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Applications of Orlicz–Pettis theorem in vector valued multiplier spaces of generalized weighted mean fractional difference operators

Abstract

In this study, we deal with some new vector valued multiplier spaces \(S_{G_{h}}(\sum_{k}z_{k})\) and \(S_{wG_{h}}(\sum_{k}z_{k})\) related with \(\sum_{k}z_{k}\) in a normed space Y. Further, we obtain the completeness of these spaces via weakly unconditionally Cauchy series in Y and \(Y^{*}\). Moreover, we show that if \(\sum_{k}z_{k}\) is unconditionally Cauchy in Y, then the multiplier spaces of \(G_{h}\)-almost convergence and weakly \(G_{h}-\)almost convergence are identical. Finally, some applications of the Orlicz–Pettis theorem with the newly formed sequence spaces and unconditionally Cauchy series \(\sum_{k}z_{k}\) in Y are given.

Introduction and preliminaries

Consider Ω as the space of real (or complex) valued sequences. Consider Y to be a sequence space with linear topology. Then Y is said to be a K-space provided that each of the maps \(p_{i}:Y\rightarrow \mathbb{R}\) defined by \(p_{i}(z)=z_{i}\) is continuous \(\forall i\in \mathbb{N}\). A K-space Y, where Y is a complete linear space, is called FK space. A normed FK space is called BK space. An FK space Y is said to have the property AK if for every sequence \(y=(y_{n})_{n\geq 1}\in Y\)

$$ y= \lim_{n\rightarrow \infty } \sum_{k=1}^{n}y_{k}e^{k}, $$

where \(e^{k}=(0,0,0,\ldots,1,0,\ldots)\) such that 1 is in the kth-position \(\forall k\in \mathbb{N}\). The spaces of bounded, convergent, and null sequences, which are denoted by \(\ell _{\infty }\), c, and \(c_{0}\), respectively, are BK spaces which are endowed with the sup norm \(\|y\|_{\infty }=\sup_{k\in \mathbb{N}}|y_{k}|\). By \(\ell _{1}\), we denote the space of absolutely summable sequences, bs and cs are the spaces consisting of all bounded and convergent series. Let Y and Z be two sequence spaces and \(\mathcal{A}=(a_{nk})_{n,k\in \mathbb{N}}\) be an infinite matrix. Then, for \(z=(z_{k})\in Y\), we have \(\mathcal{A}:Y\rightarrow Z\) which is defined as

$$ (\mathcal{A}z)_{n}=\sum_{k}a_{nk}z_{k}. $$
(1.1)

If \(\sum_{k}a_{nk}z_{k}\) converges for each \(n\in \mathbb{N}\), then we call \(\mathcal{A}z\) the \(\mathcal{A}\)-transform of z. Thus, \(\mathcal{A}\in (Y,Z)\) iff the series in (1.1) converges \(\forall n\in \mathbb{N}\) and \(\mathcal{A}z\in Z\). A sequence \(z=(z_{k})\) is called \(\mathcal{A}-\)summable to \(p\in \mathbb{C}\) (the set of complex numbers) if \((\mathcal{A}z) \) converges to p. For a detailed study about recent results in summability theory, one can refer to [8, 24, 33]. The Euler gamma functions are represented by \(\Gamma {(\gamma )}\) where \(\gamma \in (0,\infty )\) is defined as an improper integral such as \(\Gamma {(\gamma )}=\int _{0}^{\infty }e^{-t}t^{\gamma -1}\,dt\). Let \((\gamma )_{k}\) be the generalized factorial function which is defined in terms of Euler gamma function as

$$ (\gamma )_{k}=\textstyle\begin{cases} 1, & k=0, \\ \frac{\Gamma {(\gamma +k)}}{\Gamma {(\gamma )}}=\gamma (\gamma +1)( \gamma +2)(\gamma +3)\cdots(\gamma +k-1), & k \in \mathbb{N}, \end{cases} $$

where \(\mathbb{N}\) is denoted by a set of all positive integers. Kizmaz [20] gave the idea of difference sequences spaces which was generalized by Et and Colak [15]. Recently, many specialists like Ahmad and Mursaleen [2], Tripathy [32], Altay and Basar [4] studied difference sequences spaces. For a detailed study about the difference sequence spaces, one can refer to [27, 28]. Furthermore, Baliarsingh ([6, 7]) defined the generalized fractional difference operator \(\Delta ^{\gamma }\), which is given as

$$ \bigl(\Delta ^{\gamma }z \bigr)_{k}= \sum _{i=0}^{\infty } \frac{(-1)^{i}\Gamma {(\gamma +1)}}{i!\Gamma {(\gamma -i+1)}}z_{k+i}\quad (k \in \mathbb{N}_{0}), $$

where \(\mathbb{N}_{0}=\mathbb{N}\cup \{0\}\) and \(z\in \Omega \). In [25] the difference operator \(\Delta ^{\gamma }\), \(\Delta ^{(\gamma )}\), \(\Delta ^{-\gamma }\), \(\Delta ^{(- \gamma )}\) is defined from Ω to Ω as follows:

$$\begin{aligned}& \bigl(\Delta ^{\gamma }z \bigr)_{k}= \sum _{i=0}^{\infty } \frac{(-\gamma )_{i}}{i!}z_{k+i}, \end{aligned}$$
(1.2)
$$\begin{aligned}& \bigl(\Delta ^{(\gamma )}z \bigr)_{k}= \sum _{i=0}^{\infty } \frac{(-\gamma )_{i}}{i!}z_{k-i}, \end{aligned}$$
(1.3)
$$\begin{aligned}& \bigl(\Delta ^{-\gamma }z \bigr)_{k}= \sum _{i=0}^{\infty } \frac{(\gamma )_{i}}{i!}z_{k+i}, \end{aligned}$$
(1.4)
$$\begin{aligned}& \bigl(\Delta ^{(-\gamma )}z \bigr)_{k}= \sum _{i=0}^{\infty } \frac{(\gamma )_{i}}{i!}z_{k-i}. \end{aligned}$$
(1.5)

It is being assumed throughout that the above defined summations are convergent for \(z\in \Omega \). For a detailed study of fractional difference operator, one may refer to [6]. Recently, Mohiuddine et al. [23] studied linear isomorphic spaces of fractional-order difference operators. A lot of research has been made in this field, one can refer to [1, 17, 34].

Let Y be a Banach space. Then \(\sum_{k}z_{k}\in Y\) is called unconditionally convergent (uc) or unconditionally Cauchy (uC) if \(\sum_{k}z_{\pi (k)}\) is convergent (or Cauchy, resp.) for every \(\pi \in \mathbb{N}\), where π is the permutation. Further, \(\sum_{k}z_{k}\in Y\) is called weakly unconditionally Cauchy (\(wuC\)) if the sequence \((\sum_{k=1}^{n}z_{\pi (k)})\) is weakly Cauchy sequence or, alternatively, \(\sum_{k}z_{k}\) is \(wuC\) iff \(\sum_{k}|z^{\ast }(z_{k})|<\infty \) \(\forall z^{\ast }\in Y^{\ast }\), the space of all linear and bounded (continuous) functionals defined on Y. For a detailed study, one can refer to [10]. Using the completeness property of a subspace of \(\ell _{\infty }\) obtained by almost convergence, a depiction of \(wuC\) and uc series along with a new form of the Orlicz–Pettis theorem was presented by Aizpuru et al. [3]. Recently, a vector valued multiplier space through Cesàro convergence was introduced by Altay and Kama [5]. Esi [11] investigated some classes of generalized paranormed sequence spaces associated with multiplier sequences. Tripathy and Mahanta [31] also studied vector valued sequences associated with multiplier sequences. Furthermore, Karakus and Basar introduced the multiplier spaces \(S_{\Lambda }(\mathbb{T})\), \(S_{w\Lambda }(\mathbb{T})\) and studied some new multiplier spaces by using generalization of almost summability in [18, 19]. To know more about multiplier spaces, one may refer to [13, 14, 16, 29]. Lorentz proved that a sequence \(z=(z_{k})\in \ell _{\infty }\) is said to be almost convergent to \(L\in \mathbb{C}\) and is denoted by \(f-\lim z_{k}=L\) iff

$$ \lim_{m\rightarrow \infty } \sum_{k=0}^{m} \frac{z_{n+k}}{m+1}=L $$

uniformly in n. For a detailed study of almost convergence of the sequence spaces, one can refer to [12, 22, 35]. A sequence \(z=(z_{k})\in \ell _{\infty }\) is called \(F_{\mathcal{A}}\)-summable if

$$ \lim_{n\rightarrow \infty }\sum_{k=0}^{\infty }a_{nk}z_{k+m}=L $$

uniformly in \(m\in \mathbb{N}\).

Altay and Basar [4] first studied generalized weighted mean operator \(G(p,q)\) which was further enlarged to a difference operator \(G(p,q,\Delta )\) by Polat et al. [26]. Later, Demiriz and Cakan [9] introduced generalized weighted mean of order m as \(G(p,q,\Delta ^{m})\). Consider a set of all sequences U and \(p=(p_{n})\) such that \(p_{n}\neq 0\) \(\forall n\in \mathbb{N}\) and \(\frac{1}{p}= (\frac{1}{p}_{n} )\), \(\forall p\in \boldsymbol{U}\). As defined by Nayak et al. [25], the generalized weighted fractional difference mean or factorable fractional difference matrix \(G(p,q,\Delta ^{(\gamma ) })=(g_{nk}^{\Delta ^{(\gamma )}})\) is defined as follows:

$$ g_{nk}^{\Delta ^{(\gamma )}}=\textstyle\begin{cases} \sum_{i=k}^{n}p_{n}\frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}, & \text{when } 1 \leq k\leq n; \\ 0, & \text{when } k > n, \end{cases} $$

where \(i,k,n\in \mathbb{N}\) such that \(p_{n}\) depends on n and \(q_{k}\) on k.

Let us consider \(h=(h_{k})\) to be a strictly increasing sequence of positive real numbers such that

$$ 0< h_{1}< h_{2}< h_{3}< \cdots \quad \text{and}\quad \lim_{k\rightarrow \infty }h_{k}=\infty . $$
(1.6)

It is being assumed throughout that any term with a negative subscript is zero. The matrix \(G(p,q,\Delta ^{(\gamma )},h)=(g_{hnk}^{\Delta ^{(\gamma )}})\) is given by

$$ g_{hnk}^{\Delta ^{(\gamma )}}=\textstyle\begin{cases} \frac{1}{h_{n}} \sum_{i=k}^{n}p_{n} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}, & \text{when } 1 \leq k\leq n; \\ 0, & \text{when } k > n. \end{cases} $$

A sequence \(z=(z_{k})\in \Omega \) is called \(G_{h}\)-convergent to \(a\in \mathbb{R}\) if

$$\begin{aligned} \lim_{n\rightarrow \infty }\frac{1}{h_{n}} \sum _{k=1}^{n}p_{n}q_{k}\Delta ^{(\gamma )}z_{k} =&a,\quad \forall n\in \mathbb{N} \end{aligned}$$

or

$$\begin{aligned} \lim_{n\rightarrow \infty }\frac{1}{h_{n}} \sum _{k=1}^{n}p_{n} \Biggl( \sum _{i=k}^{n} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i} \Biggr)z_{k} =&a,\quad \forall n\in \mathbb{N}. \end{aligned}$$

Before going to our main results, we present some lemmas. For details, one may refer to [30].

Lemma 1.1

  1. (i)

    Let Y be a normed space. Then \(\sum_{k}z_{k}\) is said to be \(wuC\) series iff

    $$\begin{aligned} H =& \sup_{n\in \mathbb{N}} \Biggl\{ \Biggl\Vert \sum _{k=1}^{n}t_{k}z_{k} \Biggr\Vert : \vert t_{k} \vert \leq 1 \Biggr\} \\ =& \sup_{n\in \mathbb{N}} \Biggl\{ \Biggl\Vert \sum _{k=1}^{n}\epsilon _{k}z_{k} \Biggr\Vert : \vert \epsilon _{k} \vert \in \{-1,1\} \Biggr\} \\ =& \sup_{n\in \mathbb{N}} \Biggl\{ \sum_{k=1}^{n} \bigl\vert z^{*}(z_{k}) \bigr\vert : \forall z^{*}\in B_{Y^{*}} \Biggr\} , \end{aligned}$$

    where \(H\in \mathbb{R}^{+}\), where \(\mathbb{R}^{+}\) is the set of positive real numbers and \(B_{Y^{*}}\) represents the closed unit ball of \(Y^{*}\).

  2. (ii)

    Suppose that Y is a normed space. Then a formal series \(\sum_{k}z_{k}\) in Y is called uC (or \(wuC\)) iff, for any \((a_{n})\in \ell _{\infty }\), \(\sum_{k}a_{k}z_{k}\) converges, i.e., \(\sum_{k}z_{k}\) is an \(\ell _{\infty }-\)(respectively a \(c_{0}-\)) multiplier convergent series.

Main results

Definition 2.1

Consider Y to be a normed space and \(h= (h_{n})\) to be the sequence fulfilling property (1.6). Then \(z=(z_{k})\) is called \(G_{h}\)-almost convergent (or \(wG_{h}\)-almost convergent) to \(z_{0}\in Y\) if

$$\begin{aligned} \lim_{n\rightarrow \infty }\frac{1}{h_{n}} \sum _{k=m}^{m+n}p_{m+n} \Biggl( \sum _{i=k}^{m+n} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i} \Biggr)z_{k}=z_{0} \end{aligned}$$

uniformly in \(m\in \mathbb{N}\) or

$$\begin{aligned} \lim_{n\rightarrow \infty }\frac{1}{h_{n}} \sum _{k=m}^{m+n}p_{m+n} \Biggl( \sum _{i=k}^{m+n} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i} \Biggr)z^{*}(z_{k})=z^{*}({z}_{0}) \end{aligned}$$

uniformly in \(m\in \mathbb{N}\), \(\forall z^{*}\in Y^{*}\), where \(z_{0}\in Y \) is the \(G_{h}\)-limit (or weak \(G_{h}\)-limit) of \(z=(z_{k})\) and is denoted by \(G_{h}-\lim_{n\rightarrow \infty } z_{n}=z_{0}\) or \((wG_{h}-\lim_{n\rightarrow \infty } z_{n}=z_{0})\).

Let \(\Omega (Y)\) be the Y-valued sequence space. Then the spaces of all \(G_{h}\)-almost convergent and \(wG_{h}\)-almost convergent sequences in Y are denoted by \(G_{h}(Y)\) and \(wG_{h}(Y)\), respectively, which are defined as

$$\begin{aligned} G_{h}(Y) =& \Biggl\{ (z_{k})\in \Omega (Y): \lim _{n \rightarrow \infty }\frac{1}{h_{n}} \sum_{k=m}^{m+n}p_{m+n} \Biggl( \sum_{i=k}^{m+n}\frac{(-\gamma )_{i-k}}{(i-k)!}q_{i} \Biggr)z_{k}, \\ &{}\text{uniformly exists in } m\in \mathbb{N} \Biggr\} \end{aligned}$$

and

$$\begin{aligned} wG_{h}(Y) =& \Biggl\{ z^{*}(z_{k})\in \Omega (Y): \lim_{n\rightarrow \infty }\frac{1}{h_{n}} \sum_{k=m}^{m+n}p_{m+n} \Biggl( \sum_{i=k}^{m+n}\frac{(-\gamma )_{i-k}}{(i-k)!}q_{i} \Biggr)z^{*}(z_{k}), \\ &{}\text{uniformly exists in } m\in \mathbb{N} \Biggr\} . \end{aligned}$$

We may consider this definition as a generalization of almost convergence given by Lorentz [21].

Proposition 2.2

Suppose that Y is a normed space. If \(z=(z_{k})\) is \(G_{h}\)-almost convergent in Y, then \(z \in \ell _{\infty }(Y)\).

Proof

Since \(z=(z_{k})\) is an \(G_{h}\)-almost convergent sequence in Y, then \(\exists z_{0}\in Y\), \(\forall \varepsilon > 0\) and \(n^{\prime }_{0}\in \mathbb{N}\) such that

$$\begin{aligned} \Biggl\Vert \frac{1}{h_{n}} \sum_{k=m}^{m+n}p_{m+n} \Biggl( \sum_{i=k}^{m+n}\frac{(-\gamma )_{i-k}}{(i-k)!}q_{i} \Biggr)z_{k}- z_{0} \Biggr\Vert < \varepsilon , \end{aligned}$$

\(\forall m \in \mathbb{N}\) and \(n\geq n_{0}\), which implies that

$$\begin{aligned} \Biggl\Vert \frac{1}{h_{n}} \sum_{k=m}^{m+n}p_{m+n} \Biggl( \sum_{i=k}^{m+n}\frac{(-\gamma )_{i-k}}{(i-k)!}q_{i} \Biggr)z_{k} \Biggr\Vert \leq \Vert z_{0} \Vert + \varepsilon , \end{aligned}$$

\(\exists Z>0\) such that

$$\begin{aligned} \frac{p_{m}}{h_{n^{\prime }_{0}}}q_{m} \bigl\Vert \Delta ^{(\gamma )}z_{m} \bigr\Vert =& \Biggl\Vert \frac{h_{n^{\prime }_{0}+1}}{h_{n^{\prime }_{0}}}p_{m+n^{ \prime }_{0}+1} \sum _{k=m}^{m+n^{\prime }_{0}+1} \frac{q_{k}}{h_{n^{\prime }_{0}+1}}\Delta ^{(\gamma )}z_{k}- p_{m+n^{ \prime }_{0}+1} \sum _{k=m+1}^{m+n^{\prime }_{0}+1} \frac{q_{k}}{h_{n^{\prime }_{0}}}\Delta ^{(\gamma )}z_{k} \Biggr\Vert \\ \leq & \Biggl\Vert \frac{h_{n^{\prime }_{0}+1}}{h_{n^{\prime }_{0}}}p_{m+n^{\prime }_{0}+1} \sum _{k=m}^{m+n^{\prime }_{0}+1} \frac{q_{k}}{h_{n^{\prime }_{0}+1}}\Delta ^{(\gamma )}z_{k} \Biggr\Vert + \Biggl\Vert p_{m+n^{ \prime }_{0}+1} \sum_{k=m+1}^{m+n^{\prime }_{0}+1} \frac{q_{k}}{h_{n^{\prime }_{0}}}\Delta ^{(\gamma )}z_{k} \Biggr\Vert \\ \leq & \biggl(\frac{h_{n^{\prime }_{0}+1}}{h_{n^{\prime }_{0}}}+1 \biggr) \bigl( \Vert z_{0} \Vert +\varepsilon \bigr), \end{aligned}$$

which yields that

$$\begin{aligned} \bigl\Vert \Delta ^{(\gamma )}z_{m} \bigr\Vert \leq \biggl( \frac{h_{n^{\prime }_{0}+1}+h_{n^{\prime }_{0}}}{p_{m}q_{m}} \biggr) \bigl( \Vert z_{0} \Vert +\varepsilon \bigr) = Z. \end{aligned}$$

There exists an analog of Proposition 2.2 in weak topologies as, by the Banach–Mackey theorem, a weak bounded subset of Y is also bounded. □

Proposition 2.3

Let Y be the normed space. If \(z=(z_{k})\) is a \(wG_{h}\)-almost convergent sequence, then \((z_{k}) \in \ell _{\infty }(Y)\).

Definition 2.4

Suppose that Y is a normed space and \(h= (h_{n})\) is the sequence fulfilling property (1.6). Then \(\sum_{k}z_{k}\in Y\) is called \(G_{h}\)-almost convergent to \(z_{0}\in Y\) if

$$\begin{aligned} \lim_{n\rightarrow \infty } \Biggl\Vert \frac{1}{h_{n}} \sum _{k=m}^{m+n}p_{m+n}q_{k}\Delta ^{\gamma }s_{k}-z_{0} \Biggr\Vert =0 \end{aligned}$$

uniformly in \(m\in \mathbb{N}\), where \(\Delta ^{\gamma }s_{k}=\sum_{j=1}^{k}\Delta ^{\gamma }z_{j} \ \forall k \in \mathbb{N}\). So, we use the notation \(G_{h}-\sum_{k}z_{k}=z_{0}\) for \(G_{h}\)-almost convergence. By some easy calculation, we have \(G_{h}-\sum_{k}z_{k}=z_{0}\) iff

$$\begin{aligned} \lim_{n \rightarrow \infty } \Biggl[\frac{1}{h_{n}} \sum _{k=1}^{m}p_{m}q_{k}\Delta ^{(\gamma )}z_{k}+\frac{1}{h_{n}} \sum _{k=1}^{n}p_{m+n}q_{m+k}\Delta ^{(\gamma )}z_{m+k} \Biggr]=z_{0}, \end{aligned}$$

i.e.,

$$\begin{aligned} \lim_{n \rightarrow \infty } \Biggl[\frac{1}{h_{n}} \sum _{k=1}^{m}p_{m} \Biggl(\sum _{i=k}^{m} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i} \Biggr)z_{k}+\frac{1}{h_{n}} \sum_{k=1}^{n}p_{m+n} \Biggl(\sum_{i=k}^{n} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i} \Biggr)z_{m+k} \Biggr]=z_{0} \end{aligned}$$

in the norm topology, uniformly in \(m\in \mathbb{N}\) m, \(n,k \in \mathbb{N}\). We can write \(wG_{h}-\sum_{k}z_{k}=z_{0}\) if the series is weakly \(G_{h}\)-almost convergent to \(z_{0}\) in the weak topology. To obtain the definition given in [3], we will take \(h_{n}=n+1\), \(p_{n+m}=1\), \(\gamma =0\) such that \(q_{k}=\Delta q_{m+n}z_{k}\), where \(q_{n}= n\), \(\forall n\in \mathbb{N}\).

Multiplier spaces of \(G_{h}\)-almost convergence

This particular section deals with multiplier spaces of \(G_{h}-\)almost convergence and gives a theorem related to completeness through \(wuC\) series.

Definition 3.1

Suppose that Y is the normed space such that \(\sum_{k}z_{k}\) belongs to Y. Then the Y-valued multiplier space of \(G_{h}\)-almost convergence of \(\sum_{k}z_{k}\) is defined as

$$\begin{aligned} S_{G_{h}} \biggl(\sum_{k}z_{k} \biggr) =& \biggl\{ y=(y_{k})\in \ell _{ \infty }: \sum _{k}z_{k}y_{k} is G_{h} \text{-almost convergent} \biggr\} \end{aligned}$$

equipped with S (summing operator), and the sup norm is also defined by

$$ \boldsymbol{S}:S_{G_{h}} \biggl(\sum _{k}z_{k} \biggr)\rightarrow Y,\qquad y=(y_{k}) \mapsto \boldsymbol{S}(y)= G_{h}-\sum _{k}z_{k}y_{k}. $$
(3.1)

Theorem 3.2

Suppose that Y is a Banach space such that the formal series \(\sum_{k}z_{k} \) belongs to Y. Then the following are identical:

  1. (i)

    \(\sum_{k}z_{k}\) is \(wuC\).

  2. (ii)

    \(S_{G_{h}} (\sum_{k}z_{k} )\) is complete.

  3. (iii)

    \(c_{0}\subseteq S_{G_{h}} (\sum_{k}z_{k} )\).

Proof

(i) (ii) Since \(\sum_{k}z_{k}\) is \(wuC\) series in Y, then from Lemma 1.1 the following supremum is greater than zero, i.e., \(Q>0\) such that

$$\begin{aligned} Q = \sup_{n\in \mathbb{N}} \Biggl\{ \Biggl\Vert \sum _{k=1}^{n}t_{k}z_{k} \Biggr\Vert : \vert t_{k} \vert \leq 1 \Biggr\} . \end{aligned}$$

Now, let \(t^{n}\in S_{G_{h}} (\sum_{k}z_{k} )\), where \(t^{n}=(t^{n}_{k})\) such that \(\lim_{n\rightarrow \infty }\|t^{n}-t^{0}\|=0\) with \(t^{0} \in \ell _{\infty }\). We wish to prove that \(t^{0} \in S_{G_{h}} (\sum_{k}z_{k} )\). Let \(y_{n}= G_{h}-\sum_{k}t_{k}^{n}z_{k}\), then \(y_{n}\in Y\) since \((t^{n}_{k})\in S_{G_{h}} (\sum_{k}z_{k} )\). Now \(\forall \varepsilon >0\), \(\exists n^{\prime }_{0}\in \mathbb{N}\) and \(\nu _{1},\nu _{2}>n^{\prime }_{0} \) such that \(\|t^{\nu _{1}}-t^{\nu _{2}}\|<\frac{\varepsilon }{3Q}\). Therefore, for \(\nu _{1},\nu _{2}>n^{\prime }_{0}\), \(\exists n\in \mathbb{N}\) which satisfies the inequalities

$$\begin{aligned}& \Biggl\Vert y_{\nu _{1}}- \Biggl[ \sum_{k=1}^{m} \frac{p_{m+n}}{h_{n}}\sum_{i=k}^{m} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}t_{k}^{\nu _{1}}z_{k}+ \sum_{k=1}^{n}\frac{p_{m+n}}{h_{n}}\sum _{i=k}^{n} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}t_{m+k}^{\nu _{1}}z_{m+k} \Biggr] \Biggr\Vert < \frac{\varepsilon }{3}, \end{aligned}$$
(3.2)
$$\begin{aligned}& \Biggl\Vert y_{\nu _{2}}- \Biggl[ \sum_{k=1}^{m} \frac{p_{m+n}}{h_{n}}\sum_{i=k}^{m} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}t_{k}^{\nu _{2}}z_{k}+ \sum_{k=1}^{n}\frac{p_{m+n}}{h_{n}}\sum _{i=k}^{n} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}t_{m+k}^{\nu _{2}}z_{m+k} \Biggr] \Biggr\Vert < \frac{\varepsilon }{3}, \end{aligned}$$
(3.3)

and

$$\begin{aligned}& \Bigg\| \sum_{k=1}^{m}\frac{p_{m+n}}{h_{n}}\sum _{i=k}^{m} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i} \bigl(t_{k}^{\nu _{1}}-t_{k}^{\nu _{2}} \bigr)z_{k}+ \sum_{k=1}^{n} \frac{p_{m+n}}{h_{n}}\sum_{i=k}^{n} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i} \bigl(t_{m+k}^{\nu _{1}}-t_{m+k}^{\nu _{2}} \bigr)z_{m+k} ]\Bigg\| \\& \quad < \frac{\varepsilon }{3}, \end{aligned}$$
(3.4)

uniformly in \(m\in \mathbb{N}\). Thus, \(\exists n^{\prime }_{0}\in \mathbb{N}\) such that

$$\begin{aligned} \Vert y_{\nu _{1}}-y_{\nu _{2}} \Vert \leq \text{(3.2)}+ \text{(3.3)}+\text{(3.4)} < \varepsilon \end{aligned}$$

\(\forall \nu _{1}, \nu _{2} \geq n^{\prime }_{0} \). To a further extent, \(\exists y_{0} \in Y\) such that \(y_{n}\rightarrow y_{0}\) as \(n\rightarrow \infty \), as Y is complete.

Now, we also have to show that \(G_{h}-\sum_{k}t_{k}^{0}z_{k}=y_{0}\). For this, let \(\forall \varepsilon >0\), we have \(\|t^{j}-t^{0}\|< \frac{\varepsilon }{3Q}\), and for fixed j

$$ \Vert y_{j}-y_{0} \Vert < \frac{\varepsilon }{3}. $$
(3.5)

Hence, \(\exists n^{\prime }_{0}\in \mathbb{N}\) such that

$$ \Biggl\Vert y_{j}- \Biggl[ \sum _{k=1}^{m}\frac{p_{m}}{h_{n}}\sum _{i=k}^{m}\frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}t_{k}^{j}z_{k}+ \sum_{k=1}^{n}\frac{p_{m+n}}{h_{n}}\sum _{i=k}^{n} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}t_{m+k}^{j}z_{m+k} \Biggr] \Biggr\Vert < \frac{\varepsilon }{3} $$
(3.6)

\(\forall n\geq n^{\prime }_{0}\), uniformly in \(m \in \mathbb{N}\), since

$$\begin{aligned} y_{j}= G_{h}- \sum_{k}t_{k}^{j}z_{k} \quad \forall j\in \mathbb{N}. \end{aligned}$$

From Lemma 1.1, we get

$$\begin{aligned}& \Biggl[ \sum_{k=1}^{m} \frac{p_{m}}{h_{n}} \sum_{i=k}^{m} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i} \frac{(t_{k}^{j}-t_{k}^{0})}{ \Vert t^{j}-t^{0} \Vert }z_{k}+ \sum_{k=1}^{n} \frac{p_{m+n}}{h_{n}}\sum_{i=k}^{n} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i} \frac{(t_{m+k}^{j}-t_{m+k}^{0})}{ \Vert t^{j}-t^{0} \Vert }z_{m+k} \Biggr] \\& \quad \leq Q. \end{aligned}$$
(3.7)

Since \(\sum_{k}z_{k}\) is a \(wuC\) series, so \(\forall \varepsilon >0 \ \exists n^{\prime }_{0}\in \mathbb{N}\) such that

$$\begin{aligned}& \Biggl\Vert y_{0}- \Biggl[\sum_{k=1}^{m} \frac{p_{m}}{h_{n}}\sum_{i=k}^{m} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}t_{k}^{0}z_{k}+ \sum_{k=1}^{n}\frac{p_{m+n}}{h_{n}}\sum _{i=k}^{n} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}t_{m+k}^{0}z_{m+k} \Biggr] \Biggr\Vert \\& \quad \leq \text{(3.5)}+\text{(3.6)} \\& \qquad {} + \Biggl\Vert \sum _{k=1}^{m} \frac{p_{m}}{h_{n}}\sum _{i=k}^{m}\frac{(-\gamma )_{i-k}}{(i-k)!}q_{i} \bigl(t_{k}^{j}-t_{k}^{0} \bigr)z_{k}+ \sum_{k=1}^{n} \frac{p_{m+n}}{h_{n}}\sum_{i=k}^{n} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i} \bigl(t_{m+k}^{j}-t_{m+k}^{0} \bigr)z_{m+k} \Biggr\Vert \\& \quad \leq \frac{\varepsilon }{3}+\frac{\varepsilon }{3} + \bigl\Vert t^{j}-t^{0} \bigr\Vert .Q \\& \quad \leq \frac{\varepsilon }{3}+\frac{\varepsilon }{3} + \frac{\varepsilon }{3Q}.Q = \varepsilon , \end{aligned}$$

\(\forall n \geq n^{\prime }_{0}\) uniformly in \(m\in \mathbb{N}\). Therefore, \(t^{0}=(t^{0}_{k})_{k}\in S_{G_{h}}(\sum_{k}z_{k})\).

(ii) (iii) If \(S_{G_{h}}(\sum_{k}z_{k})\) is a complete space with \(t=(t_{k})\) being an arbitrary sequence in the space \(c_{0}\), then we need to show that \(t=(t_{k}) \in S_{G_{h}}(\sum_{k}z_{k})\). Now, since \(S_{G_{h}}(\sum_{k}z_{k})\) is a complete space, then it contains the space of eventually zero sequences \(c_{0}\). That is, \(\phi \subset S_{G_{h}}(\sum_{k}z_{k})\). Since \(c_{0}\) is an AK space, we have \(t^{[m]}=\sum_{k=1}^{m}t_{k}e^{k} \in S_{G_{h}}(\sum_{k}z_{k})\). Therefore, \(\lim_{m\rightarrow \infty } \|t^{[m]}-t\|_{\infty } = 0\). Thus \(t=(t_{k})\in S_{G_{h}}(\sum_{k}z_{k})\).

(iii) (i) Let us consider that a series \(\sum_{k}z_{k}\) is not \(wuC\), then \(\exists z^{*} \in B_{z^{*}}\) such that \(\sum_{k=1}^{\infty }|z^{*}(z_{k})|= +\infty \). Since \(\sum_{k=1}^{\infty }|z^{*}(z_{k})|= +\infty \), then there exists \(m_{1}\) such that \(\sum_{k=1}^{m_{1}}|z^{*}(z_{k})|> n.n\) for \(n>1\). Let us define

$$ (t_{k})=\textstyle\begin{cases} \frac{1}{n}, & \text{when } z^{*}(z_{k})\geq 0; \\ -\frac{1}{n}, & \text{when } z^{*}(z_{k})< 0,\end{cases} $$

for \(k=\{1,2,3,\ldots\}\), which implies that \(\sum_{k=1}^{m_{1}}t_{k}z^{*}(z_{k})>n\) and \(t_{k}z^{*}(z_{k})\geq 0\). Let \(m_{2}>m_{1}\) such that \(\sum_{k=m_{1}+1}^{m_{2}}t_{k}z^{*}(z_{k})> n^{2}.n^{2}\). Now, we define

$$ (t_{k})=\textstyle\begin{cases} \frac{1}{n^{2}}, & \text{when } z^{*}(z_{k})\geq 0; \\ -\frac{1}{n^{2}}, & \text{when } z^{*}(z_{k})< 0,\end{cases} $$

for \(k=\{m_{1}+1,\ldots m_{2}\}\), which shows that \(\sum_{k=m_{1}+1}^{m_{2}}t_{k}z^{*}(z_{k})>n^{2}\) and \(t_{k}z^{*}(z_{k})\geq 0 \). Thus, for arbitrary null sequences \(t=(t_{k})\in S_{G_{h}}(\sum_{k}z_{k})\), we have \(\sum_{k}t_{k}z^{*}(z_{k})\rightarrow +\infty \), which is a contradiction since the sequences of partial sums \(\{\sum_{k=1}^{\eta }t_{k}z^{*}(z_{k}) \}_{n \in \mathbb{N}}\) should be bounded by the hypothesis. Therefore, our claim is wrong, and hence the series \(\sum_{k}z_{k}\) must be \(wuC\) series.

(ii) (i) Suppose that \(S_{G_{h}}(\sum_{k}z_{k})\) is a Banach space and \(t=(t_{k})\in c_{0}(Y)\), which means \(c_{0}(Y)\subseteq S_{G_{h}}(\sum_{k}z_{k}) \) (already proved), which implies that \(\sum_{k}t_{k}z_{k}\) is almost convergent for all \(t=(t_{k})\in c_{0}(Y)\). From the monotonicity of \(c_{0}(Y)\), \(\sum_{k}t_{k}z_{k}\) is subseries almost convergent, and thus from the Orlicz–Pettis theorem, we get \(\sum_{k}t_{k}z_{k}\) is \(wuC\). □

Corollary 3.3

Let Y be the Banach space such that the formal series \(\sum_{k}z_{k}\) belongs to Y. Then \(\sum_{k}z_{k}\) is \(c_{0}\)-multiplier convergent iff \(c_{0}\subseteq S_{G_{h}}(\sum_{k}z_{k})\).

Aizpuru et al. [3] studied \(S_{AC}(\sum_{k}z_{k})\) which was given as

$$\begin{aligned} S_{AC} \biggl(\sum_{k}z_{k} \biggr)= \biggl\{ t=t_{k}\in \ell _{\infty }: AC \sum _{k}t_{k}z_{k} \text{ exists} \biggr\} . \end{aligned}$$

We have \(\sum_{k}z_{k}\) is almost convergent to \(z_{0}\in Y\). If \(AC\sum_{k}z_{k}=z_{0}\), then \(S_{AC}(\sum_{k}z_{k})\subseteq S_{G_{h}}(\sum_{k}z_{k})\).

Corollary 3.4

Suppose that Y is a Banach space such that the formal series \(\sum_{k}z_{k}\) belongs to Y. Then the following are identical:

  1. (i)

    \(\sum_{k}z_{k}\) is (\(wuC\)).

  2. (ii)

    \(c_{0}(Y)\subseteq S_{G_{h}}(\sum_{k}z_{k})\).

  3. (iii)

    \(S_{G_{h}}(\sum_{k}z_{k})\) is a Banach space.

  4. (iv)

    \(c_{0}(Y)\subseteq AC \sum_{k}t_{k}z_{k}\).

  5. (v)

    \(S_{AC}(\sum_{k}z_{k})\) is a Banach space.

Theorem 3.5

Suppose that Y is a normed space. Then Y is complete iff \(S_{G_{h}}(\sum_{k}z_{k})\) is closed in \(\ell _{\infty }\) for each \(wuC\) series \(\sum_{k}z_{k}\).

Proof

If we consider Y to be complete, then Theorem 3.2 shows that \(S_{G_{h}}(\sum_{k}z_{k})\) is complete for each \(wuC\) series \(\sum_{k}z_{k}\). Conversely, suppose that Y is not complete, then we obtain a series \(\sum_{k}z_{k}\) with \(\|z_{k}\|< \frac{1}{k2^{k}}\) and \(\sum_{k}z_{k} = z^{**}\in Y^{**}\setminus Y\). Thus, we have \(G_{h}-\sum_{k}z_{k}=z^{**}\). Let us define the series \(\sum_{k}x_{k}\), which is \(wuC\), as it is defined that \(x_{k}=kz_{k}\) for \(k \in \mathbb{N}\). Consider a sequence \(t=(t_{k})\in c_{0} \) given by \(t_{k}=\frac{1}{k} \ \forall k \in \mathbb{N}\), then we have \(G_{h}-\sum_{k}t_{k}z_{k}\in Y^{**}\setminus Y\). Therefore, \(t \notin S_{G_{h}}(\sum_{k}z_{k}) \), which implies that there exists \(\sum_{k}z_{k}\) such that \(S_{G_{h}}(\sum_{k}z_{k})\) is not complete. □

Theorem 3.6

Suppose that Y is a Banach space such that the formal series \(\sum_{k}z_{k}\) belongs to Y, then \(\sum_{k}z_{k}\) is \(wuC\) iff S defined in (3.1) is continuous.

Proof

Suppose that S is continuous and I is a set such that

$$ I= \Biggl\{ \Biggl\Vert \sum_{k=1}^{n}y_{k}z_{k} \Biggr\Vert : \Vert y_{k} \Vert \leq 1, \forall n\in \mathbb{N} \Biggr\} . $$
(3.8)

Thus, we have \(Q= \sup_{n\in \mathbb{N}}I \leq \|\boldsymbol{S}\|\) such that \(\sum_{k}z_{k}\) in Y is \(wuC\) as \(\phi \subset S_{G_{h}}(\sum_{k}z_{k})\). Conversely, let \(\sum_{k}z_{k}\) be \(wuC\) series, then \(Q= \sup_{n\in \mathbb{N}}I\), since the set I in (3.8) is bounded. If \(y=(y_{k})\in S_{G_{h}}(\sum_{k}z_{k})\), then \(\|\boldsymbol{S}(y)\|= \|G_{h}-\sum_{k}y_{k}z_{k} \| \leq Q\|y\|\). We can say that S is continuous. □

As defined in [3], the linear mapping T related with \(\sum_{k}z_{k}\) in Y is given as

$$ \boldsymbol{T}: S_{AC} \biggl(\sum_{k}z_{k} \biggr)\rightarrow Y,\qquad t=(t_{k}) \rightarrow A({t})= AC\sum _{k}a_{k}z_{k}. $$
(3.9)

Corollary 3.7

Suppose that Y is a Banach space such that the formal series \(\sum_{k}z_{k} \) belongs to Y. Then the following are identical:

  1. (i)

    \(\sum_{k}z_{k}\) is (\(wuC\)).

  2. (ii)

    \(\boldsymbol{T}: S_{AC} (\sum_{k}z_{k} )\rightarrow Y\) is continuous.

  3. (iii)

    S described in (3.1) is continuous.

Multiplier spaces of weak \(G_{h}\)-almost convergence

This particular section deals with multiplier spaces of weak \(G_{h}\)-almost convergence and build on the prior results to weak topologies.

Definition 4.1

Let us consider \(\sum_{k}z_{k}\) to be the formal series in the normed space Y. Then the Y-valued multiplier space of \(wG_{h}\)-almost convergence of \(\sum_{k}z_{k}\) is defined as

$$\begin{aligned} S_{wG_{h}} \biggl(\sum_{k}z_{k} \biggr)= \biggl\{ y=(y_{k})\in \ell _{\infty }:\sum _{k}z_{k}y_{k}\text{ is }wG_{h} \text{-almost convergent} \biggr\} , \end{aligned}$$

equipped with S (summing operator), and the sup norm is also defined by

$$ \boldsymbol{S}:S_{wG_{h}} \biggl(\sum _{k}z_{k} \biggr)\rightarrow Y,\qquad y \rightarrow \boldsymbol{S}(y)= wG_{h}-\sum_{k}z_{k}y_{k}. $$
(4.1)

Theorem 4.2

Suppose that Y is a Banach space such that the formal series \(\sum_{k}z_{k} \) belongs to Y. Then the following are identical:

  1. (i)

    \(\sum_{k}z_{k}\) is (\(wuC\)).

  2. (ii)

    \(S_{wG_{h}}(\sum_{k}z_{k})\) is a Banach space.

  3. (iii)

    \(c_{0} \subseteq S_{w G_{h}}(\sum_{k}z_{k})\).

Proof

Consider \(\sum_{k}z_{k}\) is \(wuC\) series in Y. Then Q such that \(Q=\sup_{n\in \mathbb{N}}I\) as defined in (3.8). If \((t_{k}^{n})\) is a Cauchy sequence in \(S_{wG_{h}}(\sum_{k}z_{k})\), then we have \(t^{0}=(t^{0}_{k})\in \ell _{\infty }(Y)\) such that \(t^{n}\rightarrow t^{0}\), as \(n \rightarrow \infty \). Since \(\ell _{\infty }(Y)\) is a Banach space, we wish to prove that \(t^{0}\in S_{wG_{h}}(\sum_{k}z_{k})\). Let \(y_{n}= wG_{h}-\sum_{k}t_{k}^{n}z_{k}\), then \(y_{n}\in Y\) since \((t^{n}_{k})\in S_{G_{h}} (\sum_{k}z_{k} )\) for each \(n\in \mathbb{N}\). Now, \(\forall \varepsilon >0 \ \exists n^{\prime }_{0}\in \mathbb{N}\) such that \(\|t^{\nu _{1}}-t^{\nu _{2}}\|<\frac{\varepsilon }{3Q} \ \forall \nu _{1}\), \(\nu _{2}>n^{\prime }_{0} \). Thus, for \(\nu _{1},\nu _{2}>n^{\prime }_{0} \ \exists n\in \mathbb{N}\) such that the following inequalities are satisfied for all \(y^{*}\in Y^{*}\):

$$\begin{aligned}& \Biggl\Vert y^{*}(y_{\nu _{1}})- \Biggl[ \sum _{k=1}^{m} \frac{p_{m+n}}{h_{n}}\sum _{i=k}^{m}\frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}y^{*} \bigl(t_{k}^{\nu _{1}}z_{k} \bigr) \\& \quad {}+ \sum _{k=1}^{n}\frac{p_{m+n}}{h_{n}} \sum _{i=k}^{n}\frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}y^{*} \bigl(t_{m+k}^{\nu _{1}}z_{m+k} \bigr) \Biggr] \Biggr\Vert < \frac{\varepsilon }{3}, \end{aligned}$$
(4.2)
$$\begin{aligned}& \Biggl\Vert y^{*}(y_{\nu _{2}})- \Biggl[ \sum _{k=1}^{m} \frac{p_{m+n}}{h_{n}}\sum _{i=k}^{m}\frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}y^{*} \bigl(t_{k}^{\nu _{2}}z_{k} \bigr) \\& \quad {}+ \sum _{k=1}^{n}\frac{p_{m+n}}{h_{n}} \sum _{i=k}^{n}\frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}y^{*} \bigl(t_{m+k}^{\nu _{2}}z_{m+k} \bigr) \Biggr] \Biggr\Vert < \frac{\varepsilon }{3}, \end{aligned}$$
(4.3)

and

$$\begin{aligned}& \Biggl\Vert \sum_{k=1}^{m} \frac{p_{m+n}}{h_{n}}\sum_{i=k}^{m} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}y^{*} \bigl[ \bigl(t_{k}^{\nu _{1}}-t_{k}^{\nu _{2}} \bigr)z_{k} \bigr] \\& \quad {}+ \sum_{k=1}^{n} \frac{p_{m+n}}{h_{n}}\sum_{i=k}^{n} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}y^{*} \bigl[ \bigl(t_{m+k}^{\nu _{1}}-t_{m+k}^{\nu _{2}} \bigr)z_{m+k} \bigr] \Biggr\Vert < \frac{\varepsilon }{3} \end{aligned}$$
(4.4)

uniformly in \(m\in \mathbb{N}\). Thus, \(\forall \varepsilon >0\)

$$\begin{aligned} \Vert y_{\nu _{1}}-y_{\nu _{2}} \Vert = \bigl\vert y^{*}(y_{\nu _{1}})-y^{*}(y_{\nu _{2}}) \bigr\vert \leq \text{(4.2)}+\text{(4.3)}+\text{(4.4)} < \varepsilon \end{aligned}$$

\(\forall \nu _{1}, \nu _{2} \geq n^{\prime }_{0} \) and \(y^{*}\in Y^{*}\). To a further extent, \(\exists y_{0}^{*} \in Y^{*}\) such that \(y_{n}\rightarrow y_{0}\) as \(n\rightarrow \infty \), as Y is complete.

Now, we also have to show that \(wG_{h}-\sum_{k}t_{k}^{0}z_{k}=y_{0}\). For this, let \(\forall \varepsilon >0\), we have \(\|t^{j}-t^{0}\|< \frac{\varepsilon }{3Q}\), and for fixed j and \(y^{*}\in Y^{*}\), we have

$$ \bigl\Vert y^{*}(y_{j}-y_{0}) \bigr\Vert < \frac{\varepsilon }{3}. $$
(4.5)

Hence, \(\exists n^{\prime }_{0}\in \mathbb{N}\) such that

$$\begin{aligned}& \Biggl\Vert y^{*}(y_{j})- \Biggl[ \sum _{k=1}^{m} \frac{p_{m}}{h_{n}}\sum _{i=k}^{m}\frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}y^{*} \bigl(t_{k}^{j}z_{k} \bigr)+ \sum _{k=1}^{n}\frac{p_{m+n}}{h_{n}}\sum _{i=k}^{n} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}y^{*} \bigl(t_{m+k}^{j}z_{m+k} \bigr) \Biggr] \Biggr\Vert \\& \quad < \frac{\varepsilon }{3} \end{aligned}$$
(4.6)

\(\forall n\geq n^{\prime }_{0}\), uniformly in \(m \in \mathbb{N}\), since

$$\begin{aligned} y_{j}= wG_{h}- \sum_{k}t_{k}^{j}z_{k}\quad \forall j\in \mathbb{N}. \end{aligned}$$

Now, from Lemma 1.1, we get

$$\begin{aligned}& \Biggl[ \sum_{k=1}^{m} \frac{p_{m}}{h_{n}}\sum_{i=k}^{m} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}y^{*} \frac{(t_{k}^{j}-t_{k}^{0})}{ \Vert t^{j}-t^{0} \Vert }z_{k} \\& \quad {}+ \sum_{k=1}^{n} \frac{p_{m+n}}{h_{n}}\sum_{i=k}^{n}\frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}y^{*} \frac{(t_{m+k}^{j}-t_{m+k}^{0})}{ \Vert t^{j}-t^{0} \Vert }z_{m+k} \Biggr]\leq Q. \end{aligned}$$
(4.7)

Since \(\sum_{k}z_{k}\) is \(wuC\), so \(\forall \varepsilon >0 \ \exists n^{\prime }_{0}\in \mathbb{N}\) such that

$$\begin{aligned}& \Biggl\Vert y^{*}(y_{0})- \Biggl[\sum _{k=1}^{m} \frac{p_{m}}{h_{n}}\sum _{i=k}^{m}\frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}y^{*} \bigl(t_{k}^{0}z_{k} \bigr)+\sum _{k=1}^{n}\frac{p_{m+n}}{h_{n}}\sum _{i=k}^{n} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}y^{*} \bigl(t_{m+k}^{0}z_{m+k} \bigr) \Biggr] \Biggr\Vert \\& \quad \leq (4.5)+(4.6) + \Biggl\Vert \sum_{k=1}^{m} \frac{p_{m}}{h_{n}}\sum_{i=k}^{m} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}y^{*} \bigl[ \bigl(t_{k}^{j}-t_{k}^{0} \bigr)z_{k} \bigr] \\& \qquad {}+ \sum_{k=1}^{n} \frac{p_{m+n}}{h_{n}}\sum_{i=k}^{n} \frac{(-\gamma )_{i-k}}{(i-k)!}q_{i}y^{*} \bigl[ \bigl(t_{m+k}^{j}-t_{m+k}^{0} \bigr)z_{m+k} \bigr] \Biggr\Vert \\& \quad \leq \frac{\varepsilon }{3}+\frac{\varepsilon }{3} + \bigl\Vert t^{j}-t^{0} \bigr\Vert .Q \\& \quad \leq \frac{\varepsilon }{3}+\frac{\varepsilon }{3} + \frac{\varepsilon }{3Q}.Q = \varepsilon \end{aligned}$$

\(\forall n \geq n^{\prime }_{0}\), uniformly in \(m\in \mathbb{N}\). Thus,

$$ t^{0}= \bigl(t^{0}_{k} \bigr)_{k}\in S_{wG_{h}} \biggl(\sum_{k}z_{k} \biggr). $$

(ii) (iii) If \(S_{wG_{h}}(\sum_{k}z_{k})\) is complete with \(t=(t_{k})\) being a sequence in \(c_{0}\), then we need to prove that \(t=(t_{k}) \in S_{wG_{h}}(\sum_{k}z_{k})\). Now, since \(S_{wG_{h}}(\sum_{k}z_{k})\) is a complete space, then it contains the space of eventually zero sequences \(c_{0}\). That is, \(\phi \subset S_{wG_{h}}(\sum_{k}z_{k})\). Since \(c_{0}\) is an AK space, we have \(t^{[m]}=\sum_{k=1}^{m}t_{k}e^{k} \in S_{wG_{h}}(\sum_{k}z_{k})\). Therefore, \(\lim_{m\rightarrow \infty } \|t^{[m]}-t\|_{\infty } = 0\). Thus \(t=(t_{k})\in S_{wG_{h}}(\sum_{k}z_{k})\).

(iii) (ii) We can prove this with the same example as given in Theorem 3.2.

(ii) (i) Suppose that \(S_{wG_{h}}(\sum_{k}z_{k})\) is a Banach space and \(t=(t_{k})\in c_{0}(Y)\), which means \(c_{0}(Y)\subseteq S_{wG_{h}}(\sum_{k}z_{k}) \) (already proved), which implies that \(\sum_{k}t_{k}z_{k}\) is almost convergent for all \(t=(t_{k})\in c_{0}(Y)\). Therefore, from the monotonicity of \(c_{0}(Y)\), \(\sum_{k}t_{k}z_{k}\) is subseries almost convergent, and thus we get \(\sum_{k}t_{k}z_{k}\) is \(wuC\) from the Orlicz–Pettis theorem. □

Corollary 4.3

Suppose that Y is a Banach space such that the formal series \(\sum_{k}z_{k} \) belongs to Y. Then \(\sum_{k}z_{k}\) is \(c_{0}\)-multiplier convergent iff \(c_{0}\subseteq S_{wG_{h}}(\sum_{k}z_{k})\).

\(S_{wG_{h}}(\sum_{k}z_{k})\) of almost summability related with \(\sum_{k}z_{k}\) was studied by Aizpuru et al. [3] which is given as

$$\begin{aligned} S_{wAC} \biggl(\sum_{k}z_{k} \biggr)= \biggl\{ t=(t_{k})\in \ell _{\infty }:wAC\sum _{k}t_{k}z_{k} \text{ exists} \biggr\} . \end{aligned}$$

Corollary 4.4

Suppose that Y is a Banach space such that the formal series \(\sum_{k}z_{k} \) belongs to Y. Then the following are identical:

  1. (i)

    \(\sum_{k}z_{k}\) is (\(wuC\)).

  2. (ii)

    \(c_{0}(Y)\subseteq S_{wG_{h}}(\sum_{k}z_{k})\).

  3. (iii)

    \(S_{w G_{h}}(\sum_{k}z_{k})\) is a Banach space.

  4. (iv)

    For all \(t=(t_{k})\in c_{0}\) there exists \(wAC\sum_{k}t_{k}z_{k}\).

  5. (v)

    \(S_{wAC}(\sum_{k}z_{k})\) is a Banach space.

Theorem 4.5

Suppose that Y is a normed space. Then Y is complete iff \(S_{wG_{h}}(\sum_{k}z_{k})\) is closed in \(\ell _{\infty }\) for each \(wuC\) series \(\sum_{k}z_{k}\).

Proof

The proof is similar to Theorem 3.5. So, we omit the details. □

Theorem 4.6

Suppose that Y is a Banach space such that the formal series \(\sum_{k}z_{k} \) belongs to Y, then \(\sum_{k}z_{k}\) is \(wuC\) iff S defined in (4.1) is continuous.

Proof

The proof is similar to Theorem 3.5. So, we omit the details. □

Corollary 4.7

Suppose that Y is a Banach space such that the formal series \(\sum_{k}z_{k} \) belongs to Y. Then the following are identical:

  1. (i)

    \(\sum_{k}z_{k}\) is (\(wuC\)).

  2. (ii)

    \(\boldsymbol{T}:S_{wAC}(\sum_{k}z_{k})\rightarrow Y\) is continuous.

  3. (iii)

    S described in (4.1) is continuous.

Remark 4.8

Suppose that χ is a linear space and \(\mu _{1}\) and \(\mu _{2}\) are linear topologies on χ such that \(\mu _{2}\) has a neighborhood base at 0 consisting of \(\mu _{1}\) closed sets [in a sense of Wilanski]. If \(z=(z_{i})\subset \chi \) is a Cauchy sequence converging to z in \((\chi ,\mu _{1})\), then it will converge to z in \((\chi ,\mu _{2})\).

Proposition 4.9

Let \(\sum_{k}z_{k}\) be uC in Y. Then \(S_{wG_{h}}(\sum_{k}z_{k})=S_{G_{h}}(\sum_{k}z_{k})\).

Proof

Suppose that \(y=(y_{k})\in S_{wG_{h}}(\sum_{k}z_{k})\). This implies that the partial sum of \(\sum_{k}y_{k}z_{k}\) obtains a Cauchy sequence that is again weakly \(G_{h}\)-convergent. Since the weak topology is connected with the norm topology, it will converge to the same point as in the norm topology. □

Orlicz–Pettis theorem for weak \(G_{h}\)-almost convergence

This particular section deals with a new version of the Orlicz–Pettis theorem for a Banach space Y. As noted earlier, the classical form of the Orlicz–Pettis theorem for the normed space claims that a series is subseries convergent in weak topology for the space is subseries convergent to the norm topology for the same space. In addition to that, if Y is complete, then \(\sum_{k}z_{k}\) is \(\ell _{\infty }\)-multiplier convergent. The Orlicz–Pettis theorem proportionately states that if Y is a Banach space and if \(\forall M\subset \mathbb{N}\) there exists a weakly sum \(\sum_{k\in M}z_{k}\), then \(\sum_{k}z_{k}\) is uc.

Theorem 5.1

Suppose that Y is a Banach space and sum \(\sum_{k\in M}z_{k}\) is \(wG_{h}\)-almost convergent for every \(M\subset \mathbb{N}\), then \(\sum_{k}z_{k}\) is uc.

Proof

From the previous results, we know that \(\sum_{k}z_{k}\) is \(wuC\). Let \(M\subset \mathbb{N}\), then \(wG_{h}-\sum_{k\in M}z_{k}=z_{0}\) \(\forall z_{0}\in Y\). From the classical Orlicz–Pettis theorem and the equalities given below

$$\begin{aligned} \sum_{k\in M}z^{*}(z_{k}) = G_{h}-\sum_{k\in M}z^{*}(z_{k})=z^{*}(z_{0}) \quad \forall z^{*}\in Y^{*}, \end{aligned}$$

we get \(\sum_{k}z_{k}\) is uc series. □

Corollary 5.2

Suppose that Y is a Banach space and \(\sum_{k}z_{k}\) belongs to Y. Then the given assertions are equivalent:

  1. (i)

    \(\sum_{k}z_{k}\) is uc.

  2. (ii)

    \(\ell _{\infty }\subseteq S_{G_{h}}(\sum_{k}z_{k})\).

  3. (iii)

    \(\ell _{\infty }\subseteq S_{wG_{h}}(\sum_{k}z_{k})\).

Here, we remark that if \(\sum_{k}z_{k}\) is \(wuC\) series in Y, then \(\sum_{k}y_{k}z_{k}\) is \(wuC\) series for all \(y_{k}\in \ell _{\infty }\). Thus,

$$\begin{aligned} S_{G_{h}} \biggl(\sum_{k}z_{k} \biggr) \subset S_{w} \biggl(\sum_{k}z_{k} \biggr), \end{aligned}$$

where \(S_{w}(\sum_{k}z_{k})= \{y=(y_{k})\in \ell _{\infty }:w\sum_{k}y_{k}z_{k} \text{ exists}\}\).

Availability of data and materials

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Acknowledgements

The first author thanks the Council of Scientific and Industrial Research (CSIR), India, for partial support under Grant No. 25(0288)/18/EMR-II, dated 24/05/2018.

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Raj, K., Jasrotia, S. & Mursaleen, M. Applications of Orlicz–Pettis theorem in vector valued multiplier spaces of generalized weighted mean fractional difference operators. Adv Differ Equ 2021, 370 (2021). https://doi.org/10.1186/s13662-021-03531-5

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  • DOI: https://doi.org/10.1186/s13662-021-03531-5

MSC

  • 40A05
  • 40A30

Keywords

  • Almost convergence
  • Generalized weighted mean operator \(G(u,v)\)
  • Weakly unconditionally Cauchy series
  • Unconditionally Cauchy series
  • Orlicz–Pettis theorem