1 Introduction

Any vector subspace of \(\Omega \) is called as a double sequence space, where \(\Omega \) is the set of all complex valued double sequences. If for every \(\varepsilon >0\) there exists \(n_0=n_0(\varepsilon )\in {\mathbb {N}}\) and \(l\in {\mathbb {C}}\) such that \(|x_{mn}-l|<\varepsilon \) for all \(m,n>n_0\), then we call that the double sequence x is convergent in the Pringsheim’s sense to the limit l and write \(p-\lim _{m,n\rightarrow \infty }x_{mn}=l\), where \(x=(x_{mn})\in \Omega \), \({\mathbb {N}}=\{0,1,2,\ldots \}\) and \({\mathbb {C}}\) denotes the complex, [10] field. A double sequence \(x=(x_{mn})\) is said to be bounded if \(\Vert x\Vert _{\infty }=\sup _{m,n\in {\mathbb {N}}}|x_{mn}|<\infty \). A Pringsheim convergent sequence need not to be bounded. If \(x=(x_{mn})\) is Pringsheim convergent and bounded, then it is called boundedly convergent in the Pringsheim sense. The space of all Pringsheim convergent, bounded and boundedly convergent double sequences are denoted by \(\mathcal {C}_p\), \(\mathcal {M}_u\) and \(\mathcal {C}_{bp}\), respectively. A sequence in the space \(\mathcal {C}_p\) is said to be regularly convergent if it is a single convergent sequence with respect to each index and denote the space of all such sequences by \(\mathcal {C}_r\). Also by \(\mathcal {C}_{bp0}\) and \(\mathcal {C}_{r0}\), we denote the spaces of all double sequences converging to 0 contained in the sequence spaces \(\mathcal {C}_{bp}\) and \(\mathcal {C}_{r}\), respectively. Móricz [8] proved that \(\mathcal {M}_u\), \(\mathcal {C}_{bp}\), \(\mathcal {C}_{bp0}\), \(\mathcal {C}_{r}\) and \(\mathcal {C}_{r0}\) are Banach spaces with the norm \(\Vert \cdot \Vert _\infty \).

Let us consider a double sequence \(x=(x_{mn})\) and define the sequence \(s=(s_{mn})\) via x by

$$\begin{aligned} s_{mn}=\sum _{k,l=0}^{m,n}x_{kl}, \end{aligned}$$
(1.1)

for all \(m,n\in {\mathbb {N}}\). Then, the pair of (x, s) is called a double series and is denoted by the symbol \(\sum _{k,l=0}^{\infty }x_{kl}\) or, more briefly, by \(\sum _{k,l}x_{kl}\). The double series is said to be converge to the sum \(\alpha \) if \(\sum _{k,l}x_{kl}=p-\underset{m,n\rightarrow \infty }{\lim }s_{mn}=\alpha \), where \(s_{mn}\) is defined by (1.1). The double series \(\sum _{k,l}x_{kl}\) converges absolutely if and only if \(\sum _{k,l}|x_{kl}|\) converges (see [11, Definition 138.1 and 138.2, p. 591]).

Remainder term of a double series \(\sum _{k,l}x_{kl}\) is given by Sever [12] in his master thesis, as follows:

$$\begin{aligned} r_{mn}=\sum _{k=0}^m\sum _{l=n+1}^\infty x_{kl}+\sum _{k=m+1}^\infty \sum _{l=0}^n x_{kl}+\sum _{k,l=m+1,n+1}^\infty x_{kl}. \end{aligned}$$

If a double series is convergent, then its remainder term must tend to zero; as \(m,n\rightarrow \infty \).

For every double sequence \(x=(x_{kl})\), its sections \(x^{[m,n]}\) defined by

$$\begin{aligned} x^{[m,n]}=\sum _{k=1}^{m}\sum _{l=1}^{n}x_{kl}e^{{\mathbf {kl}}} \end{aligned}$$

which are the elements of \(\varphi \) for each \(m,n\in {{\mathbb {N}}}\), where the double sequence \({\mathbf {e}}^{{\mathbf {kl}}}=({\mathbf {e}}_{mn}^{{\mathbf {kl}}})\) is defined as follows;

$$\begin{aligned} {\mathbf {e}}_{mn}^{\mathbf {kl}}:=\left\{ \begin{array}{ccl} 1&{} ,&{}(k,l)=(m,n),\\ 0&{} , &{}\text {otherwise} \\ \end{array}\right. \end{aligned}$$

for all \(k,l,m,n\in {\mathbb {N}}\).

A locally convex double sequence space \((\lambda ,\tau )\) is called a DK-space, if all of the seminorms \(r_{kl}\) defined for all \(k,l\in {\mathbb {N}}\) by

$$\begin{aligned}\begin{array}{ccccc} r_{kl} &{} : &{}\lambda &{}\longrightarrow &{}\mathbb {R} \\ &{} &{} x=(x_{ij})&{}\longmapsto &{}|x_{kl}| \end{array} \end{aligned}$$

are continuous. A DK-space with a Fréchet topology is called an FDK-space. A normed FDK-space is called a FDK-space. It is known that the spaces \(\mathcal {M}_{u}\), \(\mathcal {C}_{bp}\) and \(\mathcal {C}_{r}\) are BDK-spaces with the norm \(\Vert \cdot \Vert _\infty \).

Let \((\lambda ,\tau _\lambda )\) be a DK-space and \(\vartheta \) be a convergence notion for double sequences. Recall that \(\lambda \) is supposed to contain \(\varphi \). We consider the distinguished subspace \(S_\lambda ^{(\vartheta )}\) of \(\lambda \) given by

$$\begin{aligned} S_\lambda ^{(\vartheta )}:=\left\{ x=(x_{kl})\in \lambda :x=\vartheta -\sum _{k,l}x_{kl} \quad (\lambda ,\tau _\lambda )\right\} . \end{aligned}$$

We say that an element \(x\in \lambda \) has the \(AK(\vartheta )\)-property in \(\lambda \) if \(x\in S_\lambda ^{(\vartheta )}\). The space \(\lambda \) is called an \(AK(\vartheta )\)-space if every its element has \(AK(\vartheta )\) in \(\lambda \) or, equivalently, \(\lambda =S_\lambda ^{(\vartheta )}\), Zeltser [14].

The \(\alpha \)-dual \(\lambda ^{\alpha }\) and the \(\beta (\vartheta )\)-dual \(\lambda ^{\beta (\vartheta )}\) with respect to the \(\vartheta \)-convergence of a double sequence space \(\lambda \) are defined by

$$\begin{aligned} \lambda ^{\alpha }:= & {} \left\{ (a_{kl})\in \Omega :\sum _{k,l}|a_{kl}x_{kl}|<\infty ~ \text { for all }~ (x_{kl})\in \lambda \right\} ,\\ \lambda ^{\beta (\vartheta )}:= & {} \left\{ (a_{kl})\in \Omega :\vartheta -\sum _{k,l}a_{kl}x_{kl}~ \text { exists for all }~(x_{kl})\in \lambda \right\} , \end{aligned}$$

respectively. It is easy to see for any two spaces \(\lambda \) and \(\mu \) of double sequences that \(\mu ^\alpha \subset \lambda ^\alpha \) whenever \(\lambda \subset \mu \) and it is known that the inclusion \(\lambda ^{\alpha }\subset \lambda ^{\beta (\vartheta )}\) holds.

Define the four-dimensional summation matrix \(S=(s_{mnkl})\) and four-dimensional backward difference matrix \(\Delta =(d_{mnkl})\) by

$$\begin{aligned} s_{mnkl}:= & {} \left\{ \begin{array}{l@{\quad }l} 1,&{} 0\le k\le m,\text { }0\le l\le n, \\ 0,&{} \text {otherwise}\end{array}\right. \\ d_{mnkl}:= & {} \left\{ \begin{array}{l@{\quad }l} (-1)^{m+n-(k+l)}, &{} m-1 \le k \le m, n-1 \le l \le n, \\ 0, &{} \text{ otherwise } \end{array}\right. \end{aligned}$$

for all \(m,n,k,l\in {\mathbb {N}}\), respectively.

Milovidov and Povolotskiĭ [7] defined the spaces \({\mathcal {CS}}_\vartheta \) and \(\mathcal {BV}\), as

$$\begin{aligned} {\mathcal {CS}}_\vartheta:= & {} \left\{ x=(x_{kl})\in \Omega :\left( \sum _{k,l=0}^{m,n}x_{kl}\right) \in \mathcal {C}_\vartheta \right\} ,\\ \mathcal {BV}:= & {} \left\{ x=(x_{kl})\in \Omega : \sum _{k,l}|x_{kl}-x_{k-1,l}-x_{k,l-1}+x_{k+1,l+1}|<\infty \right\} , \end{aligned}$$

Also, they gave the \(\beta (\vartheta )\)-dual of those spaces. Then, following them Altay and Başar [1] redefined the spaces \({\mathcal {CS}}_\vartheta \) as the domain of the four-dimensional summation matrix S in the spaces \(\mathcal {C}_{\vartheta }\) and the space \(\mathcal {BV}\) as the domain of the four-dimensional backward difference matrix \(\Delta \) in the space \(\mathcal {L}_u\), where \(\mathcal {L}_u\) shows the space of the absolute converging double series, that is

$$\begin{aligned} \mathcal {L}_u:=\left\{ x=(x_{kl})\in \Omega : \sum _{k,l}|x_{kl}|<\infty \right\} . \end{aligned}$$

Also, they examined some properties of those spaces of double sequences.

The reader can refer to Başar [2], Mursaleen and Mohiuddine [9], and references therein, for relevant terminology and required details on the spaces of double sequences and related topics.

2 Main Results

In this section, we investigate whether the spaces \(\mathcal {L}_u\), \({\mathcal {CS}}_{p}\), \({\mathcal {CS}}_{bp}\), \({\mathcal {CS}}_{r}\) and \(\mathcal {BV}\) are BDK-space and \(AK(\vartheta )\)-space.

Theorem 2.1

The space \(\mathcal {L}_u\) endowed with the norm

$$\begin{aligned} \Vert x\Vert =\sum _{k,l}|x_{kl}| \end{aligned}$$
(2.1)

is a BDK-space and it is an \(AK(\vartheta )\)-space.

Proof

Since every norm (normed space) is a seminorm (seminormed space), we say that \(\mathcal {L}_u\) is a seminormed space with the seminorm (2.1). Also, we define new seminorms in the space \(\mathcal {L}_u\) by \(r_{kl}:\mathcal {L}_u\rightarrow \mathbb {R}\), \(x=(x_{ij})\mapsto |x_{kl}|\) for all \(i,j\in {\mathbb {N}}\). Now, we shall show that each one is continuous. To do this, we use the theorem given by Boos [5, Theorem 6.3.12, p. 284]. We easily find an \(M>0\) for all \(x\in \mathcal {L}_u\) such that \(r_{kl}(x)=|x_{kl}|\le M \Vert x\Vert =\sum _{i,j}|x_{ij}|\) for all \(k,l\in {\mathbb {N}}\). So, the seminorms \(r_{kl}\) are continuous for all \(k,l\in {\mathbb {N}}\), that is, the space \(\mathcal {L}_u\) is a DK-space. Also since it is a Banach space with the norm (2.1) by Theorem 2.1 of [3], it has Fréchet topology. Therefore, it is a BDK-space.

Now, we have that

$$\begin{aligned} \big \Vert x-x^{[mn]}\big \Vert =\sum _{k,l=0,n+1}^{m,\infty }|x_{kl}| +\sum _{k,l=m+1,0}^{\infty ,n}|x_{kl}|+\sum _{k,l=m+1,n+1}^\infty |x_{kl}| \end{aligned}$$

for all \(m,n\in {\mathbb {N}}\). Then,

$$\begin{aligned} \lim _{m,n\rightarrow \infty }\sum _{k,l=m+1,n+1}^\infty |x_{kl}|=0 \end{aligned}$$
(2.2)

is obvious. Define the double sequence \(y=(y_{ml})\) by \(y_{ml}=\sum _{k=0}^m|x_{kl}|\) for all \(m,l\in {\mathbb {N}}\).

$$\begin{aligned} \sum _{k=0}^m\sum _{l=n+1}^\infty |x_{kl}|=\sum _{l=n+1}^\infty \sum _{k=0}^m|x_{kl}|=\sum _{l=n+1}^\infty y_{ml}. \end{aligned}$$
(2.3)

Since \(\sum _{k,l}|x_{kl}|\) is convergent, from (2.3) the series \(\sum _{l=n+1}^\infty y_{ml}\) is too. So, the general term of this series tends to 0, as \(m,l\rightarrow \infty \). Hence,

$$\begin{aligned} \vartheta -\lim _{m,n\rightarrow \infty }\sum _{k=0}^m\sum _{l=n+1}^\infty |x_{kl}|=\vartheta -\lim _{m,n\rightarrow \infty }\sum _{l=n+1}^\infty y_{ml}=0. \end{aligned}$$

Let \(z_{kn}=\sum _{l=0}^n|x_{kl}|\) for all \(k,n\in {\mathbb {N}}\). With the similar way, we have that

$$\begin{aligned} \vartheta -\lim _{m,n\rightarrow \infty }\sum _{k=m+1}^\infty \sum _{l=0}^n|x_{kl}|=\vartheta -\lim _{m,n\rightarrow \infty }\sum _{l=0}^n\sum _{k=m+1}^\infty z_{kn}=0. \end{aligned}$$
(2.4)

By (2.2)–(2.4), we see that

$$\begin{aligned} \vartheta -\lim _{m,n\rightarrow \infty } \big \Vert x-x^{[mn]}\big \Vert =0. \end{aligned}$$

Since \(x\in \mathcal {L}_u\) is arbitrary, the space \(\mathcal {L}_u\) is an \(AK(\vartheta )\)-space. \(\square \)

Theorem 2.2

The space \({\mathcal {CS}}_{p}\) endowed with the norm

$$\begin{aligned} \Vert x\Vert _\infty =\lim _{j\rightarrow \infty }\left( \sup _{m,n\ge j}\left| \sum _{k,l=0}^{m,n}x_{kl}\right| \right) \end{aligned}$$
(2.5)

is a BDK-space and it is an \(AK(\vartheta )\)-space.

Proof

Let \(x=(x_{kl})\in {\mathcal {CS}}_{p}\) . Writing

$$\begin{aligned} |x_{kl}|= & {} \left| \sum _{i,j=0}^{k,l}x_{kl}-\sum _{i,j=0}^{k-1,l} x_{kl}-\sum _{i,j=0}^{k,l-1}x_{kl}-\sum _{i,j=0}^{k-1,l-1}x_{kl}\right| \\\le & {} 4\left| \sum _{i,j=0}^{k,l} x_{kl}\right| \le 4 \Vert x\Vert _{\infty }, \end{aligned}$$

and using the fact that the space \({\mathcal {CS}}_p\) is a Banach space with the norm (2.5) by Theorem 2.1 in [1], we conclude that this space is an BDK-space.

Now, let \(p>m\) and \(q>n\) for all \(m,n\in {\mathbb {N}}\). Then,

$$\begin{aligned} \big \Vert x-x^{[mn]}\big \Vert= & {} \lim _{j\rightarrow \infty }\sup _{p,r\ge j}\left| \sum _{k=m+1}^{p}\sum _{l=0}^{n}x_{kl}+\sum _{k=0}^{p} \sum _{l=n+1}^{q}x_{kl}\right| \nonumber \\\le & {} \lim _{j\rightarrow \infty }\left[ \sup _{p,r\ge j}\left| \sum _{k=m+1}^{p}\sum _{l=0}^{n}x_{kl}\right| +\sup _{p,r\ge j}\left| \sum _{k=0}^{p}\sum _{l=n+1}^{q}x_{kl}\right| \right] \end{aligned}$$
(2.6)

for all \(m,n\in {\mathbb {N}}\). Since the right hand of the (2.6) is the remainder term of the series \(\sum _{k,l}|x_{kl}|\), it must be zero when \(j\rightarrow \infty \). Therefore, we have

$$\begin{aligned} \vartheta -\lim _{m,n\rightarrow \infty }\big \Vert x-x^{[mn]}\big \Vert =0, \end{aligned}$$

as desired. \(\square \)

Corollary 2.3

The spaces \({\mathcal {CS}}_{bp}\) and \({\mathcal {CS}}_{r}\) endowed with the norm

$$\begin{aligned} \Vert x\Vert _{\mathcal {\infty }}= & {} \sup _{k,l\in {\mathbb {N}}}\left| \sum _{i,j=0}^{k,l}x_{kl}\right| \end{aligned}$$

are BDK-spaces.

Theorem 2.4

The space \(\mathcal {BV}\) endowed with the norm

$$\begin{aligned} \Vert x\Vert _{\mathcal {BV}}=\sum _{k,l}|x_{kl}-x_{k-1,l}-x_{k,l-1}+x_{k+1,l+1}| \end{aligned}$$
(2.7)

is a BDK-space but it is not an \(AK(\vartheta )\)-space.

Proof

Let \(x=(x_{kl})\in \mathcal {BV}\). Writing

$$\begin{aligned} |x_{kl}|= & {} \left| \sum _{i,j=0}^{k,l}(x_{kl}-x_{k-1,l}-x_{k,l-1}+x_{k+1,l+1})\right| \\\le & {} \sum _{i,j=0}^{k,l}|x_{kl}-x_{k-1,l}-x_{k,l-1}+x_{k+1,l+1}|\\\le & {} \Vert x\Vert _{\mathcal {BV}} \end{aligned}$$

and using the fact that the space \(\mathcal {BV}\) is a Banach space with the norm (2.7) by Theorem 2.8 in [1], we conclude that this space is a BDK-space.

Now, take the sequence \(x=(x_{kl})\) as \(\mathbf {e^{m+1,n+1}}\) for all \(k,l,m,n\in {\mathbb {N}}\). Then, \(\sum _{k,l}|(\Delta x)_{kl}|=2|x_{m+1,n+1}|=2\), that is, \(x\in \mathcal {BV}\) and also \(\big \Vert x-x^{[mn]}\big \Vert _{\mathcal {BV}}=2\). Therefore, we see that

$$\begin{aligned} \vartheta -\lim _{m,n\rightarrow \infty }\big \Vert x-x^{[mn]}\big \Vert _{\mathcal {BV}}=2\ne 0. \end{aligned}$$

Hence, the space \(\mathcal {BV}\) is not an \(AK(\vartheta )\)-space. \(\square \)

Köthe [6, pp. 407] gave the following definition for the spaces of single sequences:

Definition 2.5

For a given sequence space \(\lambda \), let \(\mu \) denote a subspace of \(\lambda ^\beta \) with \(\phi \subset \mu \), where \(\phi \) denotes the space of all finitely nonzero sequences. Then, \(\lambda \) and \(\mu \) form a dual system under the bilinear functional \(\langle x,y\rangle \) defined for \(x=(x_{k})\in \lambda \), \(y=(y_{k})\in \mu \) by

$$\begin{aligned} \langle x,y\rangle =\sum _{k=1}^\infty x_{k}y_{k}. \end{aligned}$$

Consider the dual system \(\langle \lambda ,\lambda ^\times \rangle \). Suppose that \(\mu \) is a subspace of \(\lambda ^\times \), \(\phi \subset \mu \). For each \(y\in \mu \), define the seminorm \(p_y\) on \(\lambda \) by

$$\begin{aligned} p_y(x)=\sum _{k=1}^\infty |x_{k}y_{k}|~\text { for }~x=(x_{k})\in \lambda . \end{aligned}$$

The locally convex topology generated by \(\{p_y:y\in \mu \}\) is called the normal or Köthe topology on \(\lambda \) and is denoted by \(\eta (\lambda ,\mu )\).

In her Ph.D. thesis, Zeltser [14] gave that a double sequence space \(\lambda \) and its \(\beta (\vartheta )\)-dual \(\lambda ^{\beta (\vartheta )}\) generate a dual pair \(\langle \lambda ,\lambda ^{\beta (\vartheta )}\rangle \) under the bilinear form

$$\begin{aligned}\begin{array}{rclrl} \langle \cdot ,\cdot \rangle &{} : &{}\lambda \times \lambda ^{\beta (\vartheta )}&{}\longrightarrow &{}{\mathbb {C}}\\ &{} &{} ~(x,a)&{}\longmapsto &{}\vartheta -\sum _{k,l}a_{kl}x_{kl}. \end{array} \end{aligned}$$

As a natural continuation of these consequences, we give the following theorem:

Theorem 2.6

Let \(\lambda \) be a double sequence space and \(\mu \) also be a double sequence space such that it is a subspace of \(\lambda ^\alpha \) and \(\varphi \subset \mu \). Consider the dual system \(\langle \lambda ,\lambda ^\alpha \rangle \) and define the seminorm \(p_a\) on \(\lambda \) by

$$\begin{aligned} p_a(x)=\sum _{k,l}|a_{kl}x_{kl}|, \end{aligned}$$

where \(a=(a_{kl})\in \mu \) and \(x=(x_{kl})\in \lambda \). Then, \((\lambda ,\eta (\lambda ,\mu ))\) is an \(AK(\vartheta )\)-space.

Proof

Let \(x=(x_{kl})\in \lambda \) and \(a=(a_{kl})\in \mu \subset \lambda ^\alpha \). Then,

$$\begin{aligned} p_a\big (x-x^{[mn]}\big )=\sum _{k=0}^m\sum _{l=n+1}^\infty |a_{kl} x_{kl}|+\sum _{k=m+1}^\infty \sum _{l=0}^n|a_{kl}x_{kl}| +\sum _{k,l=m+1,n+1}^\infty |a_{kl}x_{kl}|, \end{aligned}$$

and obviously

$$\begin{aligned} \lim _{m,n\rightarrow \infty }\sum _{k,l=m+1,n+1}^\infty |a_{kl}x_{kl}|=0. \end{aligned}$$
(2.8)

Define \(y=(y_{ml})\) be in \(\Omega \) such that \(y_{ml}=\sum _{k=0}^m|a_{kl}x_{kl}|\) for all \(m,l\in {\mathbb {N}}\). Therefore, we have

$$\begin{aligned} \sum _{k=0}^m\sum _{l=n+1}^\infty |a_{kl}x_{kl}|=\sum _{l=n+1}^\infty \sum _{k=0}^m|a_{kl}x_{kl}|=\sum _{l=n+1}^\infty y_{ml}. \end{aligned}$$
(2.9)

Since \(\sum _{k,l}|a_{kl}x_{kl}|\) is convergent, from (2.9) the series \(\sum _{l=n+1}^\infty y_{ml}\) is too. So, the general term of this series tends to 0 when \(m,n\rightarrow \infty \). Hence,

$$\begin{aligned} \vartheta -\lim _{m,n\rightarrow \infty }\sum _{k=0}^m\sum _{l=n+1}^\infty |a_{kl}x_{kl}|=\vartheta -\lim _{m,n\rightarrow \infty }\sum _{l=n+1}^\infty y_{ml}=0. \end{aligned}$$

Let \(z_{kn}=\sum _{l=0}^n|a_{kl}x_{kl}|\) for all \(k,n\in {\mathbb {N}}\). Similarly we conclude that

$$\begin{aligned} \vartheta -\lim _{m,n\rightarrow \infty }\sum _{k=m+1}^\infty \sum _{l=0}^n=\vartheta -\lim _{m,n\rightarrow \infty }\sum _{l=0}^n\sum _{k=m+1}^\infty z_{kn}=0. \end{aligned}$$
(2.10)

By (2.8)–(2.10), we observe that

$$\begin{aligned} \vartheta -\lim _{m,n\rightarrow \infty } p_a(x-x^{[mn]})=0. \end{aligned}$$

Since x is arbitrary, the space \((\lambda ,\eta (\lambda ,\mu ))\) is an \(AK(\vartheta )\)-space. \(\square \)

Corollary 2.7

The space \((\lambda ,\sigma (\lambda ,\mu ))\) is an \(AK(\vartheta )\)-space.

Proof

Since the locally convex topology generated on \(\lambda \) by the family of \(\{q_a:a\in \mu \}\) of seminorms is denoted by \(\sigma (\lambda ,\mu )\), taking \(q_a\) instead of \(p_a\) in Theorem 2.6 and using triangle inequality, we obtain the desired result, where \(q_a(x)=\left| \sum _{k,l}a_{kl}x_{kl}\right| \). \(\square \)

3 Conclusion

We essentially deal with the spaces \({{\mathcal {CS}}}_\vartheta \) and \({\mathcal {BV}}\) of double series defined as the domains of the four-dimensional summation matrix S and the four-dimensional backward difference matrix \(\Delta \) in the spaces \({\mathcal {C}}_\vartheta \) and \({\mathcal {L}}_u\), respectively. One can obtain some new results corresponding to our results by using the spaces \({\mathcal {F}}\), \({\mathcal {F}}_{0}\) and \({\mathcal {[F]}}\), \({\mathcal {[F]}}_{0}\) of almost convergent, almost null and strongly almost convergent, strongly almost null double sequences (see Yeşilkayagil and Başar [13] and Başarır [4]) instead of \({\mathcal {C}}_\vartheta \). In the special case \(\vartheta =bp\) or r, these results will be more general than the corresponding results of the present paper.