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.

1 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:

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

• (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, $n\in \mathbb{N}$,

• (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,

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

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

• (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 $(\ell (p, \Delta _{n+1}^{m}) )_{\tau }$.

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

• (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) 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:

• (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 $\mathbb{Y}$,

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

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

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

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

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

• (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:

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

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

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

• (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) 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) 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]).

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

• 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. 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. 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.

• (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 $F\subset X_{\tau }$.

• (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

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

• (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-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 $v+w\in \ell (p, \Delta _{n+1}^{m})$.

• (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 $r\in \mathbb{N}$.

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

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

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

  • (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 $\eta \in \mathbb{C}$.

  • (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.

  • (iv) Clearly from (2).

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

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

  • (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 $x\in \ell _{p}(\Delta _{n+1}^{m})$.

4 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.

5 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 $\Upsilon ( (\ell (p, \Delta _{n+1}^{m}) )_{\tau }^{ \mathcal{S}} )$. □

5.1 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. □

6 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. □