , then is called an -, thus implies . For example, the spaces , , and are -spaces.

Let and be two sequence spaces, and let be an infinite matrix of real or complex numbers. We write , provided that the series on the right converges for each . If implies that , then we say that defines a matrix transformation from into , and by , we denote the class of such matrices.

Let be a one-to-one mapping from the set of natural numbers into itself. A continuous linear functional on the space is said to be an or a - if and only if (i) if (i.e., for all ), (ii) , where , and (iii) for all .

Throughout this paper, we consider the mapping which has no finite orbits, that is, for all integer and , where denotes the th iterate of at . Note that a -mean extends the limit functional on the space in the sense that

for all , (cf. [4]). Consequently, , the set of bounded sequences all of whose -means are equal. We say that a sequence is - if and only if . Using this concept, Schaefer [5] defined and characterized -conservative, -regular, and -coercive matrices. If is translation, then is reduced to the set of almost convergent sequences [2].

The notion of -convergence for double sequences has been introduced in [6] and further studied in [7β9].

Recently, the sequence spaces and have been introduced in [10] which are related to the concept of -mean and the lacunary sequence .

In this section, we establish the necessary and sufficient conditions on the matrix which transforms -convex sequences in to the spaces and .

By a lacunary sequence, we mean an increasing integer sequence such that and as . Throughout this paper, the intervals determined by will be denoted by , and the ratio will be abbreviated by .

A bounded sequence of real numbers is said to be -*lacunary convergent* to a number if and only if , uniformly in , and let denote the set of all such sequences, that is,
In this case, is called the -limit of . We remark that(i) if , then is reduced to the space (cf. [11]),(ii).

A bounded sequence of real numbers is said to be -*lacunary bounded* if and only if , and let denote the set of all such sequences, that is,
We remark that and the spaces and are spaces with the norm

#### 2. Convex Sequence Spaces

Pati and Sinha [12] defined -convex sequences as follows: a real sequence is said to be -,, if for all , where is defined by The space of all bounded -convex sequences with is denoted by , that is, It is clear that .

It is well known that (Zygmund [13]) a bounded convex sequence is nonincreasing. It is easy to prove the identity which shows that , when . Properties of bounded -convex sequences have been investigated by Rath [14]. Note that . Recently, using the generalized difference operator , Γolak and Et [15], and Et and Γolak [16] defined and studied the sequence spaces , and . In this section, we establish the necessary and sufficient conditions on the matrix which transforms -convex sequences into the spaces and .

Write where for our convenience, we use instead of for throughout the paper.

Theorem 2.1. * if and only if*(i)*,
*(ii)* there exists a constant such that for ,*(iii)*, uniformly in , for each ,*(iv)*, uniformly in .*

*Proof. *In [17], a characterization of was given, where , in the sense of [18], is the bounded domain of a sequence of matrices . Now, by the setting
then , and the proof follows from Theorem 2.1 of [17].

Theorem 2.2. * if and only if the condition (i) of Theorem 2.1 holds and
*

*Proof. **Sufficiency*

Suppose that the conditions (i) and (2.6) hold and . Therefore, is bounded, and we have
Taking the supremum over on both sides and using (2.6), we get for .*Necessity*

Let . Condition follows as in the proof of Theorem 2.1. Write . It is easy to see that is a continuous seminorm on , since for ,
Suppose that (2.6) is not true, then there exists with . By the principle of condensation of singularities (cf. [19]), the set
is of second category in and hence nonempty, that is, there is with . But this contradicts the fact that is pointwise bounded on . Now, by the Banach-Steinhaus theorem, there is a constant such that
Now, we define a sequence by
Then, . Applying this sequence to (2.10), we get (2.6).

This completes the proof of the theorem.

#### 3. Maddox Sequence Spaces

A linear topological space over the real field is said to be a paranormed space if there is a subadditive function such that , and scalar multiplication is continuous, that is, and imply for all in and in , where is the zero vector in the linear space . Assume here and after that is a sequence such that for all . Let be a bounded sequence of positive real numbers with and . The sequence spaces were defined and studied by Et and Γolak [16] and Pati and Sinha [12]. If for some constant , then these sets reduce to , and , respectively. Note that is a linear metric space paranormed by where . and fail to be linear metric spaces because the continuity of scalar multiplication does not hold for them, but these two turn out to be linear metric spaces if and only if . is linear metric space paranormed by . All these spaces are complete in their respective topologies; however, these are not normed spaces in general (cf. [20]).

In this section, we characterize the matrix classes and .

Let be defined, then, for all , we write where and denotes the element of the matrix .

Theorem 3.1. * if and only if there exists such that for every ,*(i)*(ii)** for each , that is, uniformly in .**In this case, the -limit of is .*

*Proof. **Necessity*

We consider the case . Let . Since , the condition holds. Put , since exists for each and , therefore is a sequence of continuous real functionals on and further on . Now condition follows by arguing with uniform boundedness principle. The case can be proved similarly. *Sufficiency*

Suppose that the conditions and hold and . Now for every , we have
Therefore,
Thus, the series and converge for each and . For a given and , choose such that
where . Since (ii) holds, therefore there exists such that
for every . Hence, by the condition (ii), it follows that
is arbitrary small, and we have
uniformly in .

This completes the proof of the theorem.

Theorem 3.2. * if and only if there exists such that*(i)* for every ,*(ii)* for each , that is, uniformly in ,*(iii)* uniformly in .**In this case, the -limit of is .*

*Proof. **Necessity*

Let , then , and the conditions (ii) and (iii) follow from Theorem 3 of Schaefer [5]. Now on the contrary, suppose that (i) does not hold, then there exists such that . Therefore, by Theorem 3 of Schaefer [5], the matrix
that is, there exists such that . Now, let , then and , which contradicts that . Therefore, (i) must hold.*Sufficiency*

Suppose that the conditions hold and , then for every ,
Therefore is defined. Now arguing as in Theorem 3.1, we get , and the series and converge for . Hence, by the condition (iii), we get
uniformly in .

This completes the proof of the theorem.

Theorem 3.3. *Let for every , then if and only if there exists an integer such that*

*Proof. **Sufficiency*

Let (3.15) hold and that using the following inequality (see [21]):
for and , are two complex numbers , we have
where . Taking the supremum over on both sides and using (3.15), we get for , that is, .*Necessity*

Let . Write . It is easy to see that for , is a continuous seminorm on , and is pointwise bounded on . Suppose that (3.15) is not true, then there exists with . By the principle of condensation of singularities [19], the set
is of second category in and hence nonempty, that is, there is with . But this contradicts the fact that is pointwise bounded on . Now, by the Banach-Steinhaus theorem, there is constant such that
Now, we define a sequence by
where and
Then it is easy to see that and . Applying this sequence to (3.19), we get the condition (3.15).

This completes the proof of the theorem.

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