On I-convergent sequence spaces defined by a compact operator and a modulus function

Abstract

In this article, we introduce and study I-convergent sequence spaces ${\mathcal{S}}^I(f)$, ${\mathcal{S}}_0^I(f)$, and ${\mathcal{S}}_{\infty }^I(f)$ with the help of compact operator T on the real space $\mathbb{R}$ and a modulus function f. We study some topological and algebraic properties, and prove some inclusion relations on these spaces.

Keywords:

• compact operator
• ideal
• filter
• I-convergent sequence
• I-null sequence
• I-bounded sequence
• solid and monotone space
• symmetric space
• modulus function

AMS subject classifications:

• 41A10
• 41A25
• 41A36
• 40A30

Public Interest Statement

The term sequence has a great role in Analysis. Sequence spaces play an important role in various fields of Real Analysis, Complex Analysis, Functional Analysis, and Topology Convergence of sequences has always remained a subject of interest to the researchers. Several new types of convergence of sequences were studied by the researchers and named them as usual convergence, uniform convergence, strong convergence,week convergence, etc. Later, the idea of statistical convergence came into existence which is the generalization of usual convergence. Statistical convergence has several applications in different fields of Mathematics like Number Theory, Trigonometric Series, Summability Theory, Probability Theory, Measure Theory, Optimization, and Approximation Theory. The notion of Ideal convergence (I-convergence) is a generalization of the statistical convergence and equally considered by the researchers for their research purposes since its inception

1. Introduction and preliminaries

Let $\mathbb{N}$, $\mathbb{R}$, and $\mathbb{C}$ be the sets of all natural, real, and complex numbers, respectively. We denote$$\mathit{\omega }=\{x=(x_k):x_k\in \mathbb{R}\text{or}\mathbb{C}\}$$

the space of all real or complex sequences.

Let ${\ell }_{\infty }$, $c$, and $c_0$ be denote the Banach spaces of bounded, convergent, and null sequences of reals, respectively with norm$$\Vert x\Vert =\underset{k}{sup}\mid x_k\mid$$

Any subspace $\mathit{\lambda }$ of $\mathit{\omega }$ is called a sequence space. A sequence space $\mathit{\lambda }$ with linear topology is called a K-space provided each of maps $p_i\to \mathbb{C}$ defined by $p_i(x)=x_i$ is continuous, for all $i\in \mathbb{N}$. A space $\mathit{\lambda }$ is called an FK-space provided $\mathit{\lambda }$ is complete linear metric space. An FK-space whose topology is normable is called a BK-space.

Definition 1.1

Let $X$ and $Y$ be two normed linear spaces and $T:\mathcal{D}(T)\to Y$ be a linear operator, where $\mathcal{D}(T)\subset X$. Then, the operator $T$ is said to be bounded, if there exists a positive real $k$ such that$$\Vert Tx\Vert \le k\Vert x\Vert ,\text{for all}x\in \mathcal{D}(T)$$

The set of all bounded linear operators $\mathcal{B}(X,Y)$ is a normed linear space normed by (see Kreyszig, 1978)$$\Vert T\Vert =\underset{x\in X,\Vert x\Vert =1}{sup}\Vert Tx\Vert$$

and $\mathcal{B}(X,Y)$ is a Banach space if $Y$ is Banach space.

Definition 1.2

Let $X$ and $Y$ be two normed linear spaces. An operator $T:X\to Y$ is said to be a compact linear operator (or completely continuous linear operator), if

(1)	$T$ is linear,
(2)	$T$ maps every bounded sequence $(x_k)$ in $X$ onto a sequence $T(x_k)$ in $Y$ which has a convergent subsequence.

The set of all compact linear operators $\mathcal{C}(X,Y)$ is closed subspace of $\mathcal{B}(X,Y)$ and $\mathcal{C}(X,Y)$ is a Banach space if $Y$ is Banach space.

Following Basar and Altay (2003) and Sengönül (2009), we introduce the sequence spaces $\mathcal{S}$ and ${\mathcal{S}}_0$ with the help of compact operator $T$ on the real space $\mathbb{R}$ as follows.$$\mathcal{S}=\left\{x=(x_k)\in {\ell }_{\infty }:T(x)\in c\right\}$$

and$${\mathcal{S}}_0=\left\{x=(x_k)\in {\ell }_{\infty }:T(x)\in c_0\right\}$$

Definition 1.3

A function $f:[0,\infty )⟶[0,\infty )$ is called a modulus if

(1)	$f(t)=0$ if and only if $t=0$,
(2)	$f(t+u)\le f(t)+f(u)$ for all $t,u\ge 0$,
(3)	$f$ is increasing, and
(4)	$f$ is continuous from the right at zero.

For any modulus function $f$, we have the inequalities$$\mid f(x)-f(y)\mid \le f(\mid x-y\mid )$$

and$$f(nx)\le nf(x),\text{for all}x,y\in [0,\infty ]$$

A modulus function $f$ is said to satisfy ${\mathrm{\Delta }}_2-\text{Condition}$ for all values of u if there exists a constant $K>0$ such that $f(Lu)\le KLf(u)$ for all values of $L>1$.

The idea of modulus was introduced by Nakano (1953).

Ruckle (1967, 1968, 1973) used the idea of a modulus function $f$ to construct the sequence space$$X(f)=\left\{x=(x_k):\sum _{k=1}^{\infty }f(|x_k|)<\infty \right\}$$

This space is an $FK$-space and Ruckle (1967, 1968, 1973) proved that the intersection of all such $X(f)$ spaces is $\mathit{\varphi }$, the space of all finite sequences.

The space $X(f)$ is closely related to the space ${\ell }_1$ which is an $X(f)$ space with $f(x)=x$ for all real $x\ge 0$. Thus Ruckle (1967, 1968, 1973) proved that, for any modulus $f$.$$X(f)\subset {\ell }_1\text{and}X{(f)}^{\mathit{\alpha }}={\ell }_{\infty }$$

where$$X{(f)}^{\mathit{\alpha }}=\left\{y=(y_k)\in \mathit{\omega }:\sum _{k=1}^{\infty }f(|y_k{x}_k|)<\infty \right\}$$

Spaces of the type $X(f)$ are a special case of the spaces structured by Gramsch (1967). From the point of view of local convexity, spaces of the type $X(f)$ are quite pathological. Symmetric sequence spaces, which are locally convex have been frequently studied by Garling (1966), Köthe (1970), and Ruckle (1967, 1968, 1973).

The sequence spaces by the use of modulus function was further investigated by Maddox (1969, 1986), Khan (2005, 2006), Bhardwaj (2003), and many others.

As a generalization of usual convergence, the concept of statistical convergent was first introduced by Fast (1951) and also independently by Buck (1953) and Schoenberg (1959) for real and complex sequences. Later on, it was further investigated from sequence space point of view and linked with the Summability Theory by Fridy (1985), Šalát (1980), Tripathy (1998), Khan (2007), Khan and Sabiha (2012), Khan, Shafiq, and Rababah (2015), and many others.

Definition 1.4

A sequence $x=(x_k)\in \mathit{\omega }$ is said to be statistically convergent to a limit $L\in \mathbb{C}$ if for every $\mathit{\epsilon }>0$, we have$$\underset{k\to \infty }{lim}\frac{1}{k}|\left\{n\in \mathbb{N}:|x_n-L|\ge \mathit{\epsilon },n\le k\right\}|=0$$

where vertical lines denote the cardinality of the enclosed set. That is, if $\mathit{\delta }(A(\mathit{\epsilon }))=0$, where$$A(\mathit{ϵ})=\left\{k\in \mathbb{N}:\mid x_k-L\mid \ge \mathit{\epsilon }\right\}$$

The notation of ideal convergence (I-convergence) was introduced and studied by Kostyrko, Mačaj, Salǎt, and Wilczyński (2000). Later on, it was studied by Šalát, Tripathy, and Ziman (2004, 2005), Tripathy and Hazarika (2009, 2011), Khan and Ebadullah (2011), Khan, Ebadullah, Esi, and Shafiq (2013), and many others.

Now, we recall the following definitions:

Definition 1.5

Let $\mathbb{N}$ be a non-empty set. Then a family of sets $I\subseteq 2^{\mathbb{N}}$ (power set of $\mathbb{N}$) is said to be an ideal if

(1)	$I$ is additive i.e $\forall A,B\in I\Rightarrow A\cup B\in I$
(2)	$I$ is hereditary i.e $\forall A\in I\text{and}B\subseteq A\Rightarrow B\in I$.

Definition 1.6

A non-empty family of sets $\mathrm{\pounds }(I)\subseteq 2^{\mathbb{N}}$ is said to be filter on $\mathbb{N}$ if and only if

(1)	$\mathrm{\Phi }\notin \mathrm{\pounds }(I)$,
(2)	$\forall A,B\in \mathrm{\pounds }(I)$ we have $A\cap B\in \mathrm{\pounds }(I)$,
(3)	$\forall A\in \mathrm{\pounds }(I)$ and $A\subseteq B\Rightarrow B\in \mathrm{\pounds }(I)$.

Definition 1.7

An Ideal $I\subseteq 2^{\mathbb{N}}$ is called non-trivial if $I\ne 2^{\mathbb{N}}$.

Definition 1.8

A non-trivial ideal $I\subseteq 2^{\mathbb{N}}$ is called admissible if$$\left\{\{x\}:x\in \mathbb{N}\right\}\subseteq I$$

Definition 1.9

A non-trivial ideal $I$ is maximal if there cannot exist any non-trivial ideal $J\ne I$ containing $I$ as a subset.

Remark 1.10

For each ideal $I$, there is a filter $\mathrm{\pounds }(I)$ corresponding to $I$. i.e $\mathrm{\pounds }(I)$ = $\{K\subseteq \mathbb{N}:K^c\in I\}$, where $K^c=\mathbb{N}\backslash K$.

Definition 1.11

A sequence $x=(x_k)\in \mathit{\omega }$ is said to be $I$-convergent to a number $L$ if for every $\mathit{\epsilon }>0$, the set $\{k\in \mathbb{N}:|x_k-L|\ge \mathit{\epsilon }\}\in I.$ In this case, we write $I-lim{x}_k=L$.

Definition 1.12

A sequence $x=(x_k)\in \mathit{\omega }$ is said to be $I$-null if $L=0.$ In this case, we write $I-lim{x}_k=0$.

Definition 1.13

A sequence $x=(x_k)\in \mathit{\omega }$ is said to be $I$-Cauchy if for every $\mathit{\epsilon }>0$ there exists a number $m=m(\mathit{\epsilon })$ such that $\{k\in \mathbb{N}:|x_k-x_m|\ge \mathit{\epsilon }\}\in I$.

Definition 1.14

A sequence $x=(x_k)\in \mathit{\omega }$ is said to be $I$-bounded if there exists some $M>0$ such that $\{k\in \mathbb{N}:|x_k|\ge M\}\in I.$

Definition 1.15

A sequence space $E$ is said to be solid (normal) if $({\mathit{\alpha }}_k{x}_k)\in E$ whenever $(x_k)\in E$ and for any sequence $({\mathit{\alpha }}_k)$ of scalars with $\mid {\mathit{\alpha }}_k\mid \le 1$, for all $k\in \mathbb{N}.$

Definition 1.16

A sequence space $E$ is said to be symmetric if $(x_{\mathit{\pi }(k)})\in E$ whenever $x_k\in E.$ where $\mathit{\pi }$ is a permutation on $\mathbb{N}$.

Definition 1.17

A sequence space $E$ is said to be sequence algebra if $(x_k)\ast (y_k)=(x_k.y_k)\in E$ whenever $(x_k),(y_k)\in E$.

Definition 1.18

A sequence space $E$ is said to be convergence free if $(y_k)\in E$ whenever $(x_k)\in E$ and $x_k=0$ implies $y_k=0$, for all $k$.

Definition 1.19

Let $K=\{k_1<k_2<k_3<k_4<k_5\dots \}\subset \mathbb{N}$ and $E$ be a Sequence space. A $K$-step space of $E$ is a sequence space ${\mathit{\lambda }}_K^E=\{(x_{k_n})\in \mathit{\omega }:(x_k)\in E\}$.

Definition 1.20

A canonical pre-image of a sequence $(x_{k_n})\in {\mathit{\lambda }}_K^E$ is a sequence $(y_k)\in \mathit{\omega }$ defined by$$\begin{array}{c}\hfill y_k=\left\{\begin{array}{c}{x}_k,\text{if}k\in K,\\ 0,\text{otherwise}\end{array}\right.\end{array}$$

A canonical preimage of a step space ${\mathit{\lambda }}_K^E$ is a set of preimages all elements in ${\mathit{\lambda }}_K^E$. i.e. $y$ is in the canonical preimage of ${\mathit{\lambda }}_K^E$ iff $y$ is the canonical preimage of some $x\in {\mathit{\lambda }}_K^E.$

Definition 1.21

A sequence space $E$ is said to be monotone if it contains the canonical preimages of its step space.

Definition 1.22

(see, Khan et al., 2015; Kostyrko et al., 2000). If $I=I_f$, the class of all finite subsets of $v$. Then, $I$ is an admissible ideal in $v$ and $I_f$ convergence coincides with the usual convergence.

Definition 1.23

(see, Khan et al., 2015; Kostyrko et al., 2000). If $I=I_{\mathit{\delta }}=\{A\subseteq \mathbb{N}:\mathit{\delta }(A)=0\}$. Then, $I$ is an admissible ideal in $\mathbb{N}$ and we call the $I_{\mathit{\delta }}$-convergence as the logarithmic statistical convergence.

Definition 1.24

(see, Khan et al., 2015; Kostyrko et al., 2000). If $I=I_d=\{A\subseteq \mathbb{N}:d(A)=0\}$. Then, $I$ is an admissible ideal in $\mathbb{N}$ and we call the $I_d$-convergence as the asymptotic statistical convergence.

Remark 1.25

If $I_{\mathit{\delta }}-lim{x}_k=l$, then $I_d-lim{x}_k=l$

Definition 1.26

A map $ħ$ defined on a domain $D\subset X$ i.e $ħ:D\subset X\to IR$ is said to satisfy Lipschitz condition if $|ħ(x)-ħ(y)|\le K|x-y|$ where $K$ is known as the Lipschitz constant. The class of $K$-Lipschitz functions defined on $D$ is denoted by $ħ\in (D,K).$

Definition 1.27

A convergence field of $I$-covergence is a set$$F(I)=\{x=(x_k)\in l_{\infty }:\text{there exists}I-limx\in IR\}$$

The convergence field $F(I)$ is a closed linear subspace of $l_{\infty }$ with respect to the supremum norm, $F(I)=l_{\infty }\cap c^I$ (see Šalát et al., 2004, 2005).

Definition 1.28

Let $X$ be a linear space. A function $g:X⟶R$ is called paranorm, if for all $x,y\in X,$

• $(P_1)$     $g(x)=0ifx=\mathit{\theta },$

• $(P_2)$     $g(-x)=g(x),$

• $(P_3)$     $g(x+y)\le g(x)+g(y),$

• $(P_4)$     If $({\mathit{\lambda }}_n)$ is a sequence of scalars with ${\mathit{\lambda }}_n\to \mathit{\lambda }$   $(n\to \infty )$ and $x_n,a\in X$ with $x_n\to a$   $(n\to \infty )$ in the sense that $g(x_n-a)\to 0$   $(n\to \infty )$ , then $g({\mathit{\lambda }}_n{x}_n-\mathit{\lambda }a)\to 0$  $(n\to \infty ).$

The notation of paranorm sequence spaces was studied at the initial stage by Nakano (1953). Later on, it was further investigated by Maddox (1969), Tripathy and Hazarika (2009), Khan et al. (2013), and the references therein.

Throughout the article, we use the same techniques as used in Tripathy and Hazarika (2009, 2011).

We used the following lemmas for establishing some results of this article.

Lemma 1

(see, Tripathy & Hazarika, 2009, 2011) Every solid space is monotone.

Lemma 2

(see, Tripathy & Hazarika, 2009, 2011) If $I\subseteq 2^N$ and $M\subseteq N$. If $M\notin I$, then $M\cap N\notin I$.

Lemma 3

(see, Tripathy & Hazarika, 2009, 2011) Let $K\in \mathrm{\pounds }(I)$ and $M\subseteq N$. If $M\notin I$, then $M\cap K\notin I$.

Throughout the article, ${\mathcal{S}}^I$, ${\mathcal{S}}_0^I$, ${\mathcal{S}}_{\infty }^I$, ${\mathcal{M}}_{\mathcal{S}}^I$, and ${\mathcal{M}}_{{\mathcal{S}}_{\circ }}^I$ represent the $I$-convergent, $I$-null, $I$-Bounded , bounded $I$-convergent, and bounded $I$-null Sequences spaces defined by a compact operator $T$ on the real space $\mathbb{R}$, respectively.

2. Main results

In this article, we introduce the following classes of sequences.(3.1) $$\begin{array}{c}\hfill {\mathcal{S}}^I(f)=\{x=(x_k)\in {\ell }_{\infty }:\{k\in \mathbb{N}:f(\mid T(x_k)-L\mid )\ge \mathit{ϵ}\}\in I,\text{for some}L\in \mathbb{C}\}\end{array}$$(3.1) (3.2) $$\begin{array}{c}\hfill {\mathcal{S}}_0^I(f)=\{x=(x_k)\in {\ell }_{\infty }:\{k\in \mathbb{N}:f(\mid (T{x}_k)\mid )\ge \mathit{ϵ}\}\in I\}\end{array}$$(3.2) (3.3) $$\begin{array}{c}\hfill {\mathcal{S}}_{\infty }^I(f)=\{x=(x_k)\in {\ell }_{\infty }:\{k\in \mathbb{N}:\exists K>0\text{such that}f(\mid T(x_k)\mid )\ge K\}\in I\}\end{array}$$(3.3) (3.4) $$\begin{array}{c}\hfill {\mathcal{S}}_{\infty }(f)=\{x=(x_k)\in {\ell }_{\infty }:\underset{k}{sup}f(\mid T(x_k)\mid )<\infty \}\end{array}$$(3.4)

where $f$ is a modulus function. We also denote$${\mathcal{M}}_{\mathcal{S}}^I(f)={\mathcal{S}}_{\infty }(f)\cap {\mathcal{S}}^I(f)\text{and}{\mathcal{M}}_{{\mathcal{S}}_{\circ }}^I(f)={\mathcal{S}}_{\infty }(f)\cap {\mathcal{S}}_0^I(f)$$

Theorem 2.1

Let $f$ be a modulus function. Then, the classes of sequences ${\mathcal{S}}^I(f)$, ${\mathcal{S}}_0^I(f)$, ${\mathcal{M}}_{\mathcal{S}}^I(f)$, and ${\mathcal{M}}_{{\mathcal{S}}_{\circ }}^I(f)$ are linear spaces.

Proof

We shall prove the result for ${\mathcal{S}}^I(f)$. The proof for the other spaces will follow similarly.

For, let $x=(x_k),y=(y_k)\in {\mathcal{S}}^I(f)$ and $\mathit{\alpha },\mathit{\beta }$ be scalars. Then, for a given $\mathit{ϵ}>0$, we have(3.5) $$\begin{array}{c}\hfill \{k\in \mathbb{N}:f(\mid T(x_k)-L_1\mid )\ge \frac{\mathit{ϵ}}{2},\text{for some}{L}_1\in \mathbb{C}\}\in I\end{array}$$(3.5) (3.6) $$\begin{array}{c}\hfill \{k\in \mathbb{N}:f(\mid T(x_k)-L_2\mid )\ge \frac{\mathit{ϵ}}{2},\text{for some}{L}_1\in \mathbb{C}\}\in I\end{array}$$(3.6)

Let(3.7) $$\begin{array}{c}\hfill A_1=\{k\in \mathbb{N}:f\left(|T(x_k)-L_1|\right)<\frac{\mathit{ϵ}}{2},\text{for some}{L}_1\in \mathbb{C}\}\in \mathcal{L}I\end{array}$$(3.7) (3.8) $$\begin{array}{c}\hfill A_2=\{k\in \mathbb{N}:f\left(|T(y_k)-L_2|\right)<\frac{\mathit{ϵ}}{2},\text{for some}{L}_2\in \mathbb{C}\}\in \mathcal{L}I\end{array}$$(3.8)

be such that $A_1^c,A_2^c\in I$.

Since $f$ is a modulus function, we have$$\begin{array}{cc}\hfill A_3& =\{k\in \mathbb{N}:f\left(|(\mathit{\alpha }T(x_k)+\mathit{\beta }T(y_k)-(\mathit{\alpha }{L}_1+\mathit{\beta }{L}_2)|\right)<\mathit{ϵ}\}\hfill \\ \hfill & \supseteq [\{k\in \mathbb{N}:f\left(|\mathit{\alpha }||T(x_k)-L_1|\right)<\frac{\mathit{ϵ}}{2}\}\hfill \\ \hfill & \cap \{k\in \mathbb{N}:f\left(|\mathit{\beta }||T(y_k)-L_2|\right)<\frac{\mathit{ϵ}}{2}\}]\hfill \\ \hfill & \supseteq [\{k\in \mathbb{N}:f\left(|T(x_k)-L_1|\right)<\frac{\mathit{ϵ}}{2}\}\hfill \\ \hfill & \cap \{k\in \mathbb{N}:f\left(|T(y_k)-L_2|\right)<\frac{\mathit{ϵ}}{2}\}]\hfill \end{array}$$

Therefore,(3.9) $$\begin{array}{cc}\hfill A_3& =\{k\in \mathbb{N}:f\left(|(\mathit{\alpha }T(x_k)+\mathit{\beta }T(y_k)-(\mathit{\alpha }{L}_1+\mathit{\beta }{L}_2)|\right)<\mathit{ϵ}\}\hfill \\ \hfill & \supseteq [\{k\in \mathbb{N}:f\left(|T(x_k)-L_1|\right)<\frac{\mathit{ϵ}}{2}\}\hfill \\ \hfill & \cap \{k\in \mathbb{N}:f\left(|T(y_k)-L_2|\right)<\frac{\mathit{ϵ}}{2}\}]\hfill \end{array}$$(3.9)

implies that $A_3\in \mathrm{\pounds }(I).$ Thus, $A_3^c=A_1^c\cup A_2^c\in I.$ Therefore, $\mathit{\alpha }{x}_k+\mathit{\beta }{y}_k\in {\mathcal{S}}^I(f)$, for all scalars $\mathit{\alpha },\mathit{\beta }$, and $(x_k),(y_k)\in {\mathcal{S}}^I(f)$. Hence, ${\mathcal{S}}^I(f)$ is a linear space.

Theorem 2.2

The classes of sequences ${\mathcal{M}}_{\mathcal{S}}^I(f)$ and ${\mathcal{M}}_{{\mathcal{S}}_{\circ }}^I(f)$ are paranormed spaces, paranormed by$$g(x)=g(x_k)=\underset{k}{sup}f\left(|T(x_k)|\right)$$

Proof

Let $x=(x_k),y=(y_k)\in {\mathcal{M}}_{\mathcal{S}}^I(f).$

• $(P_1)$     It is clear that $g(x)=0$ if $x=\mathit{\theta }$, a zero vector.

• $(P_2)$     $g(x)=g(-x)$ is obvious.

• $(P_3)$     For $x=(x_k),y=(y_k)\in {\mathcal{M}}_{\mathcal{S}}^I(f),$ we have$$\begin{array}{cc}\hfill g(x+y)& =g(x_k+y_k)=\underset{k}{sup}f\left(|T(x_k+y_k)|\right)\hfill \\ \hfill & =\underset{k}{sup}f\left(|T(x_k)+T(y_k)|\right)\hfill \\ \hfill & \le \underset{k}{sup}f\left(|T(x_k)|\right)\hfill \\ \hfill & +\underset{k}{sup}f\left(|T(y_k)\right)|=g(x)+g(y)\hfill \end{array}$$ Therefore, $g(x+y)\le g(x)+g(y)$

• $(P_4)$     Let $({\mathit{\lambda }}_k)$ be a sequence of scalars with $({\mathit{\lambda }}_k)\to \mathit{\lambda }(k\to \infty )$ and

• $(x_k),L\in {\mathcal{M}}_{\mathcal{S}}^I(f)$ such that$$x_k\to L(k\to \infty )$$ in the sense that$$g(x_k-L)\to 0(k\to \infty )$$ Then, since the inequality$$g(x_k)\le g(x_k-L)+g(L)$$ holds by subadditivity of $g$, the sequence $\{g(x_k)\}$ is bounded.

Therefore,$$\begin{array}{cc}\hfill g[({\mathit{\lambda }}_k{x}_k-\mathit{\lambda }L)]& =g[({\mathit{\lambda }}_k{x}_k-\mathit{\lambda }{x}_k+\mathit{\lambda }{x}_k-\mathit{\lambda }L)]\hfill \\ \hfill & =g[({\mathit{\lambda }}_k-\mathit{\lambda })x_k+\mathit{\lambda }(x_k-L)]\hfill \\ \hfill & \le g[({\mathit{\lambda }}_k-\mathit{\lambda })x_k]+g[\mathit{\lambda }(x_k-L)]\hfill \\ \hfill & \le \mid ({\mathit{\lambda }}_k-\mathit{\lambda })\mid g(x_k)+\mid \mathit{\lambda }\mid g(x_k-L)\to 0\hfill \end{array}$$

as $(k\to \infty ).$ That is to say that scalar multiplication is continuous. Hence, ${\mathcal{M}}_{\mathcal{S}}^I(f)$ is a paranormed space. For ${\mathcal{M}}_{{\mathcal{S}}_{\circ }}^I(f)$, the result is similar.

Theorem 2.3

A sequence $x=(x_k)\in {\ell }_{\infty }$ I-converges if and only if for every $\mathit{ϵ}>0$, there exists $N_{\mathit{ϵ}}\in \mathbb{N}$ such that(3.10) $$\begin{array}{c}\hfill \{k\in \mathbb{N}:f(\mid T(x_k)-T(x_{N_{\mathit{ϵ}}})\mid )<\mathit{ϵ}\}\in \mathrm{\pounds }(I)\end{array}$$(3.10)

Proof

Let $x=(x_k)\in {\ell }_{\infty }.$

Suppose that $L=I-limx$. Then, the set$$B_{\mathit{ϵ}}=\{k\in \mathbb{N}:f(\mid T(x_k)-L\mid )<\frac{\mathit{ϵ}}{2}\}\in \mathrm{\pounds }(I)\text{for all}\mathit{ϵ}>0$$

Fix an $N_{\mathit{ϵ}}\in B_{\mathit{ϵ}}.$ Then we have,$$f(\mid T(x_k)-T(x_{N_{\mathit{ϵ}}})\mid )\le f(\mid T(x_k)-L\mid )+f(\mid T(x_{N_{\mathit{ϵ}}})-L\mid )<\frac{\mathit{ϵ}}{2}+\frac{\mathit{ϵ}}{2}=\mathit{ϵ}$$

which holds for all $k\in B_{\mathit{ϵ}}$. Hence $\{k\in \mathbb{N}:f(\mid T(x_k)-T(x_{N_{\mathit{ϵ}}})\mid )<\mathit{ϵ}\}\in \mathrm{\pounds }(I)$ Conversely, suppose that$$\{k\in \mathbb{N}:f(\mid T(x_k)-T(x_{N_{\mathit{ϵ}}})\mid )<\mathit{ϵ}\}\in \mathrm{\pounds }(I)$$

That is $\{k\in \mathbb{N}:\mid f\left(|T(x_k)|\right)-f\left(|T(x_{N_{\mathit{ϵ}}})\mid \right)\mid <\mathit{ϵ}\}\in \mathrm{\pounds }(I),$ for all $\mathit{ϵ}>0$.

Then, the set$$C_{\mathit{ϵ}}=\{k\in \mathbb{N}:f\left(|T(x_k)|\right)\in [f\le (|T(x_{N_{\mathit{ϵ}}})\mid )-\mathit{ϵ},f(|T(x_{N_{\mathit{ϵ}}})\mid )+\mathit{ϵ}]\}\in \mathrm{\pounds }(I)\text{for all}\mathit{ϵ}>0$$

Let $J_{\mathit{ϵ}}=[f(|T(x_{N_{\mathit{ϵ}}}))-\mathit{ϵ},f(|T(x_{N_{\mathit{ϵ}}}))+\mathit{ϵ}]$. If we fix an $\mathit{ϵ}>0$ then we have $C_{\mathit{ϵ}}\in \mathrm{\pounds }(I)$ as well as $C_{\frac{\mathit{ϵ}}{2}}\in \mathrm{\pounds }(I).$ Hence $C_{\mathit{ϵ}}\cap C_{\frac{\mathit{ϵ}}{2}}\in \mathrm{\pounds }(I).$ This implies that$$J=J_{\mathit{ϵ}}\cap J_{\frac{\mathit{ϵ}}{2}}\ne \mathit{\varphi }$$

That is$$\{k\in \mathbb{N}:f(|T(x_k)|)\in J\}\in \mathrm{\pounds }(I)$$

That is$$diamJ\le diam{J}_{\mathit{ϵ}}$$

where the diam of $J$ denotes the length of interval $J$. In this way, by induction, we get the sequence of closed intervals$$J_{\mathit{ϵ}}=I_0\supseteq I_1\supseteq \dots \supseteq I_k\supseteq \dots$$

with the property that $diam{I}_k\le \frac{1}{2}diam{I}_{k-1}$ for ($k=2,3,4,\dots$) and $\{k\in \mathbb{N}:f(|T(x_k)|)\in I_k\}\in \mathrm{\pounds }(I)$ for ($k=1,2,3,4,\dots$). Then, there exists a $\mathit{\xi }\in \cap I_k$ where $k\in \mathbb{N}$ such that $\mathit{\xi }=I-limf(|T(x_k)|)$ showing that $x=(x_k)\in {\ell }_{\infty }$ is I-convergent. Hence the result.

Theorem 2.4

Let $f_1$ and $f_2$ be two modulus functions and satisfying ${\mathrm{\Delta }}_2-\text{Condition}$, then

(a)	$\mathcal{X}(f_2)\subseteq \mathcal{X}(f_1{f}_2),$
(b)	$\mathcal{X}(f_1)\cap (f_2)\subseteq \mathcal{X}(f_1+f_2)$ for $\mathcal{X}$= ${\mathcal{S}}^I$, ${\mathcal{S}}_{\circ }^I$, ${\mathcal{M}}_{\mathcal{S}}^I$ and ${\mathcal{M}}_{{\mathcal{S}}_{\circ }}^I$.

Proof

(a)

Let $x=(x_k)\in {\mathcal{S}}_{\circ }^I(f_2)$ be any arbitrary element. Then, the set(3.11) $$\begin{array}{c}\hfill \{k\in \mathbb{N}:f_2(\mid T(x_k)\mid )\ge \mathit{ϵ}\}\in I\end{array}$$(3.11) Let $\mathit{ϵ}>0$ and choose $\mathit{\delta }$ with $0<\mathit{\delta }<1$ such that $f_1(t)<\mathit{ϵ},0\le t\le \mathit{\delta }.$ Let us denote$$y_k=f_2(\mid T(x_k)\mid )$$ and consider$$\underset{k}{lim}{f}_1(y_k)=\underset{y_k\le \mathit{\delta },k\in \mathbb{N}}{lim}{f}_1(y_k)+\underset{y_k>\mathit{\delta },k\in \mathbb{N}}{lim}{f}_1(y_k)$$ Now, since $f_1$ is an modulus function, we have(3.12) $$\begin{array}{c}\hfill \underset{y_k\le \mathit{\delta },k\in \mathbb{N}}{lim}{f}_1(y_k)\le f_1(2)\underset{y_k\le \mathit{\delta },k\in \mathbb{N}}{lim}(y_k)\end{array}$$(3.12) For $y_k>\mathit{\delta }$, we have$$y_k<\frac{y_k}{\mathit{\delta }}<1+\frac{y_k}{\mathit{\delta }}$$ Now, since $f_1$ is non-decreasing and modulus, it follows that$$f_1(y_k)<f_1(1+\frac{y_k}{\mathit{\delta }})<\frac{1}{2}{f}_1(2)+\frac{1}{2}{f}_1(\frac{2y_k}{\mathit{\delta }})$$ Again, since $f_1$ satisfies ${\mathrm{\Delta }}_2-\text{Condition}$, we have$$f_1(y_k)<\frac{1}{2}K\frac{(y_k)}{\mathit{\delta }}{f}_1(2)+\frac{1}{2}K\frac{(y_k)}{\mathit{\delta }}{f}_1(2)$$ Thus, $f_1(y_k)<K\frac{(y_k)}{\mathit{\delta }}{f}_1(2)$ Hence,(3.13) $$\begin{array}{c}\hfill \underset{y_k>\mathit{\delta },k\in \mathbb{N}}{lim}{f}_1(y_k)\le max\{1,K{\mathit{\delta }}^{-1}{f}_1(2)\underset{y_k>\mathit{\delta },k\in \mathbb{N}}{lim}(y_k)\end{array}$$(3.13) Therefore, from Equations 2.11–2.13, we have $(x_k)\in {\mathcal{S}}_{\circ }^I(f_1{f}_2)$ Thus, ${\mathcal{S}}_{\circ }^I(f_2)\subseteq {\mathcal{S}}_{\circ }^I(f_1{f}_2).$ Hence, $\mathcal{X}(f_2)\subseteq \mathcal{X}(f_1{f}_2)$ for $\mathcal{X}$= ${\mathcal{S}}_{\circ }^I.$ For $\mathcal{X}$= ${\mathcal{S}}^I$, ${\mathcal{M}}_{\mathcal{S}}^I$, and ${\mathcal{M}}_{{\mathcal{S}}_{\circ }}^I$ the inclusions can be established similarly.

(b)

Let $x=(x_k)\in {\mathcal{S}}_{\circ }^I(f_1)\cap {\mathcal{S}}_{\circ }^I(f_2)$. Let $\mathit{ϵ}>0$ be given. Then, the sets(3.14) $$\begin{array}{c}\hfill \{k\in \mathbb{N}:f_1(\mid T(x_k)\mid )\ge \mathit{ϵ}\}\in I\end{array}$$(3.14) and(3.15) $$\begin{array}{c}\hfill \{k\in \mathbb{N}:f_2(\mid T(x_k)\mid )\ge \mathit{ϵ}\}\in I\end{array}$$(3.15) Therefore, from Equations 2.14 and 2.15 the set$$\{k\in \mathbb{N}:(f_1+f_2)(\mid T(x_k)\mid )\ge \mathit{ϵ}\}\in I$$ Thus, $x=(x_k)\in {\mathcal{S}}_{\circ }^I(f_1+f_2).$ Hence, ${\mathcal{S}}_{\circ }^I(f_1)\cap {\mathcal{S}}_{\circ }^I(f_2)\subseteq {\mathcal{S}}_{\circ }^I(f_1+f_2)$ For $\mathcal{X}$= ${\mathcal{S}}^I$, ${\mathcal{M}}_{\mathcal{S}}^I$, and ${\mathcal{M}}_{{\mathcal{S}}_{\circ }}^I$ the inclusions are similar.

For $f_2(x)=x$ and $f_1(x)=f(x)$, $\forall$$x\in [0,\infty )$, we have the following corollary.

Corollary 2.5

$\mathcal{X}\subseteq \mathcal{X}(f)$ for $\mathcal{X}$= ${\mathcal{S}}^I$, ${\mathcal{S}}_{\circ }^I$, ${\mathcal{M}}_{\mathcal{S}}^I$ and ${\mathcal{M}}_{{\mathcal{S}}_{\circ }}^I$

Theorem 2.6

For any modulus function $f$, the spaces ${\mathcal{S}}_{\circ }^I(f)$ and ${\mathcal{M}}_{{\mathcal{S}}_{\circ }}^I(f)$ are solid and monotone.

Proof

we prove the result for the space ${\mathcal{S}}_{\circ }^I(f)$. For ${\mathcal{M}}_{{\mathcal{S}}_{\circ }}^I(f)$, the proof can be obtained similarly. For, let $(x_k)\in {\mathcal{S}}_{\circ }^I(f)$ be any arbitrary element. Then, the set(3.16) $$\begin{array}{c}\hfill \{k\in \mathbb{N}:f(\mid T(x_k)\mid )\ge \mathit{ϵ}\}\in I\end{array}$$(3.16)

Let $({\mathit{\alpha }}_k)$ be a sequence of scalars such that$$\mid {\mathit{\alpha }}_k\mid \le 1,\text{for all k}\in \mathbb{N}$$

Then the result follows from Equation 2.16 and the following inequality.$$f(\mid T({\mathit{\alpha }}_k{x}_k)\mid )=f(\mid {\mathit{\alpha }}_{k}T(x_k)\mid )\le \mid {\mathit{\alpha }}_k\mid f(\mid T(x_k)\mid )\le f(\mid T(x_k)\mid ),\text{for all k}\in \mathbb{N}$$

That the space ${\mathcal{S}}_{\circ }^I(f)$ is monotone follows from the Lemma (I). Hence ${\mathcal{S}}_{\circ }^I(f)$ is solid and monotone.

Theorem 2.7

The spaces ${\mathcal{S}}^I(f)$ and ${\mathcal{M}}_{\mathcal{S}}^I(f)$ are not neither solid nor monotone.

Proof

Here we give a counter example for the proof of this result.

Counter example. Let $I=I_f$ and $f(x)=x$ for all $x\in [0,\infty ).$ Consider the K-step ${\mathcal{Z}}_K$ of $\mathcal{Z}$ defined as follows.

Let $(x_k)\in \mathcal{Z}$ and let $(y_k)\in {\mathcal{Z}}_K$ be such that$$\begin{array}{c}\hfill y_k=\left\{\begin{array}{c}{x}_k,\text{if}k\text{is even},\\ 0,\text{otherwise}\end{array}\right.\end{array}$$

Consider the sequence $(x_k)$ defined as by $x_k=1$ for all $k\in \mathbb{N}$. Then $(x_k)\in {\mathcal{S}}^I(f)$ and ${\mathcal{M}}_{\mathcal{S}}^I(f)$ but its $K$-step preimage does not belong to ${\mathcal{S}}^I(f)$ and ${\mathcal{M}}_{\mathcal{S}}^I(f).$ Thus, ${\mathcal{S}}^I(f)$ and ${\mathcal{M}}_{\mathcal{S}}^I(f)$ are not monotone. Hence, ${\mathcal{S}}^I(f)$ and ${\mathcal{M}}_{\mathcal{S}}^I(f)$ are not solid by Lemma(I).

Theorem 2.8

If $(x=x_k)$ and $(y=y_k)$ be two sequences with $T(x·y)=T(x)T(y)$. Then, the spaces ${\mathcal{S}}^I(f)$ and ${\mathcal{S}}_{\circ }^I(f)$ are sequence algebra.

Proof

Let $(x=x_k)$ and $(y=y_k)$ be two elements of ${\mathcal{S}}_{\circ }^I(f)$ with $T(x·y)=T(x)T(y)$. Then, the sets(3.17) $$\begin{array}{c}\hfill \{k\in \mathbb{N}:f(\mid T(x_k)\mid )\ge \mathit{ϵ}\}\in I\end{array}$$(3.17)

and(3.18) $$\begin{array}{c}\hfill \{k\in \mathbb{N}:f(\mid T(y_k)\mid )\ge \mathit{ϵ}\}\in I\end{array}$$(3.18)

Therefore,$$\{k\in \mathbb{N}:f(\mid T(x_k).T(y_k)\mid )\ge \mathit{ϵ}\}\in I$$

Thus, $(x_k).(y_k)\in {\mathcal{S}}_{\circ }^I(f).$ Hence, ${\mathcal{S}}_{\circ }^I(f)$is sequence algebra. For ${\mathcal{S}}^I(f)$, the result can be proved similarly.

Theorem 2.9

Let $f$ be a modulus function.

Then, ${\mathcal{S}}_{\circ }^I(f)\subset {\mathcal{S}}^I(f)\subset {\mathcal{S}}_{\infty }^I(f).$

Proof

The inclusion ${\mathcal{S}}_{\circ }^I(f)\subset {\mathcal{S}}^I(f)$ is obvious.

Next, let $(x_k)\in {\mathcal{S}}^I(f)$. Then there exists some $L$ such that$$\{k\in \mathbb{N}:f(\mid T(x_k)-L\mid )\ge \mathit{ϵ}\}\in I$$

We have$$f(\mid T(x_k)\mid )\le \frac{1}{2}f(\mid T(x_k)-L\mid )+f(\frac{1}{2}\mid L\mid )$$

Taking supremum over $k$ on both sides, we get $(x_k)\in {\mathcal{S}}_{\infty }^I(f)$

Hence, ${\mathcal{S}}_{\circ }^I(f)\subset {\mathcal{S}}^I(f)\subset {\mathcal{S}}_{\infty }^I(f)$

Theorem 2.10

If $f(x)=x$ for all $x\in [0,\infty ]$. Then, the function $ħ:{\mathcal{M}}_{\mathcal{S}}^I(f)\to \mathbb{R}$ defined by $ħ(x)=I-limf(\mid T(x_k)\mid )$, where ${\mathcal{M}}_{\mathcal{S}}^I(f)={\mathcal{S}}_{\infty }(f)\cap {\mathcal{S}}^I(f)$ is a Lipschitz function and hence uniformly continuous.

Proof

Clearly, the function $ħ$ is well defined. Let $x=(x_k),y=(y_k)\in {\mathcal{M}}_{\mathcal{S}}^I(f),x\ne y.$ Then, the sets$$A_x=\{k\in \mathbb{N}:f(\mid T(x)-ħ(x)\mid )\ge \Vert x-y{\Vert }_{\ast }\}\in I$$$$A_y=\{k\in \mathbb{N}:f(\mid T(y)-ħ(y)\mid )\ge \Vert x-y{\Vert }_{\ast }\}\in I$$

where$$\Vert x-y{\Vert }_{\ast }=\underset{k}{sup}f(\mid T(x_k)-T(y_k)\mid )$$

Thus, the sets$$B_x=\{k\in \mathbb{N}:\mid T(x)-ħ(x)\mid <\Vert x-y{\Vert }_{\ast }\}\in \mathrm{\pounds }(I)$$$$B_y=\{k\in \mathbb{N}:\mid T(y)-ħ(y)\mid <\Vert x-y{\Vert }_{\ast }\}\in \mathrm{\pounds }(I)$$

Hence, $B=B_x\cap B_y\in \mathrm{\pounds }(I)$, so that $B\ne \mathrm{\varnothing }$ Now, taking $k\in B$, we have$$\mid ħ(x)-ħ(y)\mid \le \mid ħ(x)-T(x_k)\mid +\mid T(x_k)-T(y_k)\mid +\mid T(y_k)-ħ(y)\mid \le 3\Vert x-y{\Vert }_{\ast }$$

Therefore, $ħ$ is Lipschitz function and hence uniformly continuous.

Theorem 2.11

If $f(x)=x$ for all $x\in [0,\infty ]$ and if $x=(x_k),y=(y_k)\in {\mathcal{M}}_{\mathcal{S}}^I(f)$ with $T(x·y)=T(x)T(y)$. Then $(x·y)\in {\mathcal{M}}_{\mathcal{S}}^I(f)$ and $ħ(xy)=ħ(x)ħ(y)$ where $ħ:{\mathcal{M}}_{\mathcal{S}}^I(f)\to \mathbb{R}$ is defined by $ħ(x)=I-limf(\mid T(x_k)\mid )$.

Proof

For $\mathit{ϵ}>0$, the sets(3.19) $$\begin{array}{c}\hfill B_x=\{k\in \mathbb{N}:|T(x_k)-ħ(x)|<\mathit{ϵ}\}\in \mathrm{\pounds }(I)\end{array}$$(3.19) (3.20) $$\begin{array}{c}\hfill B_y=\{k\in \mathbb{N}:|T(y_k)-ħ(y)|<\mathit{ϵ}\}\in \mathrm{\pounds }(I)\end{array}$$(3.20)

where $\Vert x-y{\Vert }_{\ast }=\mathit{ϵ}$

Now,(3.21) $$\begin{array}{cc}\hfill |T(x_k{y}_k)-ħ(x)ħ(y)|& =|T(x_k)T(y_k)-T(x_k)ħ(y)+T(x_k)ħ(y)-ħ(x)ħ(y)|\hfill \\ \hfill & \le |T(x_k)||y_k-ħ(y)|+|ħ(y)||x_k-ħ(x)|\hfill \end{array}$$(3.21)

As ${\mathcal{M}}_{\mathcal{S}}^I(f)\subseteq {\mathcal{S}}_{\infty }(f)$, there exists an $M\in \mathbb{R}$ such that $|T(x_k)|<M$ and $|ħ(y)|<M$.

Therefore, from Equations 2.19–2.21, we have$$|T(x_k{y}_k)-ħ(x)ħ(y)|\le M\mathit{ϵ}+M\mathit{ϵ}=2M\mathit{ϵ}$$

for all $k\in B_x\cap B_y\in \mathrm{\pounds }(I)$.

Hence $(x·y)\in {\mathcal{M}}_{\mathcal{S}}^I(f)$ and $ħ(xy)=ħ(x)ħ(y)$.

Acknowledgements

The authors would like to record their gratitude to the reviewer for his careful reading and making some useful corrections which improved the presentation of the paper.

Additional information

Funding

Funding. The authors received no direct funding for this research.

Notes on contributors

Vakeel A. Khan

Vakeel A. Khan received his MPhil and PhD degrees in Mathematics from Aligarh Muslim University, Aligarh, India. Currently, he is a senior assistant professor in the same university. He is a vigorous researcher in the area of Sequence Spaces , and has published a number of research papers in reputed national and international journals, including Numerical Functional Analysis and Optimization (Taylor?s and Francis), Information Sciences (Elsevier), Applied Mathematics A Journal of Chinese Universities and Springer-Verlag (China), Rendiconti del Circolo Matematico di Palermo Springer-Verlag (Italy), Studia Mathematica (Poland), Filomat (Serbia), Applied Mathematics & Information Sciences (USA).

Mohd Shafiq

Mohd Shafiq did his MSc in Mathematics from University of Jammu, Jammu and Kashmir, India. Currently, he is a PhD scholar at Aligarh Muslim University, Aligarh, India. His research interests are Functional Analysis, sequence spaces, I-convergence, invariant means, zweier sequences, and interval numbers theory.

Rami Kamel Ahmad Rababah

Rami Kamel Ahmad Rababah is a research scholar in the Department of Mathematics, Aligarh Muslim University, Aligarh, India.