Multiplication Operators on Weighted Nakano (sss)

Abstract

In this article, we investigate the sufficient conditions on weighted Nakano sequence space to be premodular Banach (sss). We examine some topological and geometrical structures of the multiplication operators defined on weighted Nakano prequasi-normed (sss).

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1. Introduction

Summability is very important in mathematical models and has numerous implementations, such as normal series theory, approximation theory, ideal transformations, and fixed point theory. For more details, see [1–4]. By ${\mathrm{ℂ}}^{\mathrm{\mathbb{N}}}$, c, ${\ell }_{\infty }$, ${\ell }^{\text{r}}$, and c_o, we signify the spaces of each, convergent, bounded, r-absolutely summable, and convergent to zero sequences of complex numbers. $\mathrm{\mathbb{N}}$ indicates the set of nonnegative integers. Let $p$ = $p_n$ and ${{\beta }_n}_{n\in \mathbb{N}}$ be sequences of positive reals, and the weighted Nakano sequence space [5] is defined by$$\begin{array}{}\text{(1)}& {\ell }_{\beta }^{p_n}=x=x_k\in {ℂ}^{\mathbb{N}}:\sigma \lambda x<\infty ,\hspace{1em}\text{for\hspace{0.17em}some}\lambda >0,\end{array}$$where $\sigma x={\displaystyle {\sum }_{k=0}^{\infty }{\beta }_k{x_k}^{p_k}}$. And ${\ell }_{\beta }^{p_n},\cdot$ is a Banach space, where $x=\inf \eta >0:\sigma x/\eta \le 1.$

When $p_n\in {\ell }_{\infty }$, we have$$\begin{array}{}\text{(2)}& {\ell }_{\beta }^{p_n}=x=x_k\in {ℂ}^{\mathbb{N}}:\sigma \lambda x<\infty ,\hspace{1em}\text{for\hspace{0.17em}any}\lambda >0.\end{array}$$

(1)

If ${\beta }_n=1$, for all n ∈$\mathrm{\mathbb{N}}$, then ${\ell }_{\beta }^{p_n}$ will reduce to ${\ell }^{p_n}$ (see [6, 7])

(2)

If ${\beta }_n=1/p_n$, for all n ∈$\mathrm{\mathbb{N}}$, then ${\ell }_{\beta }^{p_n}$ will reduce to $\ell p_n$ which is studied by many authors (see [8–10])

By $BW,Z$, we will indicate the set of every operators which are linear and bounded between Banach spaces W and Z, and if W = Z, we write $BW$. The multiplication operators have a wide field of mathematics in functional analysis, for instance, in eigenvalue distributions theorem, geometric structure of Banach spaces, and theory of fixed point. For more details, see [11–16]. The s-numbers [17] have many examples such as the rth approximation number, denoted by α_r(V), is defined by α_r(V) = inf {$\text{V}-\text{B}$: B ∈$BW,Z$ and rank(B) ≤ r}, and the rth Kolmogorov number, denoted by d_r(V), is defined by d_r(V) = inf_dimw _≤ r sup $\omega$_≤1 inf $\nu$_∈W$V\omega -\nu$. The following notations will be used in the sequel:$$\begin{array}{c}\text{(3)}& X^s=X^{s}W,Z,\hspace{1em}\mathrm{where}{X}^{s}W,Z=V\in BW,Z{S_{j}V}_{j=0}^{\infty }\in X,X^{\text{app}}=X^{\text{app}}W,Z,\hspace{1em}\mathrm{where}{X}^{\text{app}}W,Z=V\in BW,Z{{\alpha }_{j}V}_{j=0}^{\infty }\in X,\\ X^{\text{Kol}}=X^{\text{Kol}}W,Z,\hspace{1em}\mathrm{where}{X}^{\text{Kol}}W,Z=V\in BW,Z{d_{j}V}_{J=0}^{\infty }\in X.\end{array}$$

A few of operator ideals in the class of Hilbert spaces or Banach spaces are defined by distinct scalar sequence spaces, such as the ideal of compact operators $B_c$ formed by (d_r (V)) and c₀. Pietsch [4] studied the quasi-ideals ${{\ell }^r}^{\text{app}}$ 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 $\overline{F}={{\ell }^r}^{\text{app}}$ for r ∈ [1, ∞), where $\overline{F}$ is the closed class of all finite rank operators, and the class ${{\ell }^r}^{\text{app}}$ became simple Banach and small [1]. The strictly inclusion ${{\ell }^j}^{\text{app}}$${W,Z}_{\ne }^{\subset }$$BW,Z$, whenever j > r > 0, W and Z are infinite-dimensional Banach spaces investigated through Makarov and Faried [2]. Faried and Bakery [18] gave a generalization of the class of quasi-operator ideal which is the prequasi-operator ideal, and they examined several geometric and topological structure of ${{\ell }_M}^S$ and (ces(r))^S. On sequence spaces, Mursaleen and Noman [19, 20] investigated the compact operators on some difference sequence spaces. The multiplication operators on (ces(r), ||.||) with the Luxemburg norm ||.|| elaborated by Komal et al. [21] and Bakery and Abou Elmatty [5] gave the sufficient (not necessary) conditions on ${\ell }_{\beta }^{p_n}$ such that $S_{{\ell }_{\beta }^{p_n}}$ constructed a simple Banach prequi operator ideal. The prequasi-operator ideal $S_{{\ell }_{\beta }^{p_n}}^{\text{app}}$ became small and strictly contained for different weights and powers. The aim of this article to explain some results of ${{\ell }_{\beta }^{p_n}}_{\tau }$ equipped with the prequasi-norm $\tau$. Firstly, we give the sufficient conditions on weighted Nakano sequence space to form premodular Banach (sss). Secondly, we give the necessity and sufficient conditions on weighted Nakano sequence space equipped with the prequasi-norm such that the multiplication operator defined on it is bounded, approximable, invertible, Fredholm, and closed range operator.

2. Preliminaries and Definitions

Definition 1 (see [4]).

An operator $V\in BW$ is called approximable if there are Dr ∈ $FW$, for every r ∈ $\mathrm{\mathbb{N}}$ and limr⟶∞ ||V—Dr|| = 0.

By $\mathrm{{\rm Y}}W,Z$ and $B_{c}W,Z$, we will indicate the space of all approximable and compact operators from W to Z, respectively.

Theorem 1 (see [4]).

If W is Banach space with dim (W) = ∞, then $F{W}_{\ne }^{\subset }$$\mathrm{{\rm Y}}{W}_{\ne }^{\subset }$$B_c{W}_{\ne }^{\subset }$$BW$.

Definition 2 (see [22]).

An operator V∈$BW$ is called Fredholm if dim (R(V))c < ∞, dim (kerV) < ∞, and R(V) is closed, where (R(V))c denotes the complement of range V.

The sequence e_j = (0, 0, 1, 0, 0, ...) with 1 in the jth coordinate, for every j ∈$\mathrm{\hspace{0.17em}\mathbb{N}}$, will be used in the sequel.

Definition 3 (see [3]).

The space of linear sequence spaces Y is called (sss)

(1)

If e_r ∈ Y with r ∈ $\mathrm{\mathbb{N}}$,

(2)

Let u = (u_r) ∈ ${\mathrm{ℂ}}^{\mathrm{\mathbb{N}}}$, $v$ = ($v_{\text{r}}$)∈ Y, and |u_r|≤|$v_{\text{r}}$|, for every r ∈$\mathrm{\mathbb{N}}$, then u ∈ Y. This means Y be solid,”

(3)

If (u_r) ${u_r}_{r=0}^{\infty }\in Y$, then ${u_{r/2}}_{r=0}^{\infty }$ ∈Y, wherever $r/2$ means the integral part of $r/2$

Definition 4 (see [23]).

A subspace of the (sss) $Y_{\tau }$ is called a premodular (sss) if there is a function: $Y$ ⟶ [0, ∞) verifying the conditions:

(i)

τ(y) ≥ 0, for each y ∈ Y and τ(y) = 0 ⇔ y = θ, where θ is the zero element of Y

(ii)

There exists a ≥ 1 such that τ(ηy) ≤ a|η| τ(y), for all y ∈ Y, and η ∈$\mathrm{ℂ}$

(iii)

For some b ≥ 1, τ(y + z) ≤ b(τ(y) + τ(z)), for every y, z ∈ Y

(iv)

|yr| ≤ |zr| with r ∈ $\mathrm{\mathbb{N}}$, implies τ((yr)) ≤ τ((zr))

(v)

For some b0 ≥ 1, τ ((yr)) ≤ τ($y_{r/2}$) ≤ b0τ ((yr))

(vi)

If y = ${y_r}_{r=0}^{\infty }$∈ Y, and d > 0, then there is r0 ∈ $\mathrm{\mathbb{N}}$ with τ(${y_r}_{r=0}^{\infty }$) < d

(vii)

There is t > 0 with τ (ν, 0, 0, 0, ...) ≥ t|ν| τ (1, 0, 0, 0, ...), for any ? ∈ $\mathrm{ℂ}$

Definition 5 (see [23]).

The (sss) $Y_{\tau }$ is called prequasi-normed (sss) if τ satisfies the Parts (i)–(iii) of Definition 4, and when the space Y is complete under τ, then $Y_{\tau }$ is called a prequasi-Banach (sss).

Theorem 2 (see [23]).

A prequasi-norm (sss) $Y_{\tau }$, whenever it is premodular (sss).$$

The inequality [24]: ${a_i+b_i}^{q_i}\le 2^{H-1}{a_i}^{q_i}+{b_i}^{q_i},$ where $q_i$ ≥ 0, for all i ∈ $\mathrm{\mathbb{N}}$ and $H=\max 1,{\sup }_i{q}_i$, will be used in the sequel.

3. Main Results

3.1. Prequasi-Norm on ${\ell }_{\beta }^{{\mathbf{p}}_{\mathbf{n}}}$

We investigate the sufficient conditions on the sequence space ${\ell }_{\beta }^{p_n}$ equipped with prequasi-norm $\tau$ to form prequasi-Banach and closed (sss).

Theorem 3.

${{\ell }_{\beta }^{p_n}}_{\tau }$, where $\tau x={{\displaystyle {\sum }_{k=0}^{\infty }{\beta }_k{x_k}^{p_k}}}^{1/H}$, for all x ∈ ${\ell }_{\beta }^{p_n}$, is a premodular (sss), if the following conditions are satisfied:

(a1)

The sequence $P_n$ ∈ ${ℝ}^{+\mathbb{N}}$∩ ${\ell }_{\infty }$ is increasing

(a2)

Either (β_n) is a monotonic decreasing or monotonic increasing such that there is C ≥ 1 for which β2_n + 1 ≤ Cβ_n

Proof.

Firstly, we have to prove ${\ell }_{\beta }^{p_n}$ is a (sss):

(1-i)

Let x, y ∈ ${\ell }_{\beta }^{p_n}$. Since $p_n$ is bounded, we obtain$$\begin{array}{}\text{(4)}& \tau X+Y={{\displaystyle \sum _{n=0}^{\infty }{\beta }_n{x_n+y_n}^{p_n}}}^{1/H}\le {{\displaystyle \sum _{n=0}^{\infty }{\beta }_n{x_n}^{p_n}}}^{1/H}+{{\displaystyle \sum _{n=0}^{\infty }{\beta }_n{y_n}^{p_n}}}^{1/H}=\tau X+\tau Y<\infty ,\end{array}$$

then x + y ∈ ${\ell }_{\beta }^{p_n}$.

(1-ii)

Let λ ∈ $\mathrm{ℂ}$ and x ∈ ${\ell }_{\beta }^{p_n}$. Since $p_n$ is bounded, we have$$\begin{array}{}\text{(5)}& \tau \lambda X={{\displaystyle \sum _{n=0}^{\infty }{\beta }_n{\lambda x_n}^{p_n}}}^{1/H}{\le }_n^{\sup }{\lambda }^{p_n/H}{{\displaystyle \sum _{n=0}^{\infty }{\beta }_n{x_n}^{p_n}}}^{1/H}\le a\lambda \tau X<\infty .\end{array}$$

(i)

Then, λ x ∈ ${\ell }_{\beta }^{p_n}$ Therefore, by using Parts (1-i) and (1-ii) we have that the space ${\ell }_{\beta }^{p_n}$ is linear. Also, en ∈ ${\ell }_{\beta }^{p_n}$, for all n ∈ $\mathrm{\mathbb{N}}$, since$$\begin{array}{}\text{(6)}& \tau e_n={{\displaystyle \sum _{i=0}^{\infty }{\beta }_i{e_{n}i}^{{\mathbf{p}}_i}}}^{1/H}={{\beta }_n}^{1/H}.\end{array}$$

(2)

Let $x_n\le y_n$, for all n ∈ $\mathrm{\mathbb{N}}$ and y ∈ ${\ell }_{\beta }^{p_n}$. Since β_n > 0, for all n ∈ $\mathrm{\mathbb{N}}$, then τ(x) = ${{\displaystyle {\sum }_{n=0}^{\infty }{\beta }_n{x_n}^{p_n}}}^{1/H}$ ≤ ${{\displaystyle {\sum }_{n=0}^{\infty }{\beta }_n{y_n}^{p_n}}}^{1/H}$ = τ(y) < ∞, we get x ∈ ${\ell }_{\beta }^{p_n}$.

(3)

Let $x_n$ ∈ ${\ell }_{\beta }^{p_n}$ and (β_n) be an increasing sequence. There exists C ≥ 1 such that β2_n + 1 ≤ Cβ_n and $p_n$ be an increasing; then, we have$$\begin{array}{c}\text{(7)}& \tau x_{n/2}={{\displaystyle \sum _{n=0}^{\infty }{\beta }_n{x}_{n/2}{}^{p_n}}}^{1/H}={{\displaystyle \sum _{n=0}^{\infty }{\beta }_{2n}{x_n}^{p_{2n}}}+{\displaystyle \sum _{n=0}^{\infty }{\beta }_{2n+1}{x_n}^{p_{2n+1}}}}^{1/H}\le {{\displaystyle \sum _{n=0}^{\infty }{\beta }_{2n}{x_n}^{p_n}}+{\displaystyle \sum _{n=0}^{\infty }{\beta }_{2n+1}{x_n}^{p_n}}}^{1/H}\le {2C}^{1/H}{{\displaystyle \sum _{n=0}^{\infty }{\beta }_n{x_n}^{p_n}}}^{1/H}={2C}^{1/H}\tau x_n,\end{array}$$

then $x_{n/2}$ ∈ ${\ell }_{\beta }^{p_n}$. Secondly, we show that the functional τ on ${\ell }_{\beta }^{p_n}$ is a premodular:

(i)

Evidently, τ (x) ≥ 0 and τ (x) = 0 ⇔ x = θ

(ii)

There is a steady a = max {1, supn $\lambda {}^{p_n/H-1}$} ≥ 1 such that τ (λx) ≤ a|λ| τ (x), for all x ∈ ${\ell }_{\beta }^{p_n}$and λ ∈ $\mathrm{ℂ}$

(iii)

There exists K ≥ 1 such that τ (x + y) ≤ K (τ(x) + τ(y)), for all x, y ∈ ${\ell }_{\beta }^{p_n}$

(iv)

Clearly, from the proof part (2) of Theorem 3, the condition is clear since the weighted Nakano sequence space is solid

(v)

It is obtained from (3) that $b_0={2C}^{1/H}$ ≥ 1

(vi)

It is clear that $\overline{F}$ =  ${\ell }_{\beta }^{p_n}$

(vii)

There exists a steady 0 < $\xi$ ≤ $\lambda {}^{p_0/H-1}$, for λ $\ne$ 0 or $\xi$ > 0, for λ = 0 such that τ (λ, 0, 0, 0, ...) $\ge$ $\xi$ |λ| τ (1, 0, 0, 0, ...)

Theorem 4.

If conditions (a1) and (a2) are satisfied, then ${{\ell }_{\beta }^{p_n}}_{\tau }$, where $\tau x={{\displaystyle {\sum }_{k=0}^{\infty }{\beta }_k{x_k}^{p_k}}}^{1/H}$ for all x ∈ ${\ell }_{\beta }^{p_n}$, is a prequasi-Banach (sss).

Proof.

Let the conditions be verified. By Theorem 3, the space ${{\ell }_{\beta }^{p_n}}_{\tau }$ is a premodular (sss). Therefore, from Theorem 2, the space ${{\ell }_{\beta }^{p_n}}_{\tau }$ is a prequasi-normed (sss). To prove that ${{\ell }_{\beta }^{p_n}}_{\tau }$ is a prequasi-Banach (sss), suppose x_n = ${x_k^n}_{k=0}^{\infty }$ be a Cauchy sequence in ${{\ell }_{\beta }^{p_n}}_{\tau }$, then for every ε ∈ (0, 1), there exists a number $n_0$∈$\mathrm{\hspace{0.17em}\mathbb{N}}$ such that, for all n, m ≥ $n_0$, one has$$\begin{array}{}\text{(8)}& \tau X^n-X^m={{\displaystyle \sum _{j=0}^{\infty }{\beta }_j{x_j^n-x_j^m}^{{\mathbf{p}}_{\mathbf{j}}}}}^{1/H}<\mathrm{\epsilon }.\end{array}$$

Hence, for n, m ≥ $n_0$ and j ∈ $\mathrm{\mathbb{N}}$, we get $x_j^n-x_j^m$ < ε.

So, $x_j^m$ is a Cauchy sequence in $\mathrm{ℂ}$ for fixed j ∈ $\mathrm{\mathbb{N}}$, and this gives $\underset{m⟶\infty }{\lim }{x}_j^m=x_j^0$ for j ∈ $\mathrm{\mathbb{N}}$. Hence, τ (x_n−x₀) < ε, for all n ≥ $n_0$. Finally, to prove that x₀ ∈ ${\ell }_{\beta }^{p_n}$, we have τ (x₀) = τ (x₀ − x_n + x_n) ≤ τ(x_n–x₀) + τ(x_n) < ∞.

So, x₀ ∈ ${\ell }_{\beta }^{p_n}$. This means that ${{\ell }_{\beta }^{p_n}}_{\tau }$ is a prequasi-Banach (sss).

Theorem 5.

If conditions (a1) and (a2) are satisfied, then ${{\ell }_{\beta }^{p_n}}_{\tau }$, where $\tau x={{\displaystyle {\sum }_{k=0}^{\infty }{\beta }_k{x_k}^{p_k}}}^{1/H}$, for all x ∈ ${\ell }_{\beta }^{p_n}$, is a prequasi-closed (sss).

Proof.

Let the conditions be verified. By Theorem 3, the space ${{\ell }_{\beta }^{p_n}}_{\tau }$ is a premodular (sss). Therefore, from Theorem 2, the space ${{\ell }_{\beta }^{p_n}}_{\tau }$ is a prequasi-normed (sss). To prove that ${{\ell }_{\beta }^{p_n}}_{\tau }$ is a prequasi-closed (sss), suppose $X^n={x_k^n}_{k=0}^{\infty }$∈ ${{\ell }_{\beta }^{p_n}}_{\tau }$ and limn ⟶ ∞ $\tau X^n-X^0=0$, then for every ε ∈ (0, 1), there exists a number $n_0$ ∈ $\mathrm{\mathbb{N}}$ such that for all n ≥ $n_0$, one has$$\begin{array}{}\text{(9)}& \epsilon >\tau X^n-X^0={{\displaystyle \sum _{j=0}^{\infty }{\beta }_j{x_j^n-x_j^0}^{p_j}}}^{1/H}\ge {{\beta }_j{x_j^n-x_j^0}^{p_j}}^{1/H}.\end{array}$$

Hence, for n ≥ $n_0$ and j ∈ $\mathrm{\mathbb{N}}$, we get $x_j^n-x_j^0$ < ε.

So, $x_j^n$ is a convergent sequence in $\mathrm{ℂ}$ for fixed j ∈ $\mathrm{\mathbb{N}}$, and this gives limm ⟶ ∞ $x_j^m=x_j^0$ for fixed j ∈ $\mathrm{\mathbb{N}}$. Finally, to prove that x₀ ∈ ${\ell }_{\beta }^{p_n}$, we have τ (x₀) = τ(x₀ − x_n + x_n) ≤ τ(x_n − x₀) + τ(x_n) < ∞.

So, x₀ ∈ ${\ell }_{\beta }^{p_n}$. This means that ${{\ell }_{\beta }^{p_n}}_{\tau }$ is a prequasi-closed (sss).

3.2. Bounded Multiplication Operator on ${\ell }_{\beta }^{{\text{p}}_{\text{n}}}$

Here after, we investigate some topological and geometric structure of the multiplication operator acting on ${\ell }_{\beta }^{p_n}$. In this section, we examine the sufficient conditions on the sequence space ${\ell }_{\beta }^{p_n}$ equipped with prequasi-norm τ such that the multiplication operator defined on ${{\ell }_{\beta }^{p_n}}_{\tau }$${{\ell }_{\beta }^{{\text{p}}_{\text{n}}}}_{\tau }$ is bounded and isometry.

Definition 6.

Let κ ∈ ${\mathrm{ℂ}}^{\mathrm{\mathbb{N}}}$ ∩ ℓ∞ and W_τ be a prequasi-normed (sss). An operator V_κ: W_τ ⟶ W_τ is called multiplication operator if V_κ$\omega$ = κ$\omega$  = ${K_r{\omega }_r}_{r=0}^{\infty }$ ∈ W, for all $\omega$∈W. If Vκ ∈ $BW$, we name it a multiplication operator generated by κ.

Theorem 6.

Let κ ∈ ${\mathrm{ℂ}}^{\mathrm{\mathbb{N}}}$ and conditions (a1) and (a2) be satisfied; then, κ ∈ ℓ∞, if and only if $V_K\in B{{\ell }_{\beta }^{p_n}}_{\tau }$, where $\tau x={{\displaystyle {\sum }_{k=0}^{\infty }{\beta }_k{x_k}^{p_k}}}^{1/H}$, for all x ∈ ${\ell }_{\beta }^{p_n}$.

Proof.

Let the conditions be satisfied. Assume κ ∈ ℓ∞. Therefore, there is ε > 0 with |${\kappa }_r$| ≤ ε, for every r ∈ $\mathrm{\mathbb{N}}$. For x ∈ ${{\ell }_{\beta }^{p_n}}_{\tau }$, since $p_r$ is bounded from above with $p_r$ > 0, for all r ∈ $\mathrm{\mathbb{N}}$, then$$\begin{array}{}\text{(10)}& \tau V_{K}X=\tau KX=\tau {K_r{X}_r}_{r=0}^{\infty }={{\displaystyle \sum _{r=0}^{\infty }{\beta }_r{{\kappa }_r{x}_r}^{p_r}}}^{1/H}\le {{\displaystyle \sum _{r=0}^{\infty }{\beta }_r|\epsilon x_r{|}^{p_r}}}^{1/H}\le {\sup }_r^{{\epsilon }^{{\mathbf{p}}_{\mathbf{r}}/H}}{{\displaystyle \sum }_{r=0}^{\infty }{\beta }_r|x_r{|}^{p_r}}^{1/H}=D\tau x.\end{array}$$

This gives V_κ ∈ $B{{\ell }_{\beta }^{p_n}}_{\tau }$. Conversely, assume that V_κ ∈ $B{{\ell }_{\beta }^{p_n}}_{\tau }$. Let us assume κ ∉ ℓ∞; hence, for each j ∈ $\mathrm{\mathbb{N}}$, there is $i_j$ ∈ $\mathrm{\mathbb{N}}$ such that ${\kappa }_{i_j}$ > j. Since $p_n$ ∈ ${\mathrm{ℝ}}^{+\mathrm{\mathbb{N}}}$ ∩ ℓ∞ is increasing, we have$$\begin{array}{}\text{(11)}& \tau V_K{e}_{i_j}=\tau K{e}_{i_j}=\tau {K_r{e_{i_j}}_{\tau }}_{r=0}^{\infty }={{\displaystyle \sum }_{r=0}^{\infty }{\beta }_r{{\kappa }_r{e_{i_j}}_r}^{p_r}}^{1/H}={{\beta }_{i_j}{\kappa }_{i_j}}^{p_{ij}/H}>{{\beta }_{i_j}j}^{p_{ij}/H}={{\beta }_{i_j}j}^{p_{ij}/H}\tau e_{i_j}.\end{array}$$

This shows that V_κ ∈$B{{\ell }_{\beta }^{p_n}}_{\tau }$. Therefore, κ ∈ ℓ∞.

Theorem 7.

Pick up κ ∈ ${\mathrm{ℂ}}^{\mathrm{\mathbb{N}}}$ and ${{\ell }_{\beta }^{p_n}}_{\tau }$ be a prequasi-normed (sss), with τ(x) = ${{\displaystyle {\sum }_{k=0}^{\infty }{\beta }_k{x_k}^{p_k}}}^{1/H}$, for all x ∈ ${\ell }_{\beta }^{p_n}$. Then, ${\kappa }_r$ = 1, for all r ∈ $\mathrm{\mathbb{N}}$, if and only if, Vκ is an isometry.

Proof.

Suppose |κ_r| = 1, for all r ∈ $\mathrm{\mathbb{N}}$. Hence,$$\begin{array}{}\text{(12)}& \tau V_{K}X=\tau KX=\tau {K_r{X}_r}_{r=0}^{\infty }={{\displaystyle \sum _{r=0}^{\infty }{\beta }_r{{\kappa }_r{x}_r}^{p_r}}}^{1/H}={{\displaystyle \sum _{r=0}^{\infty }{\beta }_r{x_r}^{p_r}}}^{1/H}=\tau x,\end{array}$$for all x ∈ ${{\ell }_{\beta }^{p_n}}_{\tau }$. Therefore, V_κ is an isometry. Conversely, Assume that |κi| < 1 for some i = i0. Since $p_r$ ∈ ${\mathrm{ℝ}}^{+\mathrm{\mathbb{N}}}$ ∩ ℓ∞ is increasing, we obtain$$\begin{array}{}\text{(13)}& \tau V{e}_{i_0}=\tau K{e}_{i_0}=\tau {K_r{e_{i_0}}_r}_{r=0}^{\infty }={{\displaystyle \sum _{r=0}^{\infty }{\beta }_r{{\kappa }_r{e_{i_0}}_r}^{p_r}}}^{1/H}<{{\displaystyle \sum _{r=0}^{\infty }{\beta }_r{{e_{i_0}}_r}^{p_r}}}^{1/H}=\tau e_{i_0}.\end{array}$$

When |${\kappa }_{i_0}$| > 1, we can prove that $\tau V_K{e}_{i_0}$ > $\tau e_{i_0}$. Therefore, in both cases, we have a contradiction. So, |κ_r| = 1, for every r ∈ $\mathrm{\mathbb{N}}$.

3.3. Approximable Multiplication Operator on ${\ell }_{\beta }^{p_n}$

In this section, we introduce the sufficient conditions on the sequence space ${\ell }_{\beta }^{p_n}$ equipped with prequasi-norm τ such that the multiplication operator defined on ${{\ell }_{\beta }^{p_n}}_{\tau }$ is an approximable and compact. By card (A), we indicate the cardinality of the set A.

Theorem 8.

If κ ∈ ${\mathrm{ℂ}}^{\mathrm{\mathbb{N}}}$ and ${{\ell }_{\beta }^{p_n}}_{\tau }$ is a prequasi-normed (sss), where τ(x) = ${{\displaystyle {\sum }_{k=0}^{\infty }{\beta }_k{x_k}^{p_k}}}^{1/H}$, for all x ∈ ${\ell }_{\beta }^{p_n}$, then, V_κ ∈ $\mathrm{{\rm Y}}{{\ell }_{\beta }^{p_n}}_{\tau }$ if and only if ${K_r}_{r=0}^{\infty }$ ∈ c₀.

Proof.

Assume that V_κ ∈ $\mathrm{{\rm Y}}{{\ell }_{\beta }^{p_n}}_{\tau }$. Therefore, V_κ ∈ $B$$B_c{{\ell }_{\beta }^{p_n}}_{\tau }$, to prove that the sequence ${K_r}_{r=0}^{\infty }$ belongs to c₀. Suppose ${K_r}_{r=0}^{\infty }$ ∉ c₀. Hence, there is δ > 0 such that the set $A_{\delta }$  = {r ∈$\mathrm{\mathbb{N}}$: |κ_r| ≥ δ} has card $A_{\delta }$ = ∞. Assume ai ∈$A_{\delta }$, for all i ∈$\mathrm{\mathbb{N}}$. Hence, {$e_{a_i}$: ai ∈ Aδ} is an infinite bounded set in ${{\ell }_{\beta }^{p_n}}_{\tau }$. Let$$\begin{array}{}\text{(14)}& \tau V_K{e}_{a_i}-V_K{e}_{a_j}=\tau K{e}_{a_i}-K{e}_{a_i}=\tau {{\kappa }_r{e_{a_i}}_r-{e_{a_i}}_r}_{r=0}^{\infty }={{\displaystyle \sum _{r=0}^{\infty }{\beta }_r{{\kappa }_r{e_{a_i}}_r-{e_{a_j}}_r}^{p_r}}}^{1/H}\ge {{\displaystyle \sum }_{r=0}^{\infty }{\beta }_r{\delta {e_{a_i}}_r-{e_{a_j}}_r}^{p_r}}^{1/H}\ge \begin{array}{c}\inf \\ r\end{array}{\delta }^{p_r/H}\tau e_{a_i}-e_{a_j},\end{array}$$for all $a_i,a_j$ ∈ Aδ. This shows {$e_{a_i}$: $a_i$ ∈ Bδ} ∈ ℓ∞ which cannot have a convergent subsequence under V_κ. This proves that V_k ∉ $B_c{{\ell }_{\beta }^{p_n}}_{\tau }$. Then, V_k ∉$\mathrm{{\rm Y}}{{\ell }_{\beta }^{p_n}}_{\tau }$, this gives a contradiction. So, limi ⟶ ∞ κ_i = 0. Conversely, let limi ⟶∞ κ_i = 0. Then, for each δ > 0, the set $A_{\delta }$ = {i ∈$\mathrm{\mathbb{N}}$: |κ_i| ≥ δ} has card (Aδ) < ∞. Hence, for every δ > 0, the space ${{{\ell }_{\beta }^{p_n}}_{\tau }}_{A_{\delta }}$ = {x = (xi) ∈ ${\mathrm{ℂ}}^{A_{\delta }}$} is finite dimensional. Then, Vκ|${{{\ell }_{\beta }^{p_n}}_{\tau }}_{A_{\delta }}$ is a finite rank operator. For every i ∈ $\mathrm{\mathbb{N}}$, dene κi∈ ${\mathrm{ℂ}}^{\mathrm{\mathbb{N}}}$ by (κ i)j = $\begin{array}{cc}{\kappa }_j,& j\in A_{1/i},\\ 0,& \text{otherwise}.\end{array}$

It is clear that $V_{{\kappa }_i}$ has rank $V_{{\kappa }_i}$ <∞ as dim ${{{\ell }_{\beta }^{p_n}}_{\tau }}_{A_{1/i}}$ < ∞, for each i ∈ $\mathrm{\mathbb{N}}$. Therefore, since $p_n$ ∈ ${\mathrm{ℝ}}^{+\mathrm{\mathbb{N}}}$∩ ℓ∞ is increasing, we obtain$$\begin{array}{c}\text{(15)}& \tau V_k-V_{k_i}X\tau k_j-k_{i}j{X}_{j}j{=}_0^{\infty }={{\displaystyle \sum _{j=0}^{\infty }{\beta }_j{{\kappa }_j-{{\kappa }_i}_j{x}_j}^{p_j}}}^{1/H}{{\displaystyle \sum _{j=0,j\in A_{1/i}}^{\infty }{\beta }_j{{\kappa }_j-{{\kappa }_i}_j{x}_j}^{p_j}}+{\displaystyle \sum _{j=0,j\in A_{1/i}}^{\infty }{\beta }_j{{\kappa }_i-{{\kappa }_i}_j{x}_j}^{p_j}}}^{1/H}={{\displaystyle \sum _{j=0,j\notin A_{1/i}}^{\infty }{\beta }_j{{\kappa }_j{x}_j}^{p_j}}}^{1/H}\le \begin{array}{c}\sup \\ j\end{array}{\frac{1}{j}}^{p_j/H}{{\displaystyle \sum _{j=0,j\notin A_{1/i}}^{\infty }{\beta }_j{x_i}^{p_j}}}^{p_j/H}>\begin{array}{c}\sup \\ j\end{array}{\frac{1}{j}}^{p_j/H}\tau x={\frac{1}{j}}^{p_0/H}\tau x.\end{array}$$

This implies that $B_c-V_{K_i}\le {1/j}^{p_0/H}$ and that V_κ is a limit of finite rank operators. Therefore, V_κ is an approximable operator.

Theorem 9.

Let κ ∈${\mathrm{ℂ}}^{\mathrm{\mathbb{N}}}$ and ${{\ell }_{\beta }^{p_n}}_{\tau }$ be a prequasi-normed (sss), where τ(x) = ${{\displaystyle {\sum }_{r=0}^{\infty }{\beta }_r{x_r}^{p_r}}}^{1/H}$, for all x ∈ ${{\ell }_{\beta }^{p_n}}_{\tau }$. Then, V_κ ∈ $B_c{{\ell }_{\beta }^{p_n}}_{\tau }$, if and only if, ${K_i}_{i=0}^{\infty }$ ∈ c₀.

Proof.

It is simple so overlooked.

Corollary 1.

If κ ∈ ${\mathrm{ℂ}}^{\mathrm{\mathbb{N}}}$, conditions (a1) and (a2) are satisfied, then $B_c{{{\ell }_{\beta }^{p_n}}_{\tau }}_{\ne }^{\subset }$$B{{\ell }_{\beta }^{p_n}}_{\tau }$, where τ(x) = ${{\displaystyle {\sum }_{r=0}^{\infty }{\beta }_r{x_r}^{p_r}}}^{1/H}$, for all x ∈${\ell }_{\beta }^{p_n}$.

Proof.

Since I is a multiplication operator on ${l_{\beta }^{P_n}}_{\tau }$ generated by κ = (1, 1, ...), therefore, I ∉ $B_c{{\ell }_{\beta }^{p_n}}_{\tau }$ and I ∈ $B{{\ell }_{\beta }^{p_n}}_{\tau }$.

3.4. Fredholm Multiplication Operator on ${\ell }_{\beta }^{{\mathbf{p}}_{\mathbf{n}}}$

In this section, we give the sufficient conditions on the sequence space ${\ell }_{\beta }^{p_n}$ equipped with prequasi-norm $\tau$ such that the multiplication operator defined on ${l_{\beta }^{P_n}}_{\tau }$ has closed range, invertible, and Fredholm.

Theorem 10.

If κ ∈ ${\mathrm{ℂ}}^{\mathrm{\mathbb{N}}}$, ${{\ell }_{\beta }^{p_n}}_{\tau }$ is prequasi-Banach (sss), where τ(x) = ${{\displaystyle {\sum }_{r=0}^{\infty }{\beta }_r{x_r}^{p_r}}}^{1/H}$, for all x ∈ ${{\ell }_{\beta }^{p_n}}_{\tau }$, then κ is bounded away from zero on $\ker \kappa$, if and only if, R(Vκ) is closed.

Proof.

Let the sufficient condition be satisfied. Therefore, there is $\epsilon$ > 0 with $\kappa i$ ≥ $\epsilon$, for all i ∈ $\ker \kappa$c, to show R(Vκ) is closed. Assume d be a limit point of R(Vκ). Therefore, there is V_κxi in ${{\ell }_{\beta }^{p_n}}_{\tau }$, for all i ∈ $\mathrm{\mathbb{N}}$ such that lim i⟶∞ Vκxi = d. Obviously, (V_κxi) is a Cauchy sequence. Since $p_n$ ∈ ${\mathrm{ℝ}}^{+\mathrm{\mathbb{N}}}$∩ ℓ∞ is increasing, one has$$\begin{array}{c}\text{(16)}& \tau V_k{x}_i-V_k{x}_j={{\displaystyle \sum _{r=0}^{\infty }{\beta }_r{{\kappa }_r{x_i}_r-{\kappa }_r{x_j}_r}^{p_r}}}^{1/H}={{\displaystyle \sum _{r=0,r\in {\ker \kappa }^c}^{\infty }{\beta }_r{{\kappa }_r{x_i}_r-{\kappa }_r{x_j}_r}^{p_r}}+{\displaystyle \sum _{r=0,r\notin {\ker \kappa }^c}^{\infty }{\beta }_r{{\kappa }_r{x_i}_r-{\kappa }_r{x_j}_r}^{p_r}}}^{1/H}\ge {{\displaystyle \sum _{r=0,r\in {\ker \kappa }^c}^{\infty }{\beta }_r{{\kappa }_r{x_i}_r-{\kappa }_r{x_j}_r}^{p_r}}}^{1/H}={{\displaystyle \sum _{r=0}^{\infty }{\beta }_r{{\kappa }_r{y_i}_r-{y_j}_r}^{p_r}}}^{\frac{1}{H}}{>}_r^{\inf {\epsilon }^{p_r/H}}{{\displaystyle \sum _{r=0}^{\infty }{\beta }_r{{y_i}_r-{y_j}_r}^{p_r}}}^{1/H},\end{array}$$where$$\begin{array}{}\text{(17)}& {y_i}_r=\begin{array}{cc}{x_i}_r,& r\in {\ker \kappa }^c,\\ 0,& r\notin {\ker \kappa }^c.\end{array}\end{array}$$

This shows that (yi) is a Cauchy sequence in ${{\ell }_{\beta }^{p_n}}_{\tau }$. Since ${{\ell }_{\beta }^{p_n}}_{\tau }$ is complete, there is x$\in {{\ell }_{\beta }^{p_n}}_{\tau }$ such that limi ⟶ ∞yi = x. Since V_κ is continuous, then limi ⟶ ∞ V_κyi = V_κx. However, limi ⟶ ∞ V_κxi = lim i ⟶ ∞ V_κyi = d. Hence, V_κx = d. Therefore, d ∈ R(Vκ). This shows that R(Vκ) is closed. Conversely, let R(Vκ) be closed. Therefore, V_κ be bounded away from zero on ${{{\ell }_{\beta }^{p_n}}_{\tau }}_{{\ker \kappa }^{c.}}$. Hence, there exists $\epsilon$ > 0 such that $\tau$ (V_κx) ≥ $\epsilon$$\tau$ (x), for all x ∈${{{\ell }_{\beta }^{p_n}}_{\tau }}_{{\ker \kappa }^c}$.

Assume B = {r ∈${\ker \kappa }^c$: |κ_r|> }. If B$\ne$ϕ, then for $i_0$ ∈ B, we obtain$$\begin{array}{c}\text{(18)}& \tau V_K{e}_{n_0}=\tau {K_r{e_{n_0}}_r}_{r=0}^{\infty }={{\displaystyle \sum _{r=0}^{\infty }{\beta }_r{{\kappa }_r{e}_{n_0}{}_r}^{p_r}}}^{1/H}<{{\displaystyle \sum _{r=0}^{\infty }{\beta }_r\epsilon e_{n_0}{}_r{}^{p_r}}}^{1/H}\le \underset{r}{{\displaystyle \sup {\epsilon }^{p_r/H}}}{{\displaystyle \sum _{r=0}^{\infty }{\beta }_r{{e_{n_0}}_r}^{p_r}}}^{1/H}=\underset{r}{{\displaystyle \sup {\epsilon }^{p_r/H}}}\tau e_{n_0},\end{array}$$and this gives contradiction. So, B = ϕ such that |κ_r| ≥ $\epsilon$, for all r ∈ ${\ker \kappa }^c$. This completes the proof of the theorem.

Theorem 11.

If κ ∈ ${\mathrm{ℂ}}^{\mathrm{\mathbb{N}}}$ and ${{\ell }_{\beta }^{p_n}}_{\tau }$ is a prequasi-Banach (sss), with τ$\omega$ =  ${\displaystyle {\sum }_{r=0}^{\infty }{\beta }_r{{\omega }_r}^{p_r}{}^{1/H}}$, for all $\omega$ ∈ ${\ell }_{\beta }^{p_n}$, there are b > 0 and B > 0 such that b <|κ_r| < B, for all r ∈$\mathrm{\mathbb{N}}$, if and only if, V_κ ∈ $B$${{\ell }_{\beta }^{p_n}}_{\tau }$ is invertible.

Proof.

Let the conditions be verified. Define γ ∈ CN by γr = $1/{\kappa }_r$. From Theorem 6, we have V_κ, V_γ ∈ $B$${{\ell }_{\beta }^{p_n}}_{\tau }$ and V_κ.V_γ = V_γ. V_κ = I. Then, V_γ is the inverse of V_κ. Conversely, let V_κ be invertible. Hence, R (V_κ) = ${{{\ell }_{\beta }^{p_n}}_{\tau }}_{\mathbb{N}}$. This implies R (V_κ) is closed. From Theorem 10, there is b > 0 such that |κ_r| ≥ b, for all r ∈ (ker (κ))c. Now, ker (κ) = ϕ, else ${\kappa }_{{\text{r}}_{\text{o}}}$ = 0, for several r_o ∈ $\mathrm{\mathbb{N}}$, we get ${\text{e}}_{{\text{r}}_{\text{o}}}$ ∈ ker (V_κ). This gives a contradiction, since ker (V_κ) is trivial. So, |κ_r| ≥ b, for all r ∈ $\mathrm{\mathbb{N}}$. Since Vκ is bounded, hence from Theorem 6, there is B > 0 such that |κ_r| ≤ B, for all r ∈$\mathrm{\mathbb{N}}$. Therefore, we have shown that b ≤ |κ_r| ≤ B, for all r ∈$\mathrm{\mathbb{N}}$.

Theorem 12.

If κ ∈ ${\mathrm{ℂ}}^{\mathrm{\mathbb{N}}}$ and ${{\ell }_{\beta }^{p_n}}_{\tau }$ is a prequasi-Banach (sss), where τ$\omega$ = ${\displaystyle {\sum }_{r=0}^{\infty }{\beta }_r{{\omega }_r}^{p_r}{}^{1/H}}$, for all $\omega$ ∈ ${\ell }_{\beta }^{p_n}$, then Vκ ∈ $B$${{\ell }_{\beta }^{p_n}}_{\tau }$ is Fredholm operator, if and only if, (i) card (ker(κ)) < ∞ and (ii) |κ_r| ≥ $\epsilon$, for all r ∈ (ker(κ))c.

Proof.

Let V_κ be Fredholm. If card (ker(κ)) = ∞, hence en ∈ ker (Vκ), for all n ∈ ker(κ). Since en’s are linearly independent, this gives card (ker(Vκ)) = ∞. This is a contradiction. Therefore, card (ker(κ)) < ∞. By Theorem 10, condition (ii) is satisfied. Next, if the necessary conditions are verified, to show that V_κ is Fredholm. From Theorem 10, condition (ii) gives that R(V_κ) is closed. Condition (i) indicates that dim (ker(V_κ)) < ∞ and dim ((R(V_κ))c) < ∞. Therefore, V_κ is Fredholm.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This work was funded by the University of Jeddah, Saudi Arabia, under Grant no. UJ-02-054-DR. The authors, therefore, acknowledge with thanks the University technical and financial support.

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