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  gave the idea of difference sequences spaces which was generalized by Et and Colak . Recently, many specialists like Ahmad and Mursaleen , Tripathy , Altay and Basar  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  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 . Recently, Mohiuddine et al.  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 . 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. . Recently, a vector valued multiplier space through Cesàro convergence was introduced by Altay and Kama . Esi  investigated some classes of generalized paranormed sequence spaces associated with multiplier sequences. Tripathy and Mahanta  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  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. . Later, Demiriz and Cakan  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. , 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 .

### 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 .

### 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 , 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.  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 , 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.  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.

## References

1. Ahmad, N., Sharma, S.K., Mohiuddine, S.A.: Generalized entire sequence spaces defined by fractional difference operator and sequence of modulus functions. TWMS J. Appl. Eng. Math. 10, 63–72 (2020) Special Issue

2. Ahmad, Z.U., Mursaleen, M.: Kothe-Toeplitz duals of some new sequence spaces and their matrix maps. Publ. Inst. Math. (Belgr.) 42(56), 57–61 (1987)

3. Aizpuru, A., Armario, R., Perez-Fernandez, F.J.: Almost summability and unconditionally Cauchy series. Bull. Belg. Math. Soc. Simon Stevin 15(4), 635–644 (2008)

4. Altay, B., Basar, F.: Some paranormed sequence spaces of non-absolute type derived by weighted mean. J. Math. Anal. Appl. 319(2), 494–508 (2006)

5. Altay, B., Kama, R.: On Cesàro summability of vector valued multiplier spaces and operator valued series. Positivity 22(2), 575–586 (2018)

6. Baliarsingh, P.: Some new difference sequence spaces of fractional order and their dual spaces. Appl. Math. Comput. 219(18), 9737–9742 (2013)

7. Baliarsingh, P., Dutta, S.: On the classes of fractional order difference sequence spaces and their matrix transformations. Appl. Math. Comput. 250, 665–674 (2015)

8. Basar, F.: Summability Theory and Its Applications. Bentham Sci. Pub., Istanbul (2012)

9. Demiriz, S., Çakan, C.: Some new paranormed difference sequence space and weighted core. Comput. Math. Appl. 64(6), 1726–1739 (2012)

10. Diestel, J.: Sequences and Series in Banach Spaces. Springer, New York (1984)

11. Esi, A.: On some classes of generalized paranormed sequence spaces associated with multiplier sequences. Iran. J. Sci. Technol. Trans. A, Sci. 32(4), 283–288 (2008)

12. Esi, A.: Lacunary strongly almost convergent sequences of fuzzy numbers. J. Concr. Appl. Math. 7(1), 64–69 (2009)

13. Esi, A.: Some classes of generalized difference paranormed sequence spaces associated with multiplier sequences. J. Comput. Anal. Appl. 11(3), 536–545 (2009)

14. Esi, A., Tripathy, B.C.: Strongly almost convergent generalized difference sequences associated with multiplier sequences. Math. Slovaca 57(4), 339–348 (2007)

15. Et, M., Çolak, R.: On some generalized difference sequence spaces. Soochow J. Math. 21(4), 377–386 (1995)

16. Hazarika, B., Esi, A.: On ideal convergent sequence spaces of fuzzy real numbers associated with multiplier sequences defined by a sequence of Orlicz functions. Ann. Fuzzy Math. Inform. 7(2), 289–301 (2014)

17. Jena, B.B., Paikray, S.K., Mohiuddine, S.A., Mishra, V.N.: Relatively equi-statistical convergence via deferred Nörlund mean based on difference operator of fractional-order and related approximation theorems. AIMS Math. 5(1), 650–672 (2020)

18. Karakus, M.: On certain vector valued multiplier spaces and series of operator. J. Math. Anal. 10(2), 1–11 (2019)

19. Karakus, M., Basar, F.: On generalization of almost convergence, completeness of some normed spaces with $$wuC$$ series and a version of Orlicz–Pettis theorem. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(4), 3461–3475 (2019)

20. Kizmaz, H.: On certain sequence spaces. Can. Math. Bull. 24(2), 169–176 (1981)

21. Lorentz, G.G.: A contribution to the theory of divergent sequences. Acta Math. 80, 167–190 (1948)

22. Mohiuddine, S.A.: An application of almost convergence in approximation theorems. Appl. Math. Lett. 24(11), 1856–1860 (2011)

23. Mohiuddine, S.A., Raj, K., Mursaleen, M., Alotaibi, A.: Linear isomorphic spaces of fractional-order difference operators. Alex. Eng. J. 60(1), 1155–1164 (2021)

24. Mursaleen, M.: Applied Summability Methods, Springer Briefs. Springer, New York (2014)

25. Nayak, L., Et, M., Baliarsingh, P.: On certain generalized weighted mean fractional difference sequence spaces. Proc. Natl. Acad. Sci. India Sect. A 89(1), 163–170 (2019)

26. Polat, H., Karakaya, V., Şimşek, N.: Difference sequence spaces derived by using a generalized weighted mean. Appl. Math. Lett. 24(5), 608–614 (2011)

27. Raj, K., Jamwal, S.: On some generalized statistically convergent sequence spaces. Kuwait J. Sci. 42(3), 86–104 (2015)

28. Raj, K., Sharma, S.K., Sharma, A.K.: Difference sequence spaces in n-normed spaces defined by Musielak-Orlicz function. Armen. J. Math. 3(3), 127–141 (2011)

29. Saha, S., Esi, A., Roy, S.: Some new classes of multiplier ideal convergent triple sequence spaces of fuzzy numbers defined by Orlicz functions. Palest. J. Math. 9(1), 174–186 (2020)

30. Swartz, C.: Multiplier Convergent Series. World Scientific, Singapore (2009)

31. Triapthy, B.C., Mahanta, S.: On a class of vector valued sequences associated with multiplier sequences. Acta Math. Appl. Sin. Engl. Ser. 20(3), 487–494 (2004)

32. Tripathy, B.C.: Generalized difference sequence spaces defined by Orlicz function in a locally convex space. J. Anal. Appl. 3(1), 175–192 (1995)

33. Wilansky, A.: Summability Through Functional Analysis. North-Holland Mathematics Studies, vol. 85 (1984)

34. Yaying, T., Hazarika, B., Mohiuddine, S.A.: On difference sequence spaces of fractional order involving Padovan numbers. Asian-Eur. J. Math. https://doi.org/10.1142/S1793557121500959

35. Zeltser, M., Mursaleen, M., Mohiuddine, S.A.: On almost conservative matrix methods for double sequence spaces. Publ. Math. (Debr.) 75(3–4), 387–399 (2009)

## 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.

## Author information

Authors

### Contributions

All authors have equal contributions. All authors read and approved the final manuscript.

### Corresponding author

Correspondence to M. Mursaleen.

## Ethics declarations

### Ethics approval and consent to participate

This article does not contain any studies with human participants or animals performed by any of the authors.

### Competing interests

The authors declare that they have no competing interests.

## Rights and permissions 