Almost absolute weighted summability with index k and matrix transformations

*Correspondence: mursaleenm@gmail.com 2Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan 3Al-Qaryah, Doharra, Street No.1 (West), Aligarh, 202002, UP, India Full list of author information is available at the end of the article Abstract In this paper we generalize the space ̂k of absolutely almost convergent series (J. Math. Anal. Appl. 161:50–56, 1991) via weighted mean transformations. We study some inclusion relations and their topological properties. Further we characterize certain matrix transformations.


Introduction
Let w be the set of all sequences of complex numbers. A Banach sequence space X is BK if the map p n : X → C defined by p n (x) = x n is continuous for all n ≥ 0. A BKspace X is said to have the AK property if φ ⊂ X and (e (v) ) is a basis for X, where e (v) = {0, 0, 0, . . . , 1 v th term , 0, 0, . . .} and φ = span{e (v) }. If φ is dense in X, then it is called an AD-space, so AK implies AD.
Let ∞ be the space of all bounded sequences. A sequence (x n ) ∈ ∞ is said to be almost convergent to γ if all of its Banach limits [1] coincide to γ . Lorentz [8] (see [10] for double sequences) characterized almost convergence by saying that a sequence (x n ) is almost convergent to γ if and only if 1 r + 1 r v=0 x n+v → γ as r → ∞ (uniformly in n). (1.1) This notion plays an important role in summability theory and was investigated by several authors. For example, it was later used to define and study some concepts such as conservative and regular matrices, some sequence spaces, and matrix transformations (see [2,6,7,9,11,12,15]).
Absolute almost convergence emerges naturally as an absolute analogue of almost convergence. To introduce this concept, let s n = n v=1 a v be a partial sum of a v . The series a v is said to be absolutely almost convergent series if (see [3]) The space of all absolutely almost convergent series k = a = (a n ) n∈N : ∞ m=0 |ψ m,n | k < ∞, uniformly in n, k > 0 was first defined and studied in [4]. We note an important relation between k and absolute Cesaro summability |C, 1| in Flett's notation [5], k ⊂ |C, 1| (see [4]).
The purpose of the present paper is to define an absolute almost weighted summability using some factors and weighted means and to study its topological structures. This new method of summability extends the well-known concept of absolute almost convergence of Das et al. [4], and space k of Das et al. [4] becomes a special case of our space |f (N u p )| k . We investigate relations between classical sequence spaces and show that the space |f (N u p )| k is not separable for k > 1. Also, we characterize the matrix classes (c, |f (N p )| k ) and (c, |f (N p )| k ), 1 ≤ k < ∞.

Main results
For any sequence (s n ), we define T m,n by where (p n ) is a sequence of positive real numbers with P n = p 0 + p 1 + · · · + p n → ∞ as n → ∞, P -1 = p -1 = 0.
A straightforward calculation then shows that So, we can give the following definition. Definition 2.1 Let a v be an infinite series with partial summations (s n ). Let (p n ) and (u n ) be sequences of positive real numbers. The series a v is said to be absolute almost For |f (N u p )| k , k ≥ 1, we write the set of all series summable by the method |f (N p ), u m | k . Then a v is summable |f (N p ), u m | k iff the series a v ∈ |f (N u p )| k . Note that, in the case u m = p m = 1 for m ≥ 0, it reduces to the set of absolutely almost convergent series k given by Das, Kuttner, and Nanda [3]. Further, it is clear that the space |N u p | k is derived from |f (N u p )| k by putting n = 0 [9,13,14], and also |f (N u p )| k ⊂ |N u p | k , but the converse is not true.
First we give some relations between the new method and classical sequence spaces such as bs and ∞ , which are the sets of all bounded series and bounded sequences, respectively.
Then, by the definition, there exists an integer M such that holds for all n. So, it is sufficient to show that the sequence (|F m,n (a))|) is bounded for a fixed number m. By It follows by applying (2.4) for any m ≥ M + 1 that a = (a v ) ∈ ∞ , which completes the proof.
This completes the proof.
So we have the following result in [3].
Proof It is routine to prove that the norm conditions are satisfied by (2.5). We only note that (2.5) is well defined. In fact, if a ∈ |f (N u p )| k , then, as in the proof of part (ii) of Theorem 2.2, there exists an integer M such that, for all n, This also gives that |a m 1 va m 2 v | k < ε/u 1-1/k 0 holds for all m 1 , m 2 > m 0 , i.e., the sequence (a m v ) is a Cauchy sequence in the set of complex numbers C. So, it converges to a number a v (v = 0, 1, . . .), i.e., lim m→∞ a m v = a v . Now, letting m 2 → ∞, by (2.6) we have for a m 1a |f (N u p )| k < ε for m 1 > m 0 . This means lim m→∞ a m = a. Further, since We note that if E is a BK -space such that bs ⊂ E ⊂ ∞ , then E is not separable (and hence not reflexive) (see [4]). Hence the following result at once follows from Theorem 2.2.

Matrix transformations on space |f (N p )| k
In this section we characterize certain matrix transformations on the space |f (N p )| k . First we recall some notations. Let X, Y be any subsets of ω and A = (a nv ) be an infinite matrix of complex numbers. By A(x): a nv x v are convergent for n ≥ 0. If Ax ∈ Y , whenever x ∈ X, then we say that A defines a matrix mapping from X into Y and denotes the class of all infinite matrices A such that A : X → Y by (X, Y ). Also, we denote the set of all k-absolutely convergent series by k , 1 ≤ k < ∞, i.e., which is a BK -space with respect to the norm Also we make use of the following lemma in [15].

Lemma 3.1 Suppose that A = (a nv ) is an infinite matrix with complex numbers and p = (p v ) is a bounded sequence of positive numbers such that H = sup v p v and C = max{1, 2 H-1 }. Then
Now we begin with the first theorem given the characterization of the class ( 1 , |f (N p )| k ). and On the other hand, it follows from (3.2) that, for all l, n, j ≥ 0, Now, let ε > 0. Then there exists an integer j 0 such that, for all l and n, Also, by (3.1), for each j, R(l, n, j) → 0 as l → ∞ uniformly in n, there exists an integer l 0 so that, for l ≥ l 0 and all n, So, we have, for l ≥ l 0 and all n, ∞ j=0 |x j |R(l, n, j) < ε, which implies, by (3.4), This states that This completes the proof.
In the special case p m = u m = 1 for all m ≥ 0, we have |f (N p )| = k , and so the following result follows from Theorem 3.2.  hold.
Proof Necessity. Let A ∈ (c, |f (N p )| k ). Then A(x) ∈ |f (N p )| k for every x ∈ c. Now, take x = e (j) and x = e = (1, 1, . . .). Then (3.1) and (3.7) hold, respectively. Also, it follows as in the proof of Theorem 3.2 that Let N be an arbitrary finite set of natural numbers, and define a sequence x by Now it is enough to show that the tail of this series tends to zero uniformly in n. To see that, we write