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A note on Nakano generalized difference sequence space

Abstract

In this paper, we investigate the necessary conditions on any s-type sequence space to form an operator ideal. As a result, we show that the s-type Nakano generalized difference sequence space X fails to generate an operator ideal. We investigate the sufficient conditions on X to be premodular Banach special space of sequences and the constructed prequasi-operator ideal becomes a small, simple, and closed Banach space and has eigenvalues identical with its s-numbers. Finally, we introduce necessary and sufficient conditions on X explaining some topological and geometrical structures of the multiplication operator defined on X.

Introduction

By \(\mathbb{C}^{\mathbb{N}}\), c, \(\ell _{\infty }\), \(\ell _{r}\), and \(c_{0}\), we denote the spaces of all, convergent, bounded, r-absolutely summable, and convergent to zero sequences of complex numbers, and \(\mathbb{N}\) is the set of nonnegative integers. Tripathy et al. [14] introduced and studied the forward and backward generalized difference sequence spaces \(U(\Delta _{n}^{(m)})= \{(w_{k})\in \mathbb{C}^{\mathbb{N}}:( \Delta _{n}^{(m)} w_{k})\in U \} \) and \(U(\Delta _{n}^{m})= \{(w_{k})\in \mathbb{C}^{\mathbb{N}}:( \Delta _{n}^{m} w_{k})\in U \} \), where \(m, n\in \mathbb{N}\), \(U =\ell _{\infty }\), c or \(c_{0}\), with \(\Delta _{n}^{(m)}w_{k}=\sum^{m}_{\nu =0}(-1)^{\nu } C{_{ \nu }^{m}} w_{k+\nu n}\), and \(\Delta _{n}^{m}w_{k}= \sum^{m}_{\nu =0}(-1)^{\nu } C{_{\nu }^{m}} w_{k-\nu n} \), respectively. When \(n=1\), the generalized difference sequence spaces reduced to \(U(\Delta ^{(m)})\) were defined and investigated by Et and Çolak [3]. For \(m=1\), the generalized difference sequence spaces reduced to \(U(\Delta _{n})\) were defined and investigated by Tripathy and Esi [13]. For \(n=1\) and \(m=1\), the generalized difference sequence spaces reduced to \(U(\Delta )\) were defined and studied by Kizmaz [6]. Summability is very important in mathematical models and has numerous implementations, such as normal series theory, approximation theory, ideal transformations, fixed point theory, and so forth. Let \(r=(r_{j})\in \mathbb{R^{+}}^{\mathbb{N}}\), where \(\mathbb{R^{+}}^{ \mathbb{N}}\) is the space of sequences with positive reals. We define the Nakano backward generalized difference sequence space as follows: \((\ell (r, \Delta _{n+1}^{m}) )_{\tau }= \{w=(w_{j})\in \mathbb{C}^{\mathbb{N}}:\exists \sigma >0 \text{ with } \tau (\sigma w)< \infty \} \), where \(\tau (w)=\sum^{\infty }_{j=0} |\Delta _{n+1}^{m}|w_{j}| |^{r_{j}} \), \(w_{j}=0\) for \(j<0\), \(\Delta _{n+1}^{m}|w_{j}|=\Delta _{n+1}^{m-1}|w_{j}|-\Delta _{n+1}^{m-1}|w_{j-1}|\) and \(\Delta ^{0}w_{j}=w_{j}\) for all \(j,n,m\in \mathbb{N}\). It is a Banach space with norm \(\|w\|=\inf \{\sigma >0:\tau (\frac{w}{\sigma } )\leq 1 \}\). If \((r_{j})\in \ell _{\infty }\), then \(\ell (r, \Delta _{n+1}^{m})= \{w=(w_{j})\in \mathbb{C}^{ \mathbb{N}}:\sum^{\infty }_{j=0} |\Delta _{n+1}^{m}|w_{j}| |^{r_{j}}< \infty \}\). Several geometric and topological characteristics of \(\ell (r, \Delta _{n+1}^{m})\) have been studied (see [5, 16]). By \(\mathfrak{B}(W, Z)\) we denote the set of all linear bounded operators between Banach spaces W and Z, and if \(W=Z\), then we write \(\mathfrak{B}(W)\). The multiplication operators and operator ideals have a wide field of mathematics in functional analysis, for instance, in eigenvalue distributions theorem, geometric structure of Banach spaces, theory of fixed point, and so forth. An s-number function [12] is a map defined on \(\mathfrak{B}(W, Z)\) that associates with each operator \(T\in \mathfrak{B}(W, Z)\) a nonnegative scaler sequence \((s_{n}(T))_{n=0}^{\infty }\) satisfying the following conditions:

  1. (a)

    \(\|T \|=s_{0}(T)\geq s_{1}(T)\geq s_{2}(T)\geq \cdots\geq 0\) for \(T\in \mathfrak{B}(W, Z)\),

  2. (b)

    \(s_{m+n-1}(T_{1}+T_{2})\leq s_{m}(T_{1})+s_{n}(T_{2})\) for all \(T_{1}, T_{2}\in \mathfrak{B}(W, Z)\) and m, \(n\in \mathbb{N}\),

  3. (c)

    ideal property: \(s_{n}(RVT)\leq \|R \| s_{n}(V) \|T \|\) for all \(T\in \mathfrak{B}(W_{0}, W)\), \(V\in \mathfrak{B}(W, Z)\), and \(R\in \mathfrak{B}(Z, Z_{0})\), where \(W_{0}\) and \(Z_{0}\) are arbitrary Banach spaces,

  4. (d)

    if \(G\in \mathfrak{B}(W, Z)\) and \(\lambda \in \mathbb{C}\), then \(s_{n}(\lambda G)=|\lambda |s_{n}(G)\).

  5. (e)

    rank property: If \(\operatorname{rank}(T)\leq n\), then \(s_{n}(T)=0\) for each \(T\in \mathfrak{B}(W, Z)\),

  6. (f)

    norming property: \(s_{r\geq n}(I_{n})=0\) or \(s_{r< n}(I_{n})=1\), where \(I_{n}\) is the unit operator on the n-dimensional Hilbert space \(\ell _{2}^{n}\).

The s-numbers have many examples such as the rth approximation number

$$ \alpha _{r}(V)=\inf \bigl\{ \Vert V-B \Vert :B\in \mathfrak{B}(W, Z) \text{ and }\operatorname{rank}(B)\leq r\bigr\} $$

and the rth Kolmogorov number

$$ d_{r}(V)=\inf_{\dim W\leq r}\sup_{ \Vert w \Vert \leq 1} \inf_{v\in W} \Vert Vw-v \Vert . $$

The following notations will be further used:

$$\begin{aligned}& X^{\mathcal{S}}:= \bigl\{ X^{\mathcal{S}}(W, Z) \bigr\} , \quad \text{where } X^{\mathcal{S}}(W, Z):= \bigl\{ V\in \mathfrak{B}(W, Z):(\bigl(s_{j}(V)\bigr)_{j=0}^{ \infty }\in X \bigr\} ; \\& X^{\mathrm{app}}:= \bigl\{ X^{\mathrm{app}}(W, Z) \bigr\} ,\quad \text{where } X^{\mathrm{app}}(W, Z):= \bigl\{ V\in \mathfrak{B}(W, Z):(\bigl(\alpha _{j}(V)\bigr)_{j=0}^{\infty }\in X \bigr\} ; \\& X^{\mathrm{Kol}}:= \bigl\{ X^{\mathrm{Kol}}(W, Z) \bigr\} ,\quad \text{where } X^{\mathrm{Kol}}(W, Z):= \bigl\{ V\in \mathfrak{B}(W, Z):(\bigl(d_{j}(V) \bigr)_{j=0}^{\infty }\in X \bigr\} ; \\& X^{\nu }:= \bigl\{ X^{\nu }(W, Z) \bigr\} ,\quad \text{where } \\& X^{\nu }(W, Z):= \bigl\{ V\in \mathfrak{B}(W, Z):(\bigl(\nu _{j}(V)\bigr)_{j=0}^{\infty }\in X \text{ and } \bigl\Vert V-\nu _{j}(V)I \bigr\Vert =0\text{ for all} j\in \mathbb{N} \bigr\} . \end{aligned}$$

The s-type Nakano generalized difference sequence space under \(\tau :\ell (r, \Delta _{n+1}^{m}) )\rightarrow [0,\infty )\) is defined as

$$\begin{aligned}& s\text{-type } \bigl(\ell \bigl(r, \Delta _{n+1}^{m}\bigr) \bigr)_{\tau } \\& \quad := \bigl\{ \bigl(s_{j}(V)\bigr)^{\infty }_{j=0}\in \mathbb{C}^{\mathbb{N}}: V\in \mathfrak{B}(W, Z) \text{ and } \tau \bigl( \lambda \bigl(s_{j}(V)\bigr)\bigr)_{j=0}^{ \infty }< \infty \text{ for some } \lambda >0 \bigr\} . \end{aligned}$$

If \((r_{j})\in \ell _{\infty }\), then

$$ s\text{-type } \bigl(\ell \bigl(r, \Delta _{n+1}^{m}\bigr) \bigr)_{\tau }= \Biggl\{ \bigl(s_{j}(V)\bigr)^{\infty }_{j=0}\in \mathbb{C}^{\mathbb{N}}: V\in \mathfrak{B}(W, Z) \text{ and } \sum ^{\infty }_{j=0} \bigl\vert \Delta _{n+1}^{m}s_{j}(V) \bigr\vert ^{r_{j}}< \infty \Biggr\} . $$

Some examples of s-type Nakano generalized difference sequence spaces are

$$\begin{aligned}& s\text{-}\text{type } \biggl(\ell \biggl(\biggl( \frac{j}{j+1}\biggr), \Delta _{2}^{3}\biggr) \biggr)_{ \tau } \\& \quad = \Biggl\{ \bigl(s_{j}(V) \bigr)^{\infty }_{j=0}\in \mathbb{C}^{\mathbb{N}}: V \in \mathfrak{B}(W, Z) \text{ and } \sum^{\infty }_{j=0} \bigl\vert \Delta _{2}^{3}s_{j}(V) \bigr\vert ^{\frac{j}{j+1}}< \infty \Biggr\} \end{aligned}$$

and

$$ s\text{-}\text{type } \bigl(\ell _{r}(\Delta ) \bigr)_{\tau }= \Biggl\{ \bigl(s_{j}(V)\bigr)^{ \infty }_{j=0} \in \mathbb{C}^{\mathbb{N}}: V\in \mathfrak{B}(W, Z) \text{ and } \Biggl( \sum^{\infty }_{j=0} \bigl\vert \Delta s_{j}(V) \bigr\vert ^{r} \Biggr)^{\frac{1}{r}}< \infty \Biggr\} . $$

A few operator ideals in the class of Hilbert or Banach spaces are defined by distinct scalar sequence spaces such as the ideal of compact operators \(\mathfrak{B}_{c}\) formed by \((d_{r}(V))\) and \(c_{0}\). Pietsch [12] studied the smallness of the quasi-ideals \((\ell _{r})^{\mathrm{app}}\) for \(r\in (0, \infty )\), the ideals of Hilbert–Schmidt operators between Hilbert spaces constructed by \(\ell _{2}\), and the ideals of nuclear operators generated by \(\ell _{1}\). He explained that \(\overline{\mathfrak{F}}=(\ell _{r})^{\mathrm{app}}\) for \(r\in [1, \infty )\), where \(\overline{\mathfrak{F}}\) is the closed class of all finite rank operators, and the class \((\ell _{r})^{\mathrm{app}}\) became simple Banach [11]. The strict inclusions \((\ell _{r})^{\mathrm{app}}(W, Z)\varsubsetneqq (\ell _{j})^{\mathrm{app}}(W, Z) \subsetneqq \mathfrak{B}(W, Z)\) for \(j>r>0\), where W and Z are infinite-dimensional Banach spaces, were investigated by Makarov and Faried [7]. Faried and Bakery [4] gave a generalization of the class of quasi-operator ideal, which is the prequasi-operator ideal and examined several geometric and topological structures of \((\ell _{M})^{\mathcal{S}}\) and \((\operatorname{ces}(r))^{\mathcal{S}}\). On sequence spaces, Mursaleen and Noman [10] investigated the compact operators on some difference sequence spaces. Kiliçman and Raj [5] studied the matrix transformations of Norlund–Orlicz difference sequence spaces of nonabsolute type. Yaying et al. [15] examined the operator ideal of type sequence space whose q-Cesáro matrix in \(\ell _{p}\) for all \(q\in (0,1]\) and \(1< p<\infty \). The point of this paper is explaining some results of \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }\) equipped with a prequasi-norm τ. Firstly, we give necessary conditions on any s-type sequence space to give an operator ideal. Secondly, we study some geometric and topological structures of \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{\mathcal{S}}\) such as closed, small, and simple Banach and \((\ell (p, \Delta _{n+1}^{m}) )^{\mathcal{S}}= (\ell (p, \Delta _{n+1}^{m}) )^{\nu }\). We determine a strict inclusion relation of \((\ell (p, \Delta _{n+1}^{m}) )^{\mathcal{S}}\) for different p and \(\Delta _{n+1}^{m}\). Finally, we investigate the multiplication operator defined on \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }\).

Preliminaries and definitions

Definition 2.1

([12])

An operator \(V\in \mathfrak{B}(W)\) is called approximable if there are \(D_{r}\in \mathfrak{F}(W)\) for every \(r\in \mathbb{N}\) and \(\lim_{r\rightarrow \infty }\|V-D_{r}\|=0\).

By \(\Upsilon (W, Z)\) we denote the space of all approximable operators from W to Z.

Lemma 2.2

([12])

Let \(V\in \mathfrak{B}(W, Z)\). If \(V\notin \Upsilon (W, Z)\), then there are \(G\in \mathfrak{B}(W)\) and \(B\in \mathfrak{B}(Z)\) such that \(\mathit{BVG} e_{r}=e_{r}\) for all \(r\in \mathbb{N}\).

Definition 2.3

([12])

A Banach space W is called simple if \(\mathfrak{B}(W)\) includes a unique nontrivial closed ideal.

Theorem 2.4

([12])

If W is Banach space with \(\dim (W)=\infty \), then

$$ \mathfrak{F}(W)\varsubsetneqq \Upsilon (W)\varsubsetneqq \mathfrak{B}_{c}(W) \varsubsetneqq \mathfrak{B}(W). $$

Definition 2.5

([9])

An operator \(V\in \mathfrak{B}(W)\) is called Fredholm if \(\dim (R(V))^{c}<\infty \), \(\dim (\ker V)<\infty \), and \(R(V)\) is closed, where \((R(V))^{c}\) denotes the complement of range V.

We will further use the sequence \(e_{j}=(0, 0,\dots ,1,0,0,\dots )\) with 1 in the jth coordinate for all \(j\in \mathbb{N}\).

Definition 2.6

([4])

The space of linear sequence spaces \(\mathbb{Y}\) is called a special space of sequences (sss) if

  1. (1)

    \(e_{r}\in \mathbb{Y}\) with \(r\in \mathbb{N}\),

  2. (2)

    if \(u=(u_{r})\in \mathbb{C}^{\mathbb{N}}\), \(v=(v_{r})\in \mathbb{Y}\), and \(|u_{r}|\leq |v_{r}|\) for every \(r\in \mathbb{N}\), then \(u\in \mathbb{Y}\). This means that \(\mathbb{Y}\) is “solid”,

  3. (3)

    if \((u_{r})_{r=0}^{\infty }\in \mathbb{Y}\), then \((u_{[\frac{r}{2}]})_{r=0}^{\infty }\in \mathbb{Y}\), where \([\frac{r}{2}]\) means the integral part of \(\frac{r}{2}\).

Definition 2.7

([2])

A subspace of the (sss) \(\mathbb{Y}_{\tau }\) is called a premodular (sss) if there is a function \(\tau : \mathbb{Y}\rightarrow [0,\infty )\) satisfying the following conditions:

  1. (i)

    \(\tau (y)\geq 0\) for each \(y\in \mathbb{Y}\) and \(\tau (y)=0\Leftrightarrow y=\theta \), where θ is the zero element of \(\mathbb{Y}\),

  2. (ii)

    there exists \(a\geq 1\) such that \(\tau (\eta y)\leq a|\eta |\tau ( y)\) for all \(y\in \mathbb{Y}\) and \(\eta \in \mathbb{C}\),

  3. (iii)

    for some \(b\geq 1\), \(\tau (y+z)\leq b(\tau (y)+\tau (z))\) for all \(y, z\in \mathbb{Y}\),

  4. (iv)

    \(|y_{r}|\leq |z_{r}|\) with \(r\in \mathbb{N}\), implies \(\tau ((y_{r}))\leq \tau ((z_{r}))\),

  5. (v)

    for some \(b_{0}\geq 1\), \(\tau ((y_{r}))\leq \tau ((y_{[\frac{r}{2}]}))\leq b_{0}\tau ((y_{i}))\),

  6. (vi)

    if \(y=(y_{r})_{r=o}^{\infty }\in \mathbb{Y}\) and \(d> 0\), then there is \(r_{0}\in \mathbb{N}\) with \(\tau ((y_{r})_{r=r_{0}}^{\infty })< d\),

  7. (vii)

    there is \(t>0\) with \(\tau (\nu , 0, 0, 0,\dots )\geq t|\nu |\tau (1, 0, 0, 0,\dots )\) for all \(\nu \in \mathbb{C}\).

The (sss) \(\mathbb{Y}_{\tau }\) is called prequasi-normed (sss) if τ satisfies parts (i)–(iii) of Definition 2.7, and when the space \(\mathbb{Y}\) is complete under τ, then \(\mathbb{Y}_{\tau }\) is called a prequasi-Banach (sss).

Theorem 2.8

([2])

A prequasi-norm (sss) \(\mathbb{Y}_{\tau }\), whenever it is premodular (sss).

By \(\mathfrak{B}\) we denote the class of all bounded linear operators between any pair of Banach spaces.

Definition 2.9

([2])

A class \(\mathfrak{G}\subseteq \mathfrak{B}\) is called an operator ideal if every component \(\mathfrak{G}(W, Z)=\mathfrak{G}\cap \mathfrak{B}(W, Z)\), where W and Z are Banach spaces, satisfies the following conditions:

  1. (i)

    \(\mathfrak{G}\supseteq \mathfrak{F}\), that is, the class \(\mathfrak{G}\) contains the class of all finite-rank Banach space operators \(\mathfrak{F}\).

  2. (ii)

    The space \(\mathfrak{G}(W, Z)\) is linear over \(\mathbb{C}\).

  3. (iii)

    If \(V\in \mathfrak{B}(W_{0}, W)\), \(G\in \mathfrak{G}(W, Z)\), and \(Q\in \mathfrak{B}(Z, Z_{0})\), then \(QGV\in \mathfrak{G}(W_{0}, Z_{0})\), where \(W_{0}\) and \(Z_{0}\) are Banach spaces.

Definition 2.10

([2])

A prequasi-norm on the ideal B is a function \(\zeta :B\rightarrow [0, \infty )\) that satisfies the following conditions:

  1. (1)

    For all \(V\in B(W, Z)\), \(\zeta (V)\geq 0\) and \(\zeta (V)=0\) if and only if \(V=0\),

  2. (2)

    there is \(H\geq 1\) such that \(\zeta (\eta V)\leq H|\eta |\zeta (V)\) for all \(V\in B(W, Z)\) and \(\eta \in \mathbb{C}\),

  3. (3)

    there is \(b\geq 1\) such that \(\zeta (V_{1}+V_{2})\leq b[\zeta (V_{1})+\zeta (V_{2})]\) for all \(V_{1},V_{2}\in B(W, Z)\),

  4. (4)

    there is \(D\geq 1\) such that if \(U\in \mathfrak{B}(W_{0}, W)\), \(T\in B(W, Z)\), and \(V\in \mathfrak{B}(Z, Z_{0})\), then \(\zeta (VTU)\leq D \|V \| \zeta (T) \|U \|\).

Theorem 2.11

([4])

The function \(\zeta (V)=\tau (s_{r}(V))_{r=0}^{\infty }\) forms a prequasi-norm on \(X_{\tau }^{\mathcal{S}}\) whenever \(X_{\tau }\) is a premodular (sss).

We will further use the inequality \(|a_{i}+b_{i}|^{q_{i}}\leq H(|a_{i}|^{q_{i}}+|b_{i}|^{q_{i}})\), where \(q_{i}\geq 0\) for all \(i\in \mathbb{N}\), \(H=\max \{1,2^{h-1}\}\), and \(h=\sup_{i}q_{i}\) (see [1]).

Main results

Pietsch [12] investigated the quasi-ideals \((\ell _{r})^{\mathrm{app}}\) for \(r\in (0, \infty )\). Faried and Bakery [4] introduced sufficient conditions on any linear sequence space X such that the class \(X^{\mathcal{S}}\) of all bounded linear operators between arbitrary Banach spaces with its sequence of s-numbers belongs to X generates an operator ideal. In this section, we give necessary conditions on s-type X under \(\tau :X\rightarrow [0,\infty )\) such that \(X^{\mathcal{S}}_{\tau }\) forms an operator ideal. Consequently, any none solid s-type sequence space does not form an operator ideal. We explain sufficient conditions on Nakano backward generalized difference sequence space to be premodular Banach (sss).

Theorem 3.1

For s-type \(X_{\tau }:= \{x=(s_{n}(V))\in \mathbb{C}^{\mathbb{N}}: V\in \mathfrak{B}(W, Z) \textit{ and } \tau (x)<\infty \}\), if \(X^{\mathcal{S}}_{\tau }\) is an operator ideal, then the following conditions are satisfied:

  1. 1.

    The set \(X_{\tau }\) contains F, the space of all sequences with finite nonzero numbers.

  2. 2.

    If \((s_{r}(V_{1}) )_{r=0}^{\infty }\in X_{\tau }\) and \((s_{r}(V_{2}) )_{r=0}^{\infty }\in X_{\tau }\), then \((s_{r}(V_{1}+V_{2}) )_{r=0}^{\infty }\in X_{\tau }\).

  3. 3.

    For all \(\lambda \in \mathbb{C}\) and \((s_{r}(V) )_{r=0}^{\infty }\in X_{\tau }\), we have \(|\lambda | (s_{r}(V) )_{r=0}^{\infty }\in X_{\tau }\).

  4. 4.

    The sequence space \(X_{\tau }\) is solid. This means that if \((s_{r}(V) )_{r=0}^{\infty }\in \mathbb{C}^{\mathbb{N}}\), \((s_{r}(T) )_{r=0}^{\infty }\in X_{\tau }\) and \(s_{r}(V)\leq s_{r}(T)\) for every \(r\in \mathbb{N}\) and \(T,V\in \mathfrak{B}(W, Z)\), then \((s_{r}(V) )_{r=0}^{\infty }\in X_{\tau }\).

Proof

Let \(X^{\mathcal{S}}_{\tau }\) be an operator ideal.

  1. (i)

    We have \(\mathfrak{F}(W, Z)\subset X^{\mathcal{S}}_{\tau }(W, Z)\). Hence for all \(T\in \mathfrak{F}(W, Z)\), we have \((s_{r}(V) )_{r=0}^{\infty }\in F\). This gives \((s_{r}(V) )_{r=0}^{\infty }\in X_{\tau }\). Hence \(F\subset X_{\tau }\).

  2. (ii)

    The space \(X^{\mathcal{S}}_{\tau }(W, Z)\) is linear over \(\mathbb{C}\). Hence for all \(\lambda \in \mathbb{C}\) and \(V_{1},V_{2}\in X^{\mathcal{S}}_{\tau }(W, Z)\), we have \(V_{1}+V_{2}\in X^{\mathcal{S}}_{\tau }(W, Z)\) and \(\lambda V_{1} \in X^{\mathcal{S}}_{\tau }(W, Z)\). This implies

    $$ \bigl(s_{r}(V_{1}) \bigr)_{r=0}^{\infty } \in X_{\tau } \quad \text{and}\quad \bigl(s_{r}(V_{2}) \bigr)_{r=0}^{\infty }\in X_{\tau }\quad \Rightarrow\quad \bigl(s_{r}(V_{1}+V_{2}) \bigr)_{r=0}^{\infty }\in X_{\tau } $$

    and

    $$ \lambda \in \mathbb{C} \quad \text{and}\quad \bigl(s_{r}(V_{1}) \bigr)_{r=0}^{ \infty }\in X_{\tau }\quad \Rightarrow\quad \vert \lambda \vert \bigl(s_{r}(V_{1}) \bigr)_{r=0}^{ \infty }\in X_{\tau }. $$
  3. (iii)

    If \(A\in \mathfrak{B}(W_{0}, W)\), \(B\in X^{\mathcal{S}}_{\tau }(W, Z)\), and \(D\in \mathfrak{B}(Z, Z_{0})\), then \(DBA\in X^{\mathcal{S}}_{\tau }(W_{0}, Z_{0})\), where \(W_{0}\) and \(Z_{0}\) are arbitrary Banach spaces. Therefore, if \(A\in \mathfrak{B}(W_{0}, W)\), \((s_{r}(B) )_{r=0}^{\infty }\in X_{\tau }\), and \(D\in \mathfrak{B}(Z, Z_{0})\), then \((s_{r}(DBA) )_{r=0}^{\infty }\in X_{\tau }\) since \(s_{r}(DBA)\leq \|D \| s_{r}(B) \|A \|\). By using condition 3, if \(( \|D \| \|A \|s_{r}(B) )_{r=0}^{\infty }\in X_{\tau }\), then we have \((s_{r}(DBA) )_{r=0}^{\infty }\in X_{\tau }\). This means that \(X_{\tau }\) is solid.

 □

Corollary 3.2

The s-type q-Cesáro sequence space of nonabsolute type \(\chi _{p}^{q}\) is solid for all \(q\in (0,1]\) and \(1< p<\infty \).

Proof

From Theorem 5.6 in [15], since the class of all bounded linear operators between any two Banach spaces such that its s-numbers belong to q-Cesáro sequence space of nonabsolute type forms an operator ideal if \(q\in (0,1]\) and \(1< p<\infty \). Then by Theorem 3.1 the s-type q-Cesáro sequence space of nonabsolute type is solid for all \(q\in (0,1]\) and \(1< p<\infty \). □

Theorem 3.3

The space \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{\mathcal{S}}\) is not operator ideal, where \((p_{i})\) satisfies \(0< p_{i}<\infty \) for all \(i\in \mathbb{N}\) and \(\tau (w)=\sum^{\infty }_{i=0} |\Delta _{n+1}^{m}|w_{i}| |^{p_{i}}\) for all \(w\in \ell (p, \Delta _{n+1}^{m})\).

Proof

We choose \(m=2\), \(n=1\), \(w_{k}=1\), \(v_{k}=w_{k}\) for \(k=3s\) and, otherwise, \(v_{k}=0\) for all \(s,k\in \mathbb{N}\). We have \(|v_{k}|\leq |w_{k}|\) for all \(k\in \mathbb{N}\), \(w\in (\ell (p, \Delta _{2}^{2}) )_{\tau }\), and \(v\notin (\ell (p, \Delta _{2}^{2}) )_{\tau }\). Hence the space \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }\) is not solid. This finishes the proof. □

According to Theorem 3.3, we correct Theorem 4.2 in [8], that is, the class of all bounded linear operators constructed by Musielak–Lorentz forward difference sequence spaces equipped with the Luxemburg norm and s-numbers fails to form a quasi-operator ideal, since it is not solid.

Definition 3.4

The backward generalized difference \(\Delta _{n+1}^{m}\) is called absolutely nondecreasing if from \(|x_{i}|\leq |y_{i}|\) for all \(i\in \mathbb{N}\) it follows that \(|\Delta _{n+1}^{m}|x_{i}| |\leq |\Delta _{n+1}^{m}|y_{i}| |\).

Theorem 3.5

If \((p_{i})\in \mathbb{R^{+}}^{\mathbb{N}}\cap \ell _{\infty }\) is an increasing and \(\Delta _{n+1}^{m}\) is absolutely nondecreasing, then the space \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }\) is a premodular Banach (sss), where

$$ \tau (w)= \sum^{\infty }_{i=0} \bigl\vert \Delta _{n+1}^{m} \vert w_{i} \vert \bigr\vert ^{p_{i}} \quad \textit{for all } w\in \ell \bigl(p, \Delta _{n+1}^{m}\bigr). $$

Proof

  1. (1-i)

    Suppose \(v, w\in \ell (p, \Delta _{n+1}^{m})\). Since \((p_{i})\in \ell _{\infty }\) and \(\Delta _{n+1}^{m}\) is absolutely nondecreasing, we have

    $$\begin{aligned} \tau (v+w) =&\sum^{\infty }_{i=0} \bigl\vert \Delta _{n+1}^{m} \vert v_{i}+w_{i} \vert \bigr\vert ^{p_{i}} \\ \leq& H \Biggl(\sum ^{\infty }_{i=0} \bigl\vert \Delta _{n+1}^{m} \vert v_{i} \vert \bigr\vert ^{p_{i}}+ \sum^{\infty }_{i=0} \bigl\vert \Delta _{n+1}^{m} \vert w_{i} \vert \bigr\vert ^{p_{i}} \Biggr) \\ =&H \bigl(\tau (v)+\tau (w) \bigr)< \infty , \end{aligned}$$

    where \(H=\max \{1,2^{\sup _{i} p_{i}-1}\}\). Then \(v+w\in \ell (p, \Delta _{n+1}^{m})\).

  2. (1-ii)

    Let \(\lambda \in \mathbb{C}\) and \(v\in \ell (p, \Delta _{n+1}^{m})\). Since \((p_{i})\) is bounded, we have

    $$ \tau (\lambda v)=\sum^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m} \vert \lambda v_{r} \vert \bigr\vert ^{p_{r}}\leq \sup _{r} \vert \lambda \vert ^{p_{r}}\sum ^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m} \vert v_{r} \vert \bigr\vert ^{p_{r}}=\sup_{r} \vert \lambda \vert ^{p_{r}} \tau (v)< \infty .$$

    Then \(\lambda v\in \ell (p, \Delta _{n+1}^{m})\). Hence from parts (1-i) and (1-ii) the space \(\ell (p, \Delta _{n+1}^{m})\) is linear. Since \(e_{r}\in \ell (p)\subseteq \ell (p, \Delta _{n+1}^{m})\) for all \(r\in \mathbb{N}\), we have \(e_{r}\in \ell (p, \Delta _{n+1}^{m})\) for all \(r\in \mathbb{N}\).

  3. (2)

    Suppose \(|x_{i}|\leq |y_{i}|\) for all \(i\in \mathbb{N}\) and \(y\in \ell (p, \Delta _{n+1}^{m})\). Since \(\Delta _{n+1}^{m}\) is absolutely nondecreasing. Hence we have

    $$ \tau (x)=\sum^{\infty }_{i=0} \bigl\vert \Delta _{n+1}^{m} \vert x_{i} \vert \bigr\vert ^{p_{i}} \leq \sum^{\infty }_{i=0} \bigl\vert \Delta _{n+1}^{m} \vert y_{i} \vert \bigr\vert ^{p_{i}}= \tau (y)< \infty ,$$

    so that \(x\in \ell (p, \Delta _{n+1}^{m})\).

  4. (3)

    Let \((v_{r})\in \ell (p, \Delta _{n+1}^{m})\). Since \((p_{r})\) is an increasing and \(\Delta _{n+1}^{m}\) is linear, we have

    $$\begin{aligned} \tau \bigl((v_{[\frac{r}{2}]})\bigr) =&\sum^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m} \vert v_{[ \frac{r}{2}]} \vert \bigr\vert ^{p_{r}} \\ =&\sum^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m} \vert v_{r} \vert \bigr\vert ^{p_{2r}}+ \sum^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m} \vert v_{r} \vert \bigr\vert ^{p_{2r+1}} \\ \leq& 2 \sum^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m} \vert v_{r} \vert \bigr\vert ^{p_{r}}=2\tau (v), \end{aligned}$$

    and then \((v_{[\frac{r}{2}]})\in \ell (p, \Delta _{n+1}^{m})\).

    1. (i)

      Obviously, \(\tau (w)\geq 0\) and \(\tau (w)=0\Leftrightarrow w=\theta \).

    2. (ii)

      \(a=\max \{1,\sup_{r}|\eta |^{p_{r}-1} \}\geq 1\), where \(\tau (\eta w)\leq a|\eta |\tau ( w)\) for all \(w\in \ell (p, \Delta _{n+1}^{m})\) and \(\eta \in \mathbb{C}\).

    3. (iii)

      The inequality \(\tau (v+w)\leq H(\tau (v)+\tau (w))\) for all \(v, w\in \ell (p, \Delta _{n+1}^{m})\) is satisfied.

    4. (iv)

      Clearly from (2).

    5. (v)

      From (3) we have that \(b_{0}=2\geq 1\).

    6. (vi)

      It is obvious that \(\overline{F}=\ell (p, \Delta _{n+1}^{m})\).

    7. (vii)

      There is ζ with \(0<\zeta \leq |\eta |^{p_{0}-1}\) such that \(\tau (\eta , 0, 0, 0,\ldots)\geq \zeta |\eta |\tau (1, 0, 0, 0,\ldots)\) for all \(\eta \neq 0\) and \(\zeta >0\) if \(\eta =0\).

Hence the space \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }\) is premodular (sss). To explain that \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }\) is a premodular Banach (sss). Let \(x^{i}=(x_{k}^{i})_{k=0}^{\infty }\) be a Cauchy sequence in \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }\). Then for each \(\varepsilon \in (0, 1)\), there is \(i_{0}\in \mathbb{N}\) such that for all \(i,j\geq i_{0}\), we have

$$ \begin{aligned} \tau \bigl(x^{i}-x^{j}\bigr)= & \sum^{\infty }_{k=0} \bigl\vert \Delta _{n+1}^{m} \bigl\vert x^{i}_{k}-x^{j}_{k} \bigr\vert \bigr\vert ^{p_{k}} < \varepsilon ^{\sup _{k}p_{k}}. \end{aligned} $$

Hence, for \(i,j\geq i_{0}\) and \(k\in \mathbb{N}\), we conclude

$$ \begin{aligned} \bigl\vert \Delta _{n+1}^{m} \bigr\vert x_{k}^{i} \bigl\vert -\Delta _{n+1}^{m} \bigl\vert x_{k}^{j} \bigr\vert \bigr\vert < \varepsilon . \end{aligned} $$

Therefore \((\Delta _{n+1}^{m}|x_{k}^{j}| )\) is a Cauchy sequence in \(\mathbb{C}\) for fixed \(k\in \mathbb{N}\), so \(\lim_{j\rightarrow \infty }\Delta _{n+1}^{m}x_{k}^{j}=\Delta _{n+1}^{m}x_{k}^{0}\) for fixed \(k\in \mathbb{N}\). Hence \(\tau (x^{i}-x^{0})<\varepsilon ^{\sup _{i}p_{i}}\) for all \(i\geq i_{0}\). Finally, to show that \(x^{0}\in \ell (p, \Delta _{n+1}^{m})\), we have

$$ \begin{aligned} \tau \bigl(x^{0}\bigr)=\tau \bigl(x^{0}-x^{n}+x^{n}\bigr)\leq H\bigl( \tau \bigl(x^{n}-x^{0}\bigr)+ \tau \bigl(x^{n} \bigr)\bigr)< \infty . \end{aligned} $$

Therefore \(x^{0}\in \ell (p, \Delta _{n+1}^{m})\). This gives that \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }\) is a premodular Banach (sss). □

In view of Theorem 2.8, we get the following theorem.

Theorem 3.6

If \((p_{i})\in \mathbb{R^{+}}^{\mathbb{N}}\cap \ell _{\infty }\) is increasing and \(\Delta _{n+1}^{m}\) is absolutely nondecreasing, then the space \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }\) is prequasi-Banach (sss), where

$$ \tau (x)= \sum^{\infty }_{i=0} \bigl\vert \Delta _{n+1}^{m} \vert x_{i} \vert \bigr\vert ^{p_{i}} \quad \textit{for all } x\in \ell \bigl(p, \Delta _{n+1}^{m}\bigr). $$

Corollary 3.7

If \(0< p<\infty \) and \(\Delta _{n+1}^{m}\) is absolutely nondecreasing, then \((\ell _{p}(\Delta _{n+1}^{m}) )_{\tau }\) is a premodular Banach (sss), where \(\tau (x)=\sum^{\infty }_{i=0} |\Delta _{n+1}^{m}|x_{i}| |^{p}\) for all \(x\in \ell _{p}(\Delta _{n+1}^{m})\).

Prequasi-Banach closed ideal

Pietsch [12] examined the Banach quasi-ideals \((\ell _{r})^{\mathrm{app}}\) for \(r\in (0, \infty )\) and the Banach quasi-ideals of Hilbert–Schmidt and nuclear operators between Hilbert spaces formed by \(\ell _{2}\) and \(\ell _{1}\), respectively. Yaying et al. [15] made current the Banach quasi-operator ideal of type sequence space whose q-Cesáro matrix is in \(\ell _{p}\) for all \(q\in (0,1]\) and \(1< p<\infty \). Bakery and Mohammed [2] introduced the concept of prequasi-ideal, which is more general than the class of quasi-ideals. In this section, we introduce sufficient conditions on \(\ell (p, \Delta _{n+1}^{m})\) such that the class \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{\mathcal{S}}\) is a prequasi-Banach and closed ideal.

Theorem 4.1

If \((p_{r})\in \mathbb{R^{+}}^{\mathbb{N}}\cap \ell _{\infty }\) is increasing and \(\Delta _{n+1}^{m}\) is absolutely nondecreasing, then \(( (\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{\mathcal{S}}, \zeta )\) is a prequasi-Banach operator ideal with \(\tau (w)=\sum^{\infty }_{i=0} |\Delta _{n+1}^{m}|w_{i}| |^{p_{i}}\) for all \(w\in \ell (p, \Delta _{n+1}^{m})\) and \(\zeta (V)=\tau ((s_{n}(V))_{n=0}^{\infty } )\).

Proof

By Theorems 3.5 and 2.11 the function ζ is a prequasi-norm on \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{\mathcal{S}}\). Let \((V_{j})\) be a Cauchy sequence in \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{\mathcal{S}}(W, Z)\). Since \(\mathfrak{B}(W, Z)\supseteq (\ell (p, \Delta _{n+1}^{m}) )_{ \tau }^{\mathcal{S}}(W, Z)\), we have

$$ \zeta (V_{i}-V_{j})= \sum ^{\infty }_{k=0} \bigl\vert \Delta _{n+1}^{m}s_{k}(V_{i}-V_{j}) \bigr\vert ^{p_{k}} \geq \bigl\vert \Delta _{n+1}^{m} \Vert V_{i}-V_{j} \Vert ) \bigr\vert ^{p_{0}}. $$

Therefore \((V_{j})_{j\in \mathbb{N}}\) is a Cauchy sequence in \(\mathfrak{B}(W, Z)\). Since \(\mathfrak{B}(W, Z)\) is a Banach space, \(T\in \mathfrak{B}(W, Z)\) with \(\lim_{j\rightarrow \infty } \|V_{j}-V\|=0\) and \((s_{n}(V_{i}))_{n=0}^{\infty }\in (\ell (p, \Delta _{n+1}^{m}) )_{\tau }\) for each \(i\in \mathbb{N}\). From parts (ii), (iii), and (iv) of Definition 2.7 we have

$$\begin{aligned} \zeta (V) =& \sum^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m}s_{r}(V-V_{j}+V_{j}) \bigr\vert ^{p_{r}} \\ \leq& H \Biggl( \sum^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m}s_{[ \frac{r}{2}]}(V-V_{j}) \bigr\vert ^{p_{r}}+ \sum^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m}s_{[\frac{r}{2}]}(V_{j}) \bigr\vert ^{p_{r}} \Biggr) \\ \leq& H \sum^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m} \Vert V-V_{j} \Vert \bigr\vert ^{p_{0}}+H b_{0} \sum ^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m}s_{r}(V_{j}) \bigr\vert ^{p_{r}}< \varepsilon . \end{aligned}$$

Therefore \((s_{r}(V))_{r=0}^{\infty }\in (\ell (p, \Delta _{n+1}^{m}) )_{ \tau }\). Hence \(V\in (\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{\mathcal{S}}(W, Z)\). □

Theorem 4.2

If \((p_{r})\in \mathbb{R^{+}}^{\mathbb{N}}\cap \ell _{\infty }\) is increasing and \(\Delta _{n+1}^{m}\) is absolutely nondecreasing, then \(( (\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{\mathcal{S}}, \zeta )\) is a prequasi-closed operator ideal with \(\tau (w)=\sum^{\infty }_{i=0} |\Delta _{n+1}^{m}|w_{i}| |^{p_{i}}\) for all \(w\in \ell (p, \Delta _{n+1}^{m})\) and \(\zeta (V)=\tau ((s_{n}(V))_{n=0}^{\infty } )\).

Proof

By Theorems 3.5 and 2.11 the function ζ is a prequasi-norm on \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{\mathcal{S}}\). Assume that \(V_{j}\in (\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{\mathcal{S}}(W, Z)\) for all \(j\in \mathbb{N}\) and \(\lim_{j\rightarrow \infty }\zeta (V_{j}-V)=0\). Since \(\mathfrak{B}(W, Z)\supseteq (\ell (p, \Delta _{n+1}^{m}) )_{ \tau }^{\mathcal{S}}(W, Z)\), we have

$$ \zeta (V-V_{j})= \sum^{\infty }_{k=0} \bigl\vert \Delta _{n+1}^{m}s_{k}(V-V_{j}) \bigr\vert ^{p_{k}} \geq \bigl\vert \Delta _{n+1}^{m} \Vert V-V_{j} \Vert \bigr\vert ^{p_{0}}. $$

Hence \((V_{j})_{j\in \mathbb{N}}\) is a convergent sequence in \(\mathfrak{B}(W, Z)\). Since \((s_{n}(V_{j}))_{n=0}^{\infty }\in (\ell (p, \Delta _{n+1}^{m} )_{\tau }\) for each \(j\in \mathbb{N}\), from parts (ii), (iii), and (iv) of Definition 2.7 we get

$$\begin{aligned} \zeta (V) =& \sum^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m}s_{r}(V-V_{j}+V_{j}) \bigr\vert ^{p_{r}} \\ \leq& H \Biggl( \sum^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m}s_{[ \frac{r}{2}]}(V-V_{j}) \bigr\vert ^{p_{r}}+ \sum^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m}s_{[\frac{r}{2}]}(V_{j}) \bigr\vert ^{p_{r}} \Biggr) \\ \leq &H \sum^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m} \Vert V-V_{j} \Vert \bigr\vert ^{p_{0}}+H b_{0} \sum ^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m}s_{r}(V_{j}) \bigr\vert ^{p_{r}}< \varepsilon . \end{aligned}$$

Therefore \((s_{r}(V))_{r=0}^{\infty }\in (\ell (p, \Delta _{n+1}^{m} )_{ \tau }\). This gives \(V\in (\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{\mathcal{S}}(W, Z)\). □

Corollary 4.3

\(( (\ell _{p}(\Delta _{n+1}^{m}) )^{\mathcal{S}}_{\tau }, \zeta )\) is prequasi-closed and Banach with \(\tau (w)=\sum^{\infty }_{i=0} |\Delta _{n+1}^{m}|w_{i}| |^{p}\) for all \(w\in \ell _{p}(\Delta _{n+1}^{m})\) and \(\zeta (V)=\tau ((s_{n}(V))_{n=0}^{\infty } )\) if \(0< p<\infty \) and \(\Delta _{n+1}^{m}\) is absolutely nondecreasing.

Small and simple of \((\ell (p, \Delta _{n+1}^{m}))^{\mathcal{S}}\)

Makarov and Faried [7] explained the strict inclusion \((\ell _{r})^{\mathrm{app}}(W, Z)\varsubsetneqq (\ell _{j})^{\mathrm{app}}(W, Z) \subsetneqq \mathfrak{B}(W, Z)\) for \(j>r>0\). Pietsch [11] proved that the class \((\ell _{r})^{\mathrm{app}}\) became simple and small Banach space for \(r\in [1, \infty )\) and \(r\in (0, \infty )\), respectively. In this section, we explain sufficient conditions on \(\ell (p, \Delta _{n+1}^{m})\) for the strict inclusion relation of \((\ell (p, \Delta _{n+1}^{m}) )^{\mathcal{S}}\) for different p and \(\Delta _{n+1}^{m}\). We study the conditions such that the class \((\ell (p, \Delta _{n+1}^{m}) )^{\mathrm{app}}\) is small. We also investigate sufficient conditions on \(\ell (p, \Delta _{n+1}^{m})\) such that \((\ell (p, \Delta _{n+1}^{m}) )^{\mathcal{S}}\) equals \((\ell (p, \Delta _{n+1}^{m}) )^{\nu }\). Finally, we give an answer of the following question: For which \(\ell (p, \Delta _{n+1}^{m})\), \((\ell (p, \Delta _{n+1}^{m}) )^{\mathcal{S}}\) is simple?

Theorem 5.1

Let W and Z be infinite-dimensional Banach spaces, \(0< p_{i}\leq q_{i}\) for all \(i\in \mathbb{N}\), and let \(\Delta _{n}^{m}\) be absolutely nondecreasing for all \(n,m\in \mathbb{N}\). Then

$$ \bigl(\ell \bigl(p, \Delta _{n+2}^{m}\bigr) \bigr)^{\mathcal{S}}(W, Z) \varsubsetneqq \bigl(\ell \bigl(q, \Delta _{n+1}^{m+1}\bigr) \bigr)^{\mathcal{S}}(W, Z)\subsetneqq \mathfrak{B}(W, Z). $$

Proof

If \(V\in (\ell (p, \Delta _{n+2}^{m}) )^{\mathcal{S}}(W, Z)\), then we have \((s_{i}(V))\in \ell (p, \Delta _{n+2}^{m})\). We can see that

$$ \sum^{\infty }_{j=0} \bigl\vert \Delta _{n+1}^{m+1}s_{j}(V) \bigr\vert ^{q_{j}}< \sum^{\infty }_{j=0} \bigl\vert \Delta _{n+2}^{m}s_{j}(V) \bigr\vert ^{p_{j}}< \infty . $$

Therefore \(V\in (\ell (q, \Delta _{n+1}^{m+1}) )^{\mathcal{S}}(W, Z)\). Next, if we choose \((s_{j}(V))_{j=0}^{\infty }\) such that \(\Delta _{n+2}^{m}s_{j}(V)= (j+1 )^{-\frac{1}{p_{j}}}\) for \(n,m\in \mathbb{N}\), then we can find \(V\in \mathfrak{B}(W, Z)\) with \(\sum_{j=0}^{\infty }|\Delta _{n+2}^{m}s_{j}(V)|^{p_{j}}=\sum_{j=0}^{ \infty }\frac{1}{j+1}=\infty \) and

$$ \sum^{\infty }_{j=0} \bigl( \bigl\vert \Delta _{n+2}^{m}s_{j}(V) \bigr\vert \bigr)^{q_{j}}=\sum_{j=0}^{\infty } \biggl(\frac{1}{j+1} \biggr)^{{ \frac{q_{j}}{p_{j}}}}< \infty . $$

Since \(\ell (q, \Delta _{n+2}^{m})\subseteq \ell (q, \Delta _{n+1}^{m+1})\), \(V\notin (\ell (p, \Delta _{n+2}^{m}) )^{\mathcal{S}}(W, Z)\) and \(V\in (\ell (q, \Delta _{n+1}^{m+1}) )^{\mathcal{S}}(W, Z)\). Clearly, \((\ell (q, \Delta _{n+1}^{m+1}) )^{\mathcal{S}}(W, Z)\subset \mathfrak{B}(W, Z)\). By choosing \((s_{j}(V))_{j=0}^{\infty }\) such that \(\Delta _{n+1}^{m+1}s_{j}(V)= (j+1 )^{-\frac{1}{q_{j}}}\) for \(n,m\in \mathbb{N}\), we have \(V\in \mathfrak{B}(W, Z)\) such that \(V\notin (\ell (q, \Delta _{n+1}^{m+1}) )^{\mathcal{S}}(W, Z)\). □

Corollary 5.2

For any infinite-dimensional Banach spaces W and Z, \(j\geq r>0\), and absolutely nondecreasing \(\Delta _{n}^{m}\) for all \(n,m\in \mathbb{N}\), we have

$$ \bigl(\ell _{r}\bigl(\Delta _{n+2}^{m}\bigr) \bigr)^{\mathcal{S}}(W, Z) \varsubsetneqq \bigl(\ell _{j}\bigl( \Delta _{n+1}^{m+1}\bigr) \bigr)^{\mathcal{S}}(W, Z) \subsetneqq \mathfrak{B}(W, Z). $$

Theorem 5.3

For any Banach spaces W and Z with \(\dim (W)=\dim (Z)=\infty \), let \((p_{r})\in \mathbb{R^{+}}^{\mathbb{N}}\cap \ell _{\infty }\) be increasing, and let \(\Delta _{n+1}^{m}\) be absolutely nondecreasing. Then the class \((\ell (p, \Delta _{n+1}^{m}) )^{\mathrm{app}}\) is small.

Proof

\(( (\ell (p, \Delta _{n+1}^{m}) )^{\mathrm{app}}, \zeta )\) is a prequasi-Banach operator ideal, where \(\zeta (V)= (\sum^{\infty }_{k=0} \vert \Delta _{n+1}^{m} \alpha _{k}(V) \vert ^{p_{k}} )^{\frac{1}{h}}\). Let \((\ell (p, \Delta _{n+1}^{m}) )^{\mathrm{app}}(W, Z)=\mathfrak{B}(W, Z)\). Then there is \(\delta >0\) with \(\zeta (V)\leq \delta \|V\|\) for all \(V\in \mathfrak{B}(W, Z)\). By Dvoretzky’s theorem [12] for \(j\in \mathbb{N}\), there are subspaces \(M_{j}\) and quotient spaces \(W/N_{j}\) of Z. By isomorphisms, \(A_{j}\) and \(H_{j}\) will be mapped Z onto \(\ell _{2}^{j}\) with \(\|H_{j}\|\|H_{j}^{-1}\|\leq 2\) and \(\|A_{j}\|\|A_{j}^{-1}\|\leq 2\). Let \(J_{j}\) be the natural embedding map from \(M_{j}\) into Z, and let \(Q_{j}\) be the quotient map from W onto \(W/N_{j}\). Denoting the Bernstein numbers [12] by \(u_{j}\), we have

$$\begin{aligned} 1 =&u_{k}(I_{j})=u_{k} \bigl(A_{j}A_{j}^{-1}I_{j}H_{j}H_{j}^{-1} \bigr) \\ \leq& \Vert A_{j} \Vert u_{k} \bigl(A_{j}^{-1}I_{j}H_{j} \bigr) \bigl\Vert H_{j}^{-1} \bigr\Vert \\ =& \Vert A_{j} \Vert u_{k}\bigl(J_{j}A_{j}^{-1}I_{j}H_{j} \bigr) \bigl\Vert H_{j}^{-1} \bigr\Vert \\ \leq& \Vert A_{j} \Vert d_{k} \bigl(J_{j}A_{j}^{-1}I_{j}H_{j} \bigr) \bigl\Vert H_{j}^{-1} \bigr\Vert \\ =& \Vert A_{j} \Vert d_{k}\bigl(J_{j}A_{j}^{-1}I_{j}H_{j}Q_{j} \bigr) \bigl\Vert H_{j}^{-1} \bigr\Vert \\ \leq& \Vert A_{j} \Vert \alpha _{k}\bigl(J_{j}A_{j}^{-1}I_{j}H_{j}Q_{j} \bigr) \bigl\Vert H_{j}^{-1} \bigr\Vert \end{aligned}$$

for \(0\leq k\leq i\). Therefore

$$\begin{aligned}& 1\leq \Vert A_{j} \Vert \bigl\vert \Delta _{n+1}^{m}\alpha _{k} \bigl(J_{j}A_{j}^{-1}I_{j}H_{j}Q_{j} \bigr) \bigr\vert \bigl\Vert H_{j}^{-1} \bigr\Vert \\& \quad \Rightarrow\quad (i+1)\leq \bigl( \Vert A_{j} \Vert \bigl\Vert H_{j}^{-1} \bigr\Vert \bigr)^{p_{i}} \sum _{k=0}^{i} \bigl\vert \Delta _{n+1}^{m}\alpha _{k}\bigl(J_{j}A_{j}^{-1}I_{j}H_{j}Q_{j} \bigr) \bigr\vert ^{p_{k}}. \end{aligned}$$

Hence

$$\begin{aligned}& (i+1 )^{\frac{1}{h}}\leq a \Vert A_{m} \Vert \bigl\Vert H_{m}^{-1} \bigr\Vert \Biggl[\sum_{k=0}^{i} \bigl\vert \Delta _{n+1}^{m}\alpha _{k} \bigl(J_{j}A_{j}^{-1}I_{j}H_{j}Q_{j} \bigr) \bigr\vert ^{p_{k}}\Biggr]^{\frac{1}{h}} \\& \quad \Rightarrow \quad (i+1 )^{\frac{1}{h}}\leq a \Vert A_{j} \Vert \bigl\Vert H_{j}^{-1} \bigr\Vert g\bigl(J_{j}A_{j}^{-1}I_{j}H_{j}Q_{j} \bigr) \\& \quad \Rightarrow\quad (i+1 )^{\frac{1}{h}}\leq a\delta \Vert A_{j} \Vert \bigl\Vert H_{j}^{-1} \bigr\Vert \bigl\Vert J_{j}A_{j}^{-1}I_{j}H_{j}Q_{j} \bigr\Vert \\& \quad \Rightarrow \quad (i+1 )^{\frac{1}{h}}\leq a\delta \Vert A_{j} \Vert \bigl\Vert H_{j}^{-1} \bigr\Vert \bigl\Vert J_{j}A_{j}^{-1} \bigr\Vert \Vert I_{j} \Vert \Vert H_{j}Q_{j} \Vert =L\delta \Vert A_{j} \Vert \bigl\Vert H_{j}^{-1} \bigr\Vert \bigl\Vert A_{j}^{-1} \bigr\Vert \Vert I_{j} \Vert \Vert H_{j} \Vert \\& \quad \Rightarrow\quad (i+1 )^{\frac{1}{h}}\leq 4a\delta \end{aligned}$$

for some \(a\geq 1\). Since i is arbitrary, we have a contradiction. So, W and Z cannot be infinite-dimensional while \((\ell (p, \Delta _{n+1}^{m}) )^{\mathrm{app}}(W, Z)=\mathfrak{B}(W, Z)\).

In the same manner we can prove that the class \((\ell (p, \Delta _{n+1}^{m}) )^{\mathrm{Kol}}\) is small. □

Theorem 5.4

Let W and Z be any Banach spaces with \(\dim (W)=\dim (Z)=\infty \). Let \((p_{r})\in \mathbb{R^{+}}^{\mathbb{N}}\cap \ell _{\infty }\) be increasing, and let \(\Delta _{n+1}^{m}\) be absolutely nondecreasing. Then the class \((\ell (p, \Delta _{n+1}^{m}) )^{\mathrm{Kol}}\) is small.

Theorem 5.5

Pick any Banach spaces W and Z with \(\dim (W)=\dim (Z)=\infty \). If \((p_{r}),(q_{r})\in \ell _{\infty }\) are increasing with \(1\leq p_{i}< q_{i}\) for all \(i\in \mathbb{N}\) and \(\Delta _{n}^{m}\) is absolutely nondecreasing, then

$$ \mathfrak{B} \bigl( \bigl(\ell \bigl(q, \Delta _{n+1}^{m+1} \bigr) \bigr)^{\mathcal{S}}, \bigl(\ell \bigl(p, \Delta _{n+2}^{m} \bigr) \bigr)^{\mathcal{S}} \bigr)= \Upsilon \bigl( \bigl(\ell \bigl(q, \Delta _{n+1}^{m+1}\bigr) \bigr)^{\mathcal{S}}, \bigl( \ell \bigl(p, \Delta _{n+2}^{m}\bigr) \bigr)^{\mathcal{S}} \bigr). $$

Proof

Assume that there is \(V\in \mathfrak{B} ( (\ell (q, \Delta _{n+1}^{m+1}) )^{ \mathcal{S}}, (\ell (p, \Delta _{n+2}^{m}) )^{\mathcal{S}} )\) that is not approximable. By Lemma 2.2 we have \(G\in \mathfrak{B} ( (\ell (q, \Delta _{n+1}^{m+1}) )^{ \mathcal{S}} )\) and \(B\in \mathfrak{B} ( (\ell (p, \Delta _{n+2}^{m}) )^{ \mathcal{S}} )\) with \(\mathit{BVG} I_{k}=I_{k}\). Therefore for all \(k\in \mathbb{N}\), we get

$$ \begin{aligned} \Vert I_{k} \Vert _{ (\ell (p, \Delta _{n+2}^{m}) )^{ \mathcal{S}}}&= \sum_{n=0}^{\infty } \bigl\vert \Delta _{n+2}^{m}s_{n}(I_{k}) \bigr\vert ^{p_{k}} \leq \Vert \mathit{BVG} \Vert \Vert I_{k} \Vert _{ (\ell (q, \Delta _{n+1}^{m+1}) )^{\mathcal{S}}}\leq \sum _{n=0}^{\infty } \bigl\vert \Delta _{n+1}^{m+1}s_{n}(I_{k}) \bigr\vert ^{q_{k}}. \end{aligned} $$

From Theorem 5.1 we obtain a contradiction. Hence \(V\in \Upsilon ( (\ell (q, \Delta _{n+1}^{m+1}) )^{ \mathcal{S}}, (\ell (p, \Delta _{n+2}^{m}) )^{\mathcal{S}} )\). □

Corollary 5.6

Let W and Z be any Banach spaces with \(\dim (W)=\dim (Z)=\infty \). If \((p_{r}),(q_{r})\in \ell _{\infty }\) are increasing with \(1\leq p_{i}< q_{i}\) for all \(i\in \mathbb{N}\) and \(\Delta _{n}^{m}\) is absolutely nondecreasing, then

$$ \mathfrak{B} \bigl( \bigl(\ell \bigl(q, \Delta _{n+1}^{m+1} \bigr) \bigr)^{\mathcal{S}}, \bigl(\ell \bigl(p, \Delta _{n+2}^{m} \bigr) \bigr)^{\mathcal{S}} \bigr)=\mathfrak{B}_{c} \bigl( \bigl( \ell \bigl(q, \Delta _{n+1}^{m+1}\bigr) \bigr)^{\mathcal{S}}, \bigl( \ell \bigl(p, \Delta _{n+2}^{m} \bigr) \bigr)^{\mathcal{S}} \bigr). $$

Proof

Since each approximable operator is compact, the result follows. □

Theorem 5.7

Let W and Z be any Banach spaces with \(\dim (W)=\dim (Z)=\infty \). If \((p_{r})\in \ell _{\infty }\) is increasing with \(p_{0}\geq 1\) for all \(i\in \mathbb{N}\) and \(\Delta _{n}^{m}\) is absolutely nondecreasing, then the class \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{\mathcal{S}}\) is simple.

Proof

Suppose that there is \(V\in \mathfrak{B}_{c} ( (\ell (p, \Delta _{n+1}^{m}) )_{ \tau }^{\mathcal{S}} )\) such that \(V\notin \Upsilon ( (\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{ \mathcal{S}} )\). Therefore by Lemma 2.2 one find \(A, B\in \mathfrak{B} ( (\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{ \mathcal{S}} )\) with \(BVAI_{k}=I_{k}\). This means that \(I_{ (\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{\mathcal{S}}}\in \mathfrak{B}_{c} ( (\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{ \mathcal{S}} )\). Consequently, \(\mathfrak{B} ( (\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{ \mathcal{S}} )=\mathfrak{B}_{c} ( (\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{\mathcal{S}} )\). Therefore \(\mathfrak{B} ( (\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{ \mathcal{S}} )\) includes one and only one nontrivial closed ideal \(\Upsilon ( (\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{ \mathcal{S}} )\). □

Eigenvalues of s-type \(\ell (p, \Delta _{n+1}^{m})\)

Theorem 5.8

Let W and Z be Banach spaces with \(\dim (W)=\dim (Z)=\infty \). If \((p_{r})\in \mathbb{R^{+}}^{\mathbb{N}}\cap \ell _{\infty }\) is increasing and \(\Delta _{n+1}^{m}\) is absolutely nondecreasing, then

$$ \bigl(\ell \bigl(p, \Delta _{n+1}^{m}\bigr) \bigr)^{\mathcal{S}}(W, Z)= \bigl(\ell \bigl(p, \Delta _{n+1}^{m} \bigr) \bigr)^{\nu }(W, Z). $$

Proof

Suppose \(V\in (\ell (p, \Delta _{n+1}^{m}) )^{\mathcal{S}}(W, Z)\). Then \((s_{r}(V))_{r=0}^{\infty }\in \ell (p, \Delta _{n+1}^{m})\), and we have \(\sum^{\infty }_{r=0} ( \vert \Delta _{n+1}^{m}s_{r}(V) \vert )^{p_{r}}<\infty \). Since \(\Delta _{n+1}^{m}\) is continuous, \(\lim_{r\rightarrow \infty }s_{r}(V)=0\). Let \(\|V-s_{r}(V)I\|\) be invertible for all \(r\in \mathbb{N}\). Then \(\|V-s_{r}(V)I\|^{-1}\) exists and is bounded for each \(r\in \mathbb{N}\). Therefore \(\lim_{r\rightarrow \infty }\|V-s_{r}(V)I\|^{-1}=\|V\|^{-1}\) with \(V^{-1}\in \mathfrak{B}(Z, W)\). From the prequasi-operator ideal of \(( (\ell (p, \Delta _{n+1}^{m}) )^{\mathcal{S}},\zeta )\) we have

$$ I=VV^{-1}\in \bigl(\ell \bigl(p, \Delta _{n+1}^{m} \bigr) \bigr)^{\mathcal{S}}(Z) \quad \Rightarrow\quad \bigl(s_{r}(I) \bigr)_{r=0}^{\infty }\in \ell \bigl(p, \Delta _{n+1}^{m}\bigr) \quad \Rightarrow\quad \lim_{r\rightarrow \infty }s_{r}(I)=0. $$

Since \(\lim_{r\rightarrow \infty }s_{r}(I)=1\), we have a contradiction. Then \(\|V-s_{r}(V)I\|\) is not invertible for all \(r\in \mathbb{N}\). Hence \((s_{r}(V))_{r=0}^{\infty }\) represents the eigenvalues of V. Conversely, if \(V\in (\ell (p, \Delta _{n+1}^{m}) )^{\nu }(W, Z)\), then \((\nu _{r}(V))_{r=0}^{\infty }\in \ell (p, \Delta _{n+1}^{m})\) and \(\|V-\nu _{r}(V)I\|=0\) for all \(n\in \mathbb{N}\). This gives \(V=\nu _{r}(V)I\) for all \(r\in \mathbb{N}\). Then \(s_{r}(V)=s_{r}(\nu _{r}(V)I)=|\nu _{r}(V)|\) for all \(r\in \mathbb{N}\). Therefore \((s_{r}(V))_{r=0}^{\infty }\in \ell (p, \Delta _{n+1}^{m})\), and so \(V\in (\ell (p, \Delta _{n+1}^{m}) )^{\mathcal{S}}(W, Z)\). This completes the proof. □

Multiplication operator on \(\ell (p, \Delta _{n+1}^{m})\)

Mursaleen and Noman [10] examined compact operators on some difference sequence spaces. Kiliçman and Raj [5] introduced the matrix transformations of Norlund–Orlicz difference sequence spaces of nonabsolute type. Yaying et al. [15] investigated the matrix transformations on q-Cesáro sequence spaces of nonabsolute type. In this section, we introduce some topological and geometric structures of the multiplication operator acting on \(\ell (p, \Delta _{n+1}^{m})\) such as bounded, invertible, approximable, closed range, and Fredholm operator.

Definition 6.1

Let \(\kappa \in \mathbb{C}^{\mathbb{N}}\cap \ell _{\infty }\), and let \(W_{\tau }\) be a prequasi-normed (sss). An operator \(V_{\kappa }:W_{\tau }\rightarrow W_{\tau }\) is called a multiplication operator if \(V_{\kappa }w=\kappa w= (\kappa _{r}w_{r} )_{r=0}^{\infty }\in W\) for all \(w\in W\). If \(V_{\kappa }\in \mathfrak{B}(W)\), then we call it a multiplication operator generated by κ.

Theorem 6.2

Let \(\kappa \in \mathbb{C}^{\mathbb{N}}\), \((p_{r})\in \mathbb{R^{+}}^{\mathbb{N}}\cap \ell _{\infty }\) be increasing, and let \(\Delta _{n+1}^{m}\) be absolutely nondecreasing. Then \(\kappa \in \ell _{\infty }\) if and only if, \(V_{\kappa }\in \mathfrak{B}(\ell (p, \Delta _{n+1}^{m})_{\tau })\), where \(\tau (x)=\sum^{\infty }_{r=0} |\Delta _{n+1}^{m}|x_{r}| |^{p_{r}}\) for all \(x\in \ell (p, \Delta _{n+1}^{m})\).

Proof

Let \(\kappa \in \ell _{\infty }\). Then there is \(\varepsilon >0\) with \(|\kappa _{r}|\leq \varepsilon \) for every \(r\in \mathbb{N}\). For \(x\in (\ell (p, \Delta _{n+1}^{m})_{\tau }\), since \(\Delta _{n+1}^{m}\) is absolutely nondecreasing and \((p_{r})\) is bounded from above with \(p_{r}>0\) for all \(r\in \mathbb{N}\), we have

$$\begin{aligned} \tau (V_{\kappa }x) =&\tau (\kappa x)=\tau \bigl( ( \kappa _{r}x_{r} )_{r=0}^{\infty } \bigr) \\ =&\sum^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m}\bigl( \vert \kappa _{r} \vert \vert x_{r} \vert \bigr) \bigr\vert ^{p_{r}} \\ \leq& \sum ^{ \infty }_{r=0} \bigl\vert \Delta _{n+1}^{m}\bigl(\varepsilon \vert x_{r} \vert \bigr) \bigr\vert ^{p_{r}} \\ \leq& \sup_{r} \varepsilon ^{p_{r}} \sum^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m} \vert x_{r} \vert \bigr\vert ^{p_{r}} \\ =&D \tau (x). \end{aligned}$$

This gives \(V_{\kappa }\in \mathfrak{B}(\ell (p, \Delta _{n+1}^{m})_{\tau })\). Conversely, let \(V_{\kappa }\in \mathfrak{B}(\ell (p, \Delta _{n+1}^{m})_{\tau })\). Suppose \(\kappa \notin \ell _{\infty }\). Then for each \(j\in \mathbb{N}\), there is \(i_{j}\in \mathbb{N}\) such that \(\kappa _{i_{j}}> j\). Since \(\Delta _{n+1}^{m}\) is absolutely nondecreasing, we have

$$\begin{aligned} \tau (V_{\kappa }e_{i_{j}}) =&\tau ( \kappa e_{i_{j}})= \tau \bigl( \bigl(\kappa _{r}(e_{i_{j}})_{r} \bigr)_{r=0}^{\infty }\bigr) \\ =&\sum^{ \infty }_{r=0} \bigl\vert \Delta _{n+1}^{m}\bigl( \vert \kappa _{r} \vert \bigl\vert (e_{i_{j}})_{r} \bigr\vert \bigr) \bigr\vert ^{p_{r}} \\ =& \bigl\vert \Delta _{n+1}^{m} \vert \kappa _{i_{j}} \vert \bigr\vert ^{p_{i_{j}}}> \bigl\vert \Delta _{n+1}^{m} \vert j \vert \bigr\vert ^{p_{i_{j}}} \\ = &\bigl\vert \Delta _{n+1}^{m} \vert j \vert \bigr\vert ^{p_{i_{j}}}\tau (e_{i_{j}}). \end{aligned}$$

This shows that \(V_{\kappa }\notin \mathfrak{B}(\ell (p, \Delta _{n+1}^{m})_{\tau })\). Therefore \(\kappa \in \ell _{\infty }\). □

Theorem 6.3

Let \(\kappa \in \mathbb{C}^{\mathbb{N}}\), and let \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }\) be a prequasi-normed (sss) with \(\tau (x)=\sum^{\infty }_{r=0} |\Delta _{n+1}^{m}|x_{r}| |^{p_{r}}\) for all \(x\in \ell (p, \Delta _{n+1}^{m})\). Then \(|\kappa _{r}|=1\) for all \(r\in \mathbb{N}\) if and only if \(V_{\kappa }\) is an isometry.

Proof

Suppose \(|\kappa _{r}|=1\) for all \(r\in \mathbb{N}\). Then

$$\begin{aligned} \tau (V_{\kappa }x) =&\tau (\kappa x)=\tau \bigl( (\kappa _{r} x_{r} )_{r=0}^{ \infty }\bigr) \\ =&\sum ^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m}\bigl( \vert \kappa _{r} \vert \vert x_{r} \vert \bigr) \bigr\vert ^{p_{r}}=\sum ^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m} \vert x_{r} \vert \bigr\vert ^{p_{r}}= \tau (x) \end{aligned}$$

for all \(x\in (\ell (p, \Delta _{n+1}^{m}))_{\tau }\). Therefore \(V_{\kappa }\) is an isometry. Conversely, assume that \(|\kappa _{i}|<1\) for some \(i = i_{0}\). Since \(\Delta _{n+1}^{m}\) is absolutely nondecreasing, we obtain

$$\begin{aligned} \tau (V_{\kappa }e_{i_{0}}) =&\tau (\kappa e_{i_{0}})= \tau \bigl( \bigl(\kappa _{r} (e_{i_{0}})_{r} \bigr)_{r=0}^{\infty }\bigr) \\ =& \sum^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m}\bigl( \vert \kappa _{r} \vert \bigl\vert (e_{i_{0}})_{r} \bigr\vert \bigr) \bigr\vert ^{p_{r}} \\ < &\sum ^{ \infty }_{r=0} \bigl\vert \Delta _{n+1}^{m} \bigl\vert (e_{i_{0}})_{r} \bigr\vert \bigr\vert ^{p_{r}}= \tau (e_{i_{0}}). \end{aligned}$$

When \(|\kappa _{i_{0}}|>1\), we can prove that \(\tau (V_{\kappa }e_{i_{0}})>\tau (e_{i_{0}})\). Therefore, in both cases, we have a contradiction. So \(|\kappa _{r}|=1\) for every \(r\in \mathbb{N}\).

By \(\operatorname{card} (A)\) we denote the cardinality of a set A. □

Theorem 6.4

If \(\kappa \in \mathbb{C}^{\mathbb{N}}\) and \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }\) is a prequasi-normed (sss), where \(\tau (x)=\sum^{\infty }_{r=0} |\Delta _{n+1}^{m}|x_{r}| |^{p_{r}}\) for all \(x\in \ell (p, \Delta _{n+1}^{m})\). Then \(V_{\kappa }\in \Upsilon ((\ell (p, \Delta _{n+1}^{m}))_{\tau } )\) if and only if \((\kappa _{r})_{r=0}^{\infty }\in c_{0}\).

Proof

Let \(V_{\kappa }\in \Upsilon ((\ell (p, \Delta _{n+1}^{m}))_{\tau } )\). Therefore \(V_{\kappa }\in \mathfrak{B}_{c} ((\ell (p, \Delta _{n+1}^{m}))_{ \tau } )\). To prove that the sequence \((\kappa _{r})_{r=0}^{\infty }\) belongs to \(c_{0}\), suppose \((\kappa _{r})_{r=0}^{\infty }\notin c_{0}\). Then there is \(\delta > 0\) such that the set \(A_{\delta }=\{r\in \mathbb{N}: |\kappa _{r}|\geq \delta \}\) has \(\operatorname{card} (A_{ \delta })=\infty \). Assume that \(a_{i}\in A_{\delta }\) for all \(i\in \mathbb{N}\). Hence \(\{e_{a_{i}}:a_{i}\in A_{\delta }\}\) is an infinite bounded set in \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }\). Let

$$\begin{aligned} \tau (V_{\kappa }e_{a_{i}}-V_{\kappa }e_{a_{j}}) =& \tau ( \kappa e_{a_{i}}-\kappa e_{a_{j}}) \\ =&\tau \bigl( \bigl( \kappa _{r}\bigl((e_{a_{i}})_{r}-(e_{a_{j}})_{r} \bigr) \bigr)_{r=0}^{\infty }\bigr) =\sum ^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m} \bigr\vert \kappa _{r} \bigl((e_{a_{i}})_{r}-(e_{a_{j}})_{r} \bigr) \big\vert \big\vert ^{p_{r}} \\ \geq& \sum^{\infty }_{r=0} \bigl\vert \Delta _{n+1}^{m} \bigr\vert \delta \bigl((e_{a_{i}})_{r}-(e_{a_{j}})_{r} \bigr) \big\vert \big\vert ^{p_{r}}=\tau (\delta e_{a_{i}}-\delta e_{a_{j}}) \end{aligned}$$

for all \(a_{i}, a_{j}\in A_{\delta }\). This shows that \(\{e_{a_{i}}: a_{i}\in B_{\delta }\}\in \ell _{\infty }\), which cannot have a convergent subsequence under \(V_{\kappa }\). This proves that \(V_{\kappa }\notin \mathfrak{B}_{c} ((\ell (p, \Delta _{n+1}^{m}))_{ \tau } )\). Then \(V_{\kappa }\notin \Upsilon ((\ell (p, \Delta _{n+1}^{m}))_{\tau } )\), a contradiction. So, \(\lim_{i\rightarrow \infty }\kappa _{i}=0\). Conversely, let \(\lim_{i\rightarrow \infty }\kappa _{i}=0\). Then for each \(\delta > 0\), the set \(A_{\delta }=\{i\in \mathbb{N}:|\kappa _{i}|\geq \delta \}\) has \(\operatorname{card} (A_{ \delta })<\infty \). Hence, for every \(\delta > 0\), the space

$$ \bigl(\bigl(\ell \bigl(p, \Delta _{n+1}^{m}\bigr) \bigr)_{\tau } \bigr)_{A_{\delta }}= \bigl\{ x=(x_{i}) \in \bigl(\ell \bigl(p, \Delta _{n+1}^{m}\bigr) \bigr)_{\tau }: i\in A_{\delta } \bigr\} $$

is finite-dimensional. Then \(V_{\kappa }| ((\ell (p, \Delta _{n+1}^{m}))_{\tau } )_{A_{\delta }}\) is a finite rank operator. For every \(i\in \mathbb{N}\), define \(\kappa _{i}\in \mathbb{C}^{\mathbb{N}}\) by

$$ (\kappa _{i})_{j} = \textstyle\begin{cases} \kappa _{j}, & j\in A_{\frac{1}{i}}, \\ 0 & \text{otherwise.} \end{cases} $$

It is clear that \(V_{\kappa _{i}}\) has \(\operatorname{rank} (V_{\kappa _{i}})<\infty \) as \(\dim ((\ell (p, \Delta _{n+1}^{m}))_{\tau } )_{A_{\frac{1}{i}}}< \infty \) for \(i\in \mathbb{N}\). Therefore, since \(\Delta _{n+1}^{m}\) is absolutely nondecreasing, we get

$$\begin{aligned} \tau \bigl((V_{\kappa }-V_{\kappa _{i}})x \bigr) =&\tau \bigl( \bigl(\bigl( \kappa _{j}-(\kappa _{i})_{j}\bigr)x_{j} \bigr)_{j=0}^{\infty } \bigr) \\ =&\sum^{ \infty }_{j=0} \bigl\vert \Delta _{n+1}^{m} \bigl( \bigl\vert \bigl(\kappa _{j}-(\kappa _{i})_{j} \bigr)x_{j} \bigr\vert \bigr) \bigr\vert ^{p_{j}} \\ =&\sum^{\infty }_{j=0, j\in A_{\frac{1}{i}}} \bigl\vert \Delta _{n+1}^{m} \bigl( \bigl\vert \bigl(\kappa _{j}-(\kappa _{i})_{j} \bigr)x_{j} \bigr\vert \bigr) \bigr\vert ^{p_{j}}+\sum ^{ \infty }_{j=0, j\notin A_{\frac{1}{i}}} \bigl\vert \Delta _{n+1}^{m} \bigl( \bigl\vert \bigl( \kappa _{j}-(\kappa _{i})_{j} \bigr)x_{j} \bigr\vert \bigr) \bigr\vert ^{p_{j}} \\ =&\sum^{\infty }_{j=0, j\notin A_{\frac{1}{i}}} \bigl\vert \Delta _{n+1}^{m} \vert \kappa _{j}x_{j} \vert \bigr\vert ^{p_{j}} \\ \leq& \frac{1}{i}\sum ^{\infty }_{j=0, j \notin A_{\frac{1}{i}}} \bigl\vert \Delta _{n+1}^{m} \vert x_{j} \vert \bigr\vert ^{p_{j}} < \frac{1}{i}\sum^{\infty }_{j=0} \bigl\vert \Delta _{n+1}^{m} \vert x_{j} \vert \bigr\vert ^{p_{j}}= \frac{1}{i}\tau (x). \end{aligned}$$

This implies that \(\|V_{\kappa }-V_{\kappa _{i}}\|\leq \frac{1}{i}\) and that \(V_{\kappa }\) is a limit of finite rank operators. Therefore \(V_{\kappa }\) is an approximable operator. □

Theorem 6.5

Let \(\kappa \in \mathbb{C}^{\mathbb{N}}\), and let \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }\) be a prequasi-normed (sss), where \(\tau (x)=\sum^{\infty }_{r=0} |\Delta _{n+1}^{m}|x_{r}| |^{p_{r}}\) for \(x\in \ell (p, \Delta _{n+1}^{m})\). Then \(V_{\kappa }\in \mathfrak{B}_{c} ((\ell (p, \Delta _{n+1}^{m}))_{ \tau } )\) if and only if \((\kappa _{i})_{i=0}^{\infty }\in c_{0}\).

Proof

It is simple and so overlooked. □

Corollary 6.6

If \(\kappa \in \mathbb{C}^{\mathbb{N}}\), \((p_{r})\in \mathbb{R^{+}}^{\mathbb{N}}\cap \ell _{\infty }\) is increasing, and \(\Delta _{n+1}^{m}\) is absolutely nondecreasing, then \(\mathfrak{B}_{c} ((\ell (p, \Delta _{n+1}^{m}))_{\tau } ) \varsubsetneqq \mathfrak{B} ((\ell (p, \Delta _{n+1}^{m}))_{\tau } )\), where \(\tau (x)=\sum^{\infty }_{r=0} |\Delta _{n+1}^{m}|x_{r}| |^{p_{r}}\) for all \(x\in \ell (p, \Delta _{n+1}^{m})\).

Proof

Since I is a multiplication operator on \((\ell (p, \Delta _{n+1}^{m}))_{\tau }\) generated by \(\kappa =(1, 1,\ldots )\), \(I\notin \mathfrak{B}_{c}((\ell (p, \Delta _{n+1}^{m}))_{\tau })\) and \(I\in \mathfrak{B}((\ell (p, \Delta _{n+1}^{m}))_{\tau })\). □

Theorem 6.7

If \(\kappa \in \mathbb{C}^{\mathbb{N}}\), then \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }\) is prequasi-Banach (sss), where \(\tau (x)= \sum^{\infty }_{r=0} |\Delta _{n+1}^{m}|x_{r}| |^{p_{r}}\) for all \(x\in \ell (p, \Delta _{n+1}^{m})\), and \(V_{\kappa }\in \mathfrak{B}((\ell (p, \Delta _{n+1}^{m}))_{\tau })\). Then κ is bounded away from zero on \((\ker (\kappa ) )^{c}\) if and only if \(R(V_{\kappa })\) is closed.

Proof

Let the sufficient condition be satisfied. Then there is \(\epsilon >0\) with \(|\kappa _{i}|\geq \epsilon \) for all \(i\in (\ker (\kappa ) )^{c}\). To show that \(R(V_{\kappa })\) is closed, let d be a limit point of \(R(V_{\kappa })\). Therefore there is \(V_{\kappa }x_{i}\) in \((\ell (p, \Delta _{n+1}^{m}))_{\tau }\) for all \(i\in \mathbb{N}\) such that \(\lim_{i\rightarrow \infty }V_{\kappa }x_{i}=d\). Obviously, \((V_{\kappa }x_{i})\) is a Cauchy sequence. Since \(\Delta _{n+1}^{m}\) is absolutely nondecreasing, we have

$$\begin{aligned}& \tau (V_{\kappa }x_{i}-V_{\kappa }x_{j}) \\& \quad = \sum_{r=0}^{ \infty } \bigl\vert \Delta _{n+1}^{m} \bigl\vert \kappa _{r}(x_{i})_{r}- \kappa _{r}(x_{j})_{r} \bigr\vert \bigr\vert ^{p_{r}} \\& \quad =\sum_{r=0, r\in (\ker (\kappa ) )^{c}}^{\infty } \bigl\vert \Delta _{n+1}^{m} \bigl\vert \kappa _{r}(x_{i})_{r}- \kappa _{r}(x_{j})_{r} \bigr\vert \bigr\vert ^{p_{r}}+ \sum_{r=0, r\notin (\ker (\kappa ) )^{c}}^{\infty } \bigl\vert \Delta _{n+1}^{m} \bigl\vert \kappa _{r}(x_{i})_{r}-\kappa _{r}(x_{j})_{r} \bigr\vert \bigr\vert ^{p_{r}} \\& \quad \geq \sum_{r=0, r\in (\ker (\kappa ) )^{c}}^{\infty } \bigl\vert \Delta _{n+1}^{m} \bigl( \vert \kappa _{r} \vert \bigl\vert (x_{i})_{r}-(x_{j})_{r} \bigr\vert \bigr) \bigr\vert ^{p_{r}} =\sum _{r=0}^{\infty } \bigl\vert \Delta _{n+1}^{m} \bigl( \vert \kappa _{r} \vert \bigl\vert (y_{i})_{r}-(y_{j})_{r} \bigr\vert \bigr) \bigr\vert ^{p_{r}} \\& \quad >\epsilon \sum_{r=0}^{\infty } \bigl\vert \Delta _{n+1}^{m} \bigl\vert (y_{i})_{r}-(y_{j})_{r} \bigr\vert \bigr\vert ^{p_{r}} =\epsilon \tau (y_{n}-y_{m} ), \end{aligned}$$

where

$$ (y_{i})_{r}= \textstyle\begin{cases} (x_{i})_{r}, & r\in (\ker (\kappa ) )^{c}, \\ 0, & r\notin (\ker (\kappa ) )^{c}. \end{cases} $$

This shows that \((y_{i})\) is a Cauchy sequence in \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }\). Since \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }\) is complete, there is \(x\in (\ell (p, \Delta _{n+1}^{m}) )_{\tau }\) such that \(\lim_{i\rightarrow \infty }y_{i}=x\). Since \(V_{\kappa }\) is continuous, \(\lim_{i\rightarrow \infty }V_{\kappa }y_{i}=V_{\kappa }x\). But \(\lim_{i\rightarrow \infty }V_{\kappa }x_{i}=\lim_{i\rightarrow \infty }V_{\kappa }y_{i}=d\). Hence \(V_{\kappa }x=d\). Therefore \(d\in R(V_{\kappa })\). This shows that \(R(V_{\kappa })\) is closed. Conversely, let \(R(V_{\kappa })\) be closed. Then \(V_{\kappa }\) is bounded away from zero on \(((\ell (p, \Delta _{n+1}^{m}))_{\tau } )_{ (\ker (\kappa ) )^{c}}\). Hence there exists \(\epsilon >0\) such that \(\tau (V_{\kappa }x)\geq \epsilon \tau (x)\) for all \(x\in ((\ell (p, \Delta _{n+1}^{m}))_{\tau } )_{ (\ker ( \kappa ) )^{c}}\).

Let \(B= \{r\in (\ker (\kappa ) )^{c}:|\kappa _{r}|<\epsilon \}\). If \(B\neq \phi \), then for \(i_{0}\in B\), we obtain

$$ \tau (V_{\kappa }e_{i_{0}})=\tau \bigl( \bigl(\kappa _{r}(e_{i_{0}})_{r} \bigr)_{r=0}^{\infty } \bigr)= \sum_{r=0}^{\infty } \bigl\vert \Delta _{n+1}^{m} \bigl\vert \kappa _{r}(e_{n_{0}})_{r} \bigr\vert \bigr\vert ^{p_{r}} < \sum_{r=0}^{\infty } \bigl\vert \Delta _{n+1}^{m} \bigl\vert \epsilon (e_{n_{0}})_{r} \bigr\vert \bigr\vert ^{p_{r}}=\epsilon \tau (e_{n_{0}}), $$

which gives a contradiction. So, \(B=\phi \) such that \(|\kappa _{r}|\geq \epsilon \) for all \(r\in (\ker (\kappa ) )^{c}\). This completes the proof of the theorem. □

Theorem 6.8

Let \(\kappa \in \mathbb{C}^{\mathbb{N}}\), and let \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }\) be a prequasi-Banach (sss) with \(\tau (w)=\sum^{\infty }_{r=0} |\Delta _{n+1}^{m}|w_{r}| |^{p_{r}}\) for all \(w\in \ell (p, \Delta _{n+1}^{m})\). Then there are \(b>0\) and \(B>0\) such that \(b<\kappa _{r}<B\) for all \(r\in \mathbb{N}\) if and only if \(V_{\kappa }\in \mathfrak{B} ((\ell (p, \Delta _{n+1}^{m}))_{\tau } )\) is invertible.

Proof

Define \(\gamma \in \mathbb{C}^{\mathbb{N}}\) by \(\gamma _{r}=\frac{1}{\kappa _{r}}\). From Theorem 6.2 we have \(V_{\kappa }, V_{\gamma }\in \mathfrak{B} ((\ell (p, \Delta _{n+1}^{m}))_{ \tau } )\) and \(V_{\kappa }.V_{\gamma }=V_{\gamma }.V_{\kappa }=I\). Then \(V_{\gamma }\) is the inverse of \(V_{\kappa }\). Conversely, let \(V_{\kappa }\) be invertible. Then \(R(V_{\kappa })= ((\ell (p, \Delta _{n+1}^{m}))_{\tau } )_{ \mathbb{N}}\). This implies that \(R(V_{\kappa })\) is closed. By Theorem 6.7 there is \(b>0\) such that \(|\kappa _{r}|\geq b\) for all \(r\in (\ker (\kappa ) )^{c}\). Now \(\ker (\kappa )=\phi \), else \(\kappa _{r_{0}}=0\) for several \(r_{0}\in \mathbb{N}\), and we get \(e_{r_{0}}\in \ker (V_{\kappa })\). This gives a contradiction, since \(\ker (V_{\kappa })\) is trivial. So, \(|\kappa _{r}|\geq a\) for all \(r\in \mathbb{N}\). Since \(V_{\kappa }\) is bounded, by Theorem 6.2 there is \(B>0\) such that \(|\kappa _{r}|\leq B\) for all \(r\in \mathbb{N}\). Therefore we have shown that \(b\leq |\kappa _{r}|\leq B\) for all \(r\in \mathbb{N}\). □

Theorem 6.9

Let \(\kappa \in \mathbb{C}^{\mathbb{N}}\), and let \((\ell (p, \Delta _{n+1}^{m}) )_{\tau }\) be a prequasi-Banach (sss), where \(\tau (w)=\sum^{\infty }_{r=0} |\Delta _{n+1}^{m}|w_{r}| |^{p_{r}}\) for all \(w\in \ell (p, \Delta _{n+1}^{m})\). Then \(V_{\kappa }\in \mathfrak{B} ( (\ell (p, \Delta _{n+1}^{m}) )_{ \tau } )\) is a Fredholm operator if and only if (i) \(\operatorname{card} (\ker ( \kappa ))<\infty \) and (ii) \(|\kappa _{r}|\geq \epsilon \) for all \(r\in (\ker (\kappa ) )^{c}\).

Proof

Let \(V_{\kappa }\) be Fredholm. If \(\operatorname{card} (\ker (\kappa ))= \infty \), then \(e_{n}\in \ker (V_{\kappa })\) for all \(n\in \ker (\kappa )\). Since \(e_{n}\) are linearly independent, this gives \(\operatorname{card} (\ker (V_{\kappa })=\infty \), a contradiction. Therefore \(\operatorname{card} (\ker ( \kappa ))<\infty \). By Theorem 6.7 condition (ii) is satisfied. Next, if the necessary conditions are satisfied, then \(V_{\kappa }\) is Fredholm. Indeed, by Theorem 6.7 condition (ii) gives that \(R(V_{\kappa })\) is closed. Condition (i) indicates that \(\dim (\ker (V_{\kappa }))<\infty \) and \(\dim ((R(V_{\kappa }))^{c})<\infty \), and therefore \(V_{\kappa }\) is Fredholm. □

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Acknowledgements

The authors thank the anonymous referees for their constructive suggestions and helpful comments, which led to significant improvement of the original manuscript of this paper.

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Bakery, A.A., Elmatty, A.R.A. A note on Nakano generalized difference sequence space. Adv Differ Equ 2020, 620 (2020). https://doi.org/10.1186/s13662-020-03082-1

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Keywords

  • Premodular
  • Generalized difference
  • Simple Banach space
  • Multiplication operator
  • Approximable operator
  • Fredholm operator