Theory and Modern Applications

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

Not applicable.

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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.

None.

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Correspondence to M. Mursaleen.

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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

• 40A05
• 40A30

### Keywords

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