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.

1 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

• (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 $Y^{*}$.

• (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.

2 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 $q_{n}= n$, $\forall n\in \mathbb{N}$.

3 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:

• (i) $\sum_{k}z_{k}$ is $wuC$.

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

• (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:

• (i) $\sum_{k}z_{k}$ is ($wuC$).

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

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

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

• (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:

• (i) $\sum_{k}z_{k}$ is ($wuC$).

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

• (iii) S described in (3.1) is continuous.

4 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:

• (i) $\sum_{k}z_{k}$ is ($wuC$).

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

• (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:

• (i) $\sum_{k}z_{k}$ is ($wuC$).

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

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

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

• (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:

• (i) $\sum_{k}z_{k}$ is ($wuC$).

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

• (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. □

5 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:

• (i) $\sum_{k}z_{k}$ is uc.

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

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