Operator ideal of Norlund-type sequence spaces

Abstract

Let E be a Norlund sequence space which is invariant under the doubling operator

$$D:x=(x_{0}, x_{1}, x_{2},\ldots )\mapsto y=(x_{0}, x_{0}, x_{1}, x_{1}, x_{2}, x_{2},\ldots ). $$

Using the approximation numbers $(\alpha_{n}(T))^{\infty}_{n=0}$ of operators from a Banach space X into a Banach space Y, we give the sufficient (not necessary) conditions on E such that the components

$$U_{E}^{\mathrm {app}}(X, Y):= \bigl\{ T\in L(X, Y):\bigl( \alpha_{n}(T)\bigr)_{n=0}^{\infty}\in E \bigr\} $$

form an operator ideal, the finite rank operators are dense in the complete space of operators $U_{E}^{\mathrm {app}}(X, Y)$ which is a longstanding open problem. Finally we give an answer for Rhoades (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 59(3-4):238-241, 1975) about the linearity of E-type spaces $(U_{E}^{\mathrm {app}}(X, Y))$, and we conclude under a few conditions that every compact operator would be approximated by finite rank operators. Our results agree with those in (J. Inequal. Appl., 2013, doi:10.1186/1029-242x-2013-186) for the space $\operatorname {ces}((p_{n}))$, where $(p_{n})$ is a sequence of positive reals.

1 Introduction

As an aftereffect of the enormous applications in geometry of Banach spaces, spectral hypothesis, hypothesis of eigenvalue dispersions and fixed point hypothesis and so on, the hypothesis of operator ideal goals possesses an uncommon essentialness in useful examination. A large portion of the administrator goals in the family of Banach spaces or normed spaces in straight practical examination are characterized by diverse scalar grouping spaces. All through the paper

$$L(X, Y)= \{T:X\rightarrow Y; T \text{ is bounded and linear; }X \text{ and } Y \text{ are Banach spaces} \}. $$

And $\mathbb{N}=\{0, 1, 2,\ldots \}$, by w we denote the space of all real sequences and θ is the zero vector of E. In [1], by using the approximation numbers and p-absolutely summable sequences of real numbers $\ell^{p}$ ($0< p<\infty$), Pietsch formed the operator ideals. In [2], Faried and Bakery considered the space $\operatorname {ces}((p_{n}))$, where $(p_{n})$ is a sequence of positive reals and $\ell_{M}$, when $M(t)=t^{p}$ ($0< p<\infty$), which match in special with $\ell^{p}$. Bakery [3, 4] took some different mean of Cesaro type spaces involving Lacunary sequence $\operatorname {Ces}(\theta, p)$ defined in [5] to form an operator ideal with its approximation numbers.

2 Definitions and preliminaries

Definition 2.1

An s-function is a map allocating to each operator $T\in L(X, Y)$ a non-negative scalar sequence $(s_{n}(T))_{n=0}^{\infty}$, and the number $s_{n}(T)$ is called the nth s-number of T assuming that the following conditions are verified:

• (a) $\Vert T \Vert =s_{0}(T)\geq s_{1}(T)\geq s_{2}(T)\geq\cdots\geq0$ for $T\in L(X, Y)$;

• (b) $s_{n}(S_{1}+S_{2})\leq s_{n}(S_{1})+ \Vert S_{2} \Vert$ for all $S_{1}, S_{2}\in L(X, Y)$;

• (c) ideal property: $s_{n}(RVT)\leq \Vert R \Vert s_{n}(V) \Vert T \Vert$ for all $T\in L(X_{0}, X)$, $V\in L(X, Y)$ and $R\in L(Y, Y_{0})$, where $X_{0}$ and $Y_{0}$ are arbitrary Banach spaces;

• (d) if $G\in L(X, Y)$ and $\lambda\in\mathbb{R}$, we obtain $s_{n}(\lambda G)=\vert \lambda \vert s_{n}(G)$;

• (e) rank property: If $\operatorname {rank}(T)\leq n$, then $s_{n}(T)=0$ for each $T\in L(X, Y)$;

• (f) norming property: $s_{r\geq n}(I_{n})=0$ or $s_{r< n}(I_{n})=1$, where $I_{n}$ represents the unit operator on the n-dimensional Hilbert space $\ell_{2}^{n}$.

There a few cases would define s-numbers, we specify close approximation numbers $\alpha_{n}(S)$, Gelfand numbers $c_{n}(S)$, Kolmogorov numbers $d_{n}(S)$ and Tichomirov numbers $d_{n}^{*}(S)$. The sum of these numbers fulfills the next condition:

• (g) Additivity: $s_{n+m}(T_{1}+T_{2})\leq s_{n}(T_{1})+s_{m}(T_{2})$ for all $T_{1}, T_{2}\in L(X, Y)$.

Definition 2.2

Let L be the class of all bounded linear operators between any arbitrary Banach spaces. A subclass U of L is called an operator ideal if each element $U(X, Y)=U\cap L(X, Y)$ fulfills the accompanying conditions:

• (i) $I_{K}\in U$, where K represents a Banach space of one dimension.

• (ii) The space $U(X, Y)$ is a linear space over $\mathbb{R}$.

• (iii) If $T\in L(X_{0}, X)$, $V\in U(X, Y)$ and $R\in L(Y, Y_{0})$, then $RVT\in U(X_{0}, Y_{0})$. See [6] and [7].

Definition 2.3

An Orlicz function is a function $M:[0,\infty)\rightarrow[0,\infty)$ which is convex, positive, non-decreasing, continuous with $M(0)= 0$ and $\lim_{x\rightarrow\infty}M(x)=\infty$.

See [8] and [9]. In the event that convexity of M is supplanted by $M(x+y)\leq M (x) + M (y)$, it is known as a modulus function, presented by Nakano [10].

Definition 2.4

An Orlicz function M is said to fulfill $\Delta_{2}$-condition for all estimations of $x\geq0$ if there exists a steady $k>0$ such that $M(2x)\leq k M(x)$. The $\Delta_{2}$-condition is compared to $M(lx)\leq k l M(x)$ for all estimations of x and for $l>1$.

Lindentrauss and Tzafriri [11] used the idea of an Orlicz function to define the following sequence spaces:

$$ \ell_{M}= \bigl\{ x\in\omega:\exists\lambda>0 \text{ with } \rho(\lambda x)< \infty \bigr\} , \quad \text{where } \rho(x)=\sum^{\infty}_{k=1}M \bigl(\vert x_{k}\vert \bigr), $$

which is a Banach space with the Luxemburg norm defined by

$$ \Vert x\Vert =\inf \biggl\{ \lambda>0:\rho\biggl(\frac{x}{\lambda}\biggr) \leq1 \biggr\} . $$

The space $\ell_{M}$ is directly related to the space $\ell^{p}$, which is an Orlicz sequence space with $M (x) = x^{p}$ for $1\leq p <\infty$.

For $\rho(x)=\sum^{\infty}_{n=0} (\frac{ \sum^{n}_{k=0}q_{k}\vert x_{k}\vert }{\sum^{n}_{k=0}q_{k}} )^{p_{n}}$, the Norlund sequence spaces considered by Wang [12] are defined as

$$ \operatorname {ces}\bigl((p_{n}), (q_{n})\bigr)= \bigl\{ x=(x_{k}) \in\omega :\exists\lambda>0 \text{ with } \rho(\lambda x)< \infty \bigr\} , $$

where $(p_{n})$ and $(q_{n})$ are sequences of positive reals, $p_{n}\geq1$ for all $n\in\mathbb{N}$. A Norlund sequence space is a Banach space with the Luxemburg norm defined by

$$ \Vert x \Vert =\inf \biggl\{ \lambda>0:\rho\biggl(\frac{x}{\lambda}\biggr) \leq1 \biggr\} . $$

If $(p_{n})$ is bounded, we might essentially compose

$$ \operatorname {ces}\bigl((p_{n}), (q_{n})\bigr)= \Biggl\{ x=(x_{k}) \in\omega :\sum^{\infty}_{n=0} \biggl( \frac{\sum^{n}_{k=0}q_{k} \vert x_{k}\vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}< \infty \Biggr\} . $$

Remark 2.5

• (1) Taking $q_{n}=1$ for all $n\in\mathbb{N}$, then $\operatorname {ces}((p_{n}), (q_{n}))$ is reduced to $\operatorname {ces}((p_{n}))$ studied by Sanhan and Suantai [13].

• (2) Taking $q_{n}=1$ and $p_{n}=p$ for all $n\in\mathbb{N}$, then $\operatorname {ces}((p_{n}), (q_{n}))$ is reduced to $\operatorname {ces}_{p}$ studied by many authors (see [14, 15] and [16]).

In order to give full knowledge to the reader, we add the article [17] on the paranormed Nörlund sequence spaces and the textbook [18] containing five chapters on the sequence spaces.

Definition 2.6

Let E be a linear space of sequences, then E is called a (sss) if:

• (1) For $n\in\mathbb{N}$, $e_{n}\in E$;

• (2) E is solid, i.e., assuming $x=(x_{n})\in w$, $y=(y_{n})\in E$ and $\vert x_{n}\vert \leq \vert y_{n}\vert$ for all $n\in\mathbb{N}$, then $x\in E$;

• (3) $(x_{[\frac{n}{2}]})_{n=0}^{\infty}\in E$, where $[\frac {n}{2}]$ indicates the integral part of $\frac{n}{2}$, whenever $(x_{n})_{n=0}^{\infty}\in E$.

Example 2.7

$\ell^{p}$ is a (sss) if $p\in(0, \infty)$.

Example 2.8

Let M be an Orlicz function with $\Delta_{2}$-condition, then $\ell _{M}$ is a (sss).

Example 2.9

$\operatorname {ces}_{p}$ is a (sss) if $p\in(1, \infty)$.

Example 2.10

Pick an increasing sequence $(p_{n})$ with $\lim_{n\rightarrow\infty}\inf p_{n}>1$ and $\lim_{n\rightarrow \infty}\sup p_{n}<\infty$, hence $\operatorname {ces}((p_{n}))$ will be a (sss).

Definition 2.11

A subclass of the (sss) is called a pre-modular (sss) assuming that we have a map $\rho: E\rightarrow[0,\infty[$ with the following:

• (i) For $x\in E$, $x=\theta\Leftrightarrow\rho(x)=0$ with $\rho (x)\geq0$;

• (ii) For each $x\in E$ and scalar λ, we get a real number $L\geq1$ for which $\rho(\lambda x)\leq L\vert \lambda \vert \rho( x)$;

• (iii) $\rho(x+y)\leq K(\rho(x)+\rho(y))$ for each $x, y\in E$ holds for a few numbers $K\geq1$;

• (iv) For $n\in\mathbb{N}$, $\vert x_{n}\vert \leq \vert y_{n}\vert$, we obtain $\rho ((x_{n}))\leq\rho((y_{n}))$;

• (v) The inequality $\rho((x_{n}))\leq\rho((x_{[\frac {n}{2}]}))\leq K_{0}\rho((x_{n}))$ holds for some numbers $K_{0}\geq1$;

• (vi) $\overline{F}=E_{\rho}$, where F is the space of any finite sequences;

• (vii) There is steady $\xi>0$ such that $\rho(\lambda, 0, 0, 0,\ldots)\geq\xi \vert \lambda \vert \rho(1, 0, 0, 0,\ldots)$ for any $\lambda\in\mathbb{R}$.

Condition (ii) is equivalent to $\rho(x)$ is continuous at θ. The pre-modular ρ characterizes a metric topology in E, and the linear space E enriched with this topology will be indicated further by $E_{\rho}$. Moreover, the condition (i) of Definition 2.6 and the condition (vi) of Definition 2.11 explain that $(e_{n})_{n\in\mathbb {N}}$ is a Schauder basis of $E_{\rho}$.

Example 2.12

$\ell^{p}$ is a pre-modular (sss) if $p\in(0, \infty)$.

Example 2.13

Suppose that M is an Orlicz function with $\Delta_{2}$-condition, hence $\ell_{M}$ is a pre-modular (sss).

Example 2.14

$\operatorname {ces}_{p}$ is a pre-modular (sss) if $p\in(1, \infty)$.

Example 2.15

$\operatorname {ces}((p_{n}))$ is a pre-modular (sss) if $(p_{n})$ is an increasing sequence, $\lim_{n\rightarrow\infty}\inf p_{n}>1$ and $\lim_{n\rightarrow\infty}\sup p_{n}<\infty$.

Definition 2.16

$$ \begin{aligned} &U_{E}^{\mathrm {app}}:= \bigl\{ U_{E}^{\mathrm {app}}(V, W); V \text{ and } W \text{ are Banach spaces}\bigr\} , \quad \text{and its components} \\ & U_{E}^{\mathrm {app}}(V, W):= \bigl\{ S\in L(V, W):\bigl( \alpha_{n}(S)\bigr)_{n=0}^{\infty}\in E \bigr\} . \end{aligned} $$

Theorem 2.17

If E is a (sss), then $U_{E}^{\mathrm {app}}$ is an operator ideal.

Now and in what follows, $(p_{n})$ and $(q_{n})$ are assumed to be bounded sequences of positive reals. We define $e_{n}=\{0, 0,\ldots,1,0,0,\ldots\}$, where 1 appears at the nth place for all $n\in \mathbb{N}$, and the given inequality will be used in the sequel: $\vert a_{n}+b_{n}\vert ^{p_{n}}\leq H(\vert a_{n}\vert ^{p_{n}}+\vert b_{n}\vert ^{p_{n}})$, where $H=\max\{1, 2^{h-1}\}$, $h=\sup_{n}p_{n}$ and $p_{n}\geq1$ for all $n\in\mathbb{N}$. See [19].

3 Main results

3.1 Linear problem

We study here the operator ideals generated by the approximation numbers and Norlund sequence spaces such that the class of all bounded linear operators between any arbitrary Banach spaces with nth approximation numbers of the bounded linear operators in these sequence spaces form an operator ideal.

Theorem 3.1

$\operatorname {ces}((p_{n}), (q_{n}))$ is a (sss) if the following conditions are satisfied:

• (b1) The sequence $(p_{n})$ is increasing;

• (b2) The sequence $(q_{n})$ with $\sum^{\infty }_{n=0} (\sum^{n}_{k=0}q_{k} )^{-p_{n}}<\infty$;

• (b3) $(q_{n})$ is either monotone decreasing or monotone increasing, and there exists a constant $C\geq1$ such that $q_{2n+1}\leq C q_{n}$.

Proof

(1-i) Suppose $x, y\in \operatorname {ces}((p_{n}), (q_{n}))$. Since

$$ \sum^{\infty}_{n=0} \biggl(\frac{\sum^{n}_{k=0}q_{k}\vert x_{k}+y_{k}\vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}} \leq H \Biggl[\sum^{\infty}_{n=0} \biggl(\frac{\sum^{n}_{k=0}q_{k}\vert x_{k}\vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}+ \sum ^{\infty}_{n=0} \biggl(\frac{\sum^{n}_{k=0}q_{k}\vert y_{k}\vert }{\sum^{n}_{k=0}\vert q_{k}\vert } \biggr)^{p_{n}} \Biggr]< \infty, $$

$x+y\in \operatorname {ces}((p_{n}), (q_{n}))$.

(1-ii) Let $x\in \operatorname {ces}((p_{n}), (q_{n}))$ and $\lambda\in\mathbb{R}$. Since $(p_{n})$ is bounded, then we have

$$ \sum^{\infty}_{n=0} \biggl(\frac {\sum^{n}_{k=0}q_{k}\vert \lambda x_{k}\vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}\leq\sup_{n}\vert \lambda \vert ^{p_{n}} \sum^{\infty}_{n=0} \biggl( \frac{\sum^{n}_{k=0}q_{k}\vert x_{k}\vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}< \infty, $$

that is, $\lambda x\in \operatorname {ces}((p_{n}), (q_{n}))$. Hence, by (1-i) and (1-ii), $\operatorname {ces}((p_{n}), (q_{n}))$ is a linear space. Let us show that $\{ e_{m}\}_{m\in\mathbb{N}}\subseteq \operatorname {ces}((p_{n}), (q_{n}))$. Since $(p_{n})$ is bounded and $\sum^{\infty}_{n=0} (\sum^{n}_{k=0}q_{k} )^{-p_{n}}<\infty$, we get

$$ \sum^{\infty}_{n=0} \biggl(\frac {\sum^{n}_{k=0}q_{k}\vert e_{m}(k)\vert }{\sum^{n}_{k=0} q_{k}} \biggr)^{p_{n}}= \sum^{\infty}_{n=m} \biggl( \frac{q_{m}}{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}\leq\sup_{n}(q_{m})^{p_{n}} \sum^{\infty}_{n=m} \Biggl(\sum ^{n}_{k=0}q_{k} \Biggr)^{-p_{n}}< \infty. $$

Hence $e_{m}\in \operatorname {ces}((p_{n}), (q_{n}))$ for all $m\in\mathbb{N}$.

(2) Let $y\in \operatorname {ces}((p_{n}), (q_{n}))$ and $\vert x_{n}\vert \leq \vert y_{n}\vert$ for every $n\in\mathbb{N}$. Since $q_{n}>0$ for every $n\in\mathbb{N}$,

$$ \sum^{\infty}_{n=0} \biggl(\frac {\sum^{n}_{k=0}q_{k}\vert x_{k}\vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}\leq\sum^{\infty}_{n=0} \biggl(\frac{\sum^{n}_{k=0}q_{k}\vert y_{k}\vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}< \infty, $$

we get $x\in \operatorname {ces}((p_{n}), (q_{n}))$.

(3) Let $(x_{n})\in \operatorname {ces}((p_{n}), (q_{n}))$, $(p_{n})$ be an increasing sequence and $(q_{n})$ be increasing with a constant $C\geq 1$ such that $q_{2n+1}\leq Cq_{n}$. Then we have

$$\begin{aligned} &\sum^{\infty}_{n=0} \biggl(\frac{\sum^{n}_{k=0}q_{k}\vert x_{[\frac{k}{2}]}\vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}} \\ &\quad =\sum ^{\infty}_{n=0} \biggl(\frac{ \sum^{2n}_{k=0}q_{k}\vert x_{[\frac{k}{2}]}\vert }{\sum^{2n}_{k=0}q_{k}} \biggr)^{p_{2n}}+ \sum^{\infty}_{n=0} \biggl(\frac{\sum^{2n+1}_{k=0}q_{k}\vert x_{[\frac{k}{2}]}\vert }{\sum^{2n+1}_{k=0}q_{k}} \biggr)^{p_{2n+1}} \\ &\quad \leq\sum^{\infty}_{n=0} \biggl( \frac{ \sum^{2n}_{k=0}q_{k}\vert x_{[\frac{k}{2}]}\vert }{\sum^{2n}_{k=0}q_{k}} \biggr)^{p_{n}}+ \sum^{\infty}_{n=0} \biggl(\frac{\sum^{2n+1}_{k=0}q_{k}\vert x_{[\frac{k}{2}]}\vert }{\sum^{2n+1}_{k=0}q_{k}} \biggr)^{p_{n}} \\ &\quad =\sum^{\infty}_{n=0} \biggl(\frac{\sum^{n}_{k=0}(q_{2k}+q_{2k+1})\vert x_{k}\vert +q_{2n}\vert x_{n}\vert }{\sum^{2n}_{k=0}q_{k}} \biggr)^{p_{n}}+ \sum^{\infty}_{n=0} \biggl( \frac{\sum^{n}_{k=0}(q_{2k}+q_{2k+1})\vert x_{k}\vert }{\sum^{2n+1}_{k=0}q_{k}} \biggr)^{p_{n}} \\ &\quad \leq\sum^{\infty}_{n=0} \biggl( \frac{\sum^{n}_{k=0}(q_{2k}+q_{2k+1})\vert x_{k}\vert +q_{2n}\vert x_{n}\vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}+ \sum^{\infty}_{n=0} \biggl(\frac{\sum^{n}_{k=0}(q_{2k}+q_{2k+1})\vert x_{k}\vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}} \\ &\quad \leq 2^{h-1} \Biggl[\sum^{\infty}_{n=0} \biggl(\frac{2C \sum^{n}_{k=0}q_{k}\vert x_{k}\vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}+ \sum ^{\infty}_{n=0} \biggl(\frac{C\sum^{n}_{k=0}q_{k}\vert x_{k}\vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}} \Biggr]\\ &\qquad {}+ \sum^{\infty}_{n=0} \biggl(\frac{2C\sum^{n}_{k=0}q_{k}\vert x_{k}\vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}} \\ &\quad \leq 2^{h-1}\bigl(2^{h}+1\bigr)C^{h}\sum ^{\infty}_{n=0} \biggl(\frac {\sum^{n}_{k=0}q_{k}\vert x_{k}\vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}+ 2^{h}C^{h}\sum ^{\infty}_{n=0} \biggl(\frac{\sum^{n}_{k=0}q_{k}\vert x_{k}\vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}} \\ &\quad \leq \bigl(2^{2h-1}+2^{h-1}+2^{h} \bigr)C^{h}\sum^{\infty}_{n=0} \biggl( \frac {\sum^{n}_{k=0}q_{k}\vert x_{k}\vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}< \infty, \end{aligned}$$

then $(x_{[\frac{n}{2}]})\in \operatorname {ces}((p_{n}), (q_{n}))$. □

By Theorem 2.17, we derive the following corollaries.

Corollary 3.2

$U_{\operatorname {ces}((p_{n}), (q_{n}))}^{\mathrm {app}}$ is an ideal operator if conditions (b1), (b2) and (b3) are satisfied.

Corollary 3.3

$U_{\operatorname {ces}((p_{n}))}^{\mathrm {app}}$ is an ideal operator if $(p_{n})$ is increasing, $\lim_{n\rightarrow\infty}\inf p_{n}>1$ and $\lim_{n\rightarrow\infty}\sup p_{n}<\infty$.

Corollary 3.4

If $1< p<\infty$, then $U_{\operatorname {ces}_{p}}^{\mathrm {app}}$ is an operator ideal.

3.2 Topological problem

For a Norlund sequence space E, the ideal of the finite rang operators in the class of Banach spaces is dense in $U_{E}^{\mathrm {app}}(X, Y)$, which gives a negative answer to Rhoades [20] open problem about the linearity of E-type spaces $(U_{E}^{\mathrm {app}}(X, Y))$.

Theorem 3.5

$U_{\operatorname {ces}((p_{n}), (q_{n}))}^{\mathrm {app}}=\overline{F(X, Y)}$, assuming conditions (b1), (b2) and (b3) are fulfilled, and the converse is not true, in general.

Proof

Firstly, we substantiate that each finite operator $T\in F(X, Y)$ belongs to $U_{\operatorname {ces}((p_{n}), (q_{n}))}^{\mathrm {app}}(X, Y)$. Since $e_{m}\in \operatorname {ces}((p_{n}), (q_{n}))$ for each $m\in\mathbb{N}$ and $\operatorname {ces}((p_{n}), (q_{n}))$ is a linear space, for each finite operator $T\in F(X, Y)$, i.e., we obtain that $(\alpha_{n}(T))_{n=0}^{\infty}$ holds main finitely a significant unique number in relation to zero. Currently we substantiate that $U_{\operatorname {ces}((p_{n}), (q_{n}))}^{\mathrm {app}}\subseteq\overline{F(X, Y)}$. Let $(q_{n})$ be a monotonic increasing sequence such that there exists a constant $C\geq1$ for which $q_{2n+1}\leq C q_{n}$, then we have for $n\geq s$ that

$$\begin{aligned} q_{2s+n}\leq q_{2s+2n+1}\leq C q_{s+n}\leq C q_{2n}\leq C q_{2n+1}\leq C^{2} q_{n}. \end{aligned}$$

(1)

By taking $T\in U_{\operatorname {ces}((p_{n}), (q_{n}))}^{\mathrm {app}}(X, Y)$, we obtain $(\alpha_{n}(T))_{n=0}^{\infty}\in \operatorname {ces}((p_{n}), (q_{n}))$, and while $\rho((\alpha_{n}(T))_{n=0}^{\infty})<\infty$, let $\varepsilon\in(0, 1)$, at that point there exists $s\in\mathbb{N}-\{0\}$ such-and-such $\rho((\alpha_{n}(T))_{n=s}^{\infty})<\frac{\varepsilon}{2^{h+2}\delta C}$, where $\delta=\max \{1, \sum_{n=s}^{\infty }(\sum^{n}_{k=0}q_{k})^{-p_{n}} \}$. As $\alpha_{n}(T)$ is decreasing for all $n\in\mathbb{N}$, we obtain

$$ \begin{aligned}[b] \sum_{n=s+1}^{2s} \biggl(\frac{\sum_{k=0}^{n}q_{k}\alpha_{2s}(T)}{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}&\leq\sum_{n=s+1}^{2s} \biggl(\frac{\sum_{k=0}^{n}q_{k}\alpha_{k}(T)}{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}\\ &\leq \sum _{n=s}^{\infty} \biggl(\frac{\sum_{k=0}^{n}q_{k}\alpha_{k}(T)}{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}< \frac{\varepsilon}{2^{h+2}\delta C}. \end{aligned} $$

(2)

Hence, there exists $A\in F_{2s}(X, Y)$ such that $\operatorname {rank}A\leq2s$ and

$$ \sum_{n=2s+1}^{3s} \biggl(\frac{\sum_{k=0}^{n}q_{k}\Vert T-A\Vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}} \leq\sum_{n=s+1}^{2s} \biggl(\frac{\sum_{k=0}^{n}q_{k}\Vert T-A\Vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}} < \frac{\varepsilon}{2^{h+2}\delta C}, $$

(3)

since the sequence $(p_{n})$ is bounded, then on considering

$$ \sup_{n\geq s\in\mathbb{N}} \Biggl(\sum_{k=0}^{2s-1}q_{k} \Vert T-A\Vert \Biggr)^{p_{n}}< \frac{\varepsilon}{2^{h}\delta}, $$

(4)

and from the definition of approximation numbers, there exists $N\in \mathbb{N}-\{0\}$, $A_{N}$ with $\operatorname {rank}(A_{N})\leq N$ and $\Vert T-A \Vert \leq 2\alpha_{n}(T)$. Since $\lim_{n\rightarrow\infty}\alpha _{n}(T)=0$, then $\lim_{N\rightarrow\infty} \Vert T-A_{N}\Vert =0$, hence we put

$$ \sum_{n=0}^{s} \biggl(\frac{\sum_{k=0}^{n}q_{k}\Vert T-A\Vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}< \frac{\varepsilon}{2^{h+3}\delta C}. $$

(5)

Since $(p_{n})$ is increasing and $(\frac{1}{\sum^{n}_{k=0}q_{k}} )$ is decreasing for each $n\in\mathbb{N}$, we have by using (1), (2), (3), (4) and (5) that

$$\begin{aligned} &d(A, T)\\ &\quad =\rho \bigl(\alpha_{n}(T-A) \bigr)_{n=0}^{\infty} \\ &\quad =\sum_{n=0}^{3s-1} \biggl(\frac{\sum_{k=0}^{n}q_{k}\alpha_{k}(T-A)}{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}+ \sum_{n=3s}^{\infty} \biggl(\frac{\sum_{k=0}^{n}q_{k}\alpha_{k}(T-A)}{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}} \\ &\quad \leq\sum_{n=0}^{3s} \biggl( \frac{\sum_{k=0}^{n}q_{k}\Vert T-A\Vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}+ \sum_{n=s}^{\infty} \biggl(\frac{\sum_{k=0}^{n+2s}q_{k}\alpha_{k}(T-A)}{\sum^{n+2s}_{k=0}q_{k}} \biggr)^{p_{n+2s}} \\ &\quad \leq\sum_{n=0}^{3s} \biggl( \frac{\sum_{k=0}^{n}q_{k}\Vert T-A\Vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}+ \sum_{n=s}^{\infty} \biggl(\frac{\sum_{k=0}^{n+2s}q_{k}\alpha_{k}(T-A)}{\sum^{n+2s}_{k=0}q_{k}} \biggr)^{p_{n}} \\ &\quad \leq\sum_{n=0}^{3s} \biggl( \frac{\sum_{k=0}^{n}q_{k}\Vert T-A\Vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}+ \sum_{n=s}^{\infty} \biggl(\frac{\sum_{k=0}^{n+2s}q_{k}\alpha_{k}(T-A)}{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}} \\ &\quad \leq3\sum_{n=0}^{s} \biggl( \frac{\sum_{k=0}^{n}q_{k}\Vert T-A\Vert }{\sum^{n}_{k=0}q_{k}} \biggr)^{p_{n}}+ \sum_{n=s}^{\infty} \biggl(\frac{\sum_{k=0}^{2s-1}q_{k}\alpha_{k}(T-A)+ \sum_{k=2s}^{n+2s}q_{k}\alpha_{k}(T-A)}{\sum_{k=0}^{n}q_{k}} \biggr)^{p_{n}} \\ &\quad \leq3\sum_{n=0}^{s} \biggl( \frac{\sum_{k=0}^{n}q_{k}\Vert T-A\Vert }{\sum_{k=0}^{n}q_{k}} \biggr)^{p_{n}}\\ &\qquad {}+ 2^{h-1} \Biggl[\sum _{n=s}^{\infty} \biggl(\frac{\sum_{k=0}^{2s-1}q_{k}\alpha_{k}(T-A)}{\sum_{k=0}^{n}q_{k}} \biggr)^{p_{n}}+ \sum_{n=s}^{\infty} \biggl( \frac{\sum_{k=2s}^{n+2s}q_{k}\alpha_{k}(T-A)}{\sum_{k=0}^{n}q_{k}} \biggr)^{p_{n}} \Biggr] \\ &\quad \leq3\sum_{n=0}^{s} \biggl( \frac{\sum_{k=0}^{n}q_{k}\Vert T-A\Vert }{\sum_{k=0}^{n}q_{k}} \biggr)^{p_{n}}\\ &\qquad {}+ 2^{h-1} \Biggl[\sum _{n=s}^{\infty} \biggl(\frac{\sum_{k=0}^{2s-1}q_{k}\Vert T-A\Vert }{\sum_{k=0}^{n}q_{k}} \biggr)^{p_{n}}+ \sum_{n=s}^{\infty} \biggl( \frac{\sum_{k=0}^{n}q_{k+2s}\alpha_{k+2s}(T-A)}{\sum_{k=0}^{n}q_{k}} \biggr)^{p_{n}} \Biggr] \\ &\quad \leq3\sum_{n=0}^{s} \biggl( \frac{\sum_{k=0}^{n}q_{k}\Vert T-A\Vert }{\sum_{k=0}^{n}q_{k}} \biggr)^{p_{n}}\\ &\qquad {} +2^{h-1}\sup_{n=s}^{\infty} \Biggl(\sum_{k=0}^{2s-1}q_{k} \Vert T-A\Vert \Biggr)^{p_{n}} \sum_{n=s}^{\infty} \Biggl(\sum^{n}_{k=0}q_{k} \Biggr)^{-p_{n}}+2^{h-1} \sum_{n=s}^{\infty} \biggl(\frac{\sum_{k=0}^{n}C^{2}q_{k}\alpha_{k}(T)}{\sum_{k=0}^{n}q_{k}} \biggr)^{p_{n}} \\ &\quad \leq3\sum_{n=0}^{s} \biggl( \frac{\sum_{k=0}^{n}q_{k}\Vert T-A\Vert }{\sum_{k=0}^{n}q_{k}} \biggr)^{p_{n}}\\ &\qquad {} +2^{h-1}\sup_{n=s}^{\infty} \Biggl(\sum_{k=0}^{2s-1}q_{k} \Vert T-A\Vert \Biggr)^{p_{n}} \sum_{n=s}^{\infty} \Biggl(\sum_{k=0}^{n}q_{k} \Biggr)^{-p_{n}}+2^{h-1}C^{2h} \sum _{n=s}^{\infty} \biggl(\frac{\sum_{k=0}^{n}q_{k}\alpha_{k}(T)}{\sum_{k=0}^{n}q_{k}} \biggr)^{p_{n}} \\ &\quad < \varepsilon. \end{aligned}$$

Since $I_{3}\in U^{\mathrm {app}}_{\operatorname {ces}((1), (1))}$ but condition (b2) is not satisfied, which gives a counter example for the converse part. This completes the proof. From Theorem 3.5, we can say that if (b1), (b2) and (b3) are satisfied, then every compact operator would be approximated by finite rank operators and the converse does not hold, in general. □

Corollary 3.6

$U_{\operatorname {ces}((p_{n}))}^{\mathrm {app}}=\overline{F(X, Y)}$ if $(p_{n})$ is increasing,

$$\lim_{n\rightarrow\infty}\inf p_{n}>1 \quad \textit{and} \quad \lim_{n\rightarrow\infty}\sup p_{n}< \infty. $$

Corollary 3.7

$U_{\operatorname {ces}_{p}}^{\mathrm {app}}=\overline{F(X, Y)}$, where $1< p<\infty$.

3.3 Completeness of the ideal components

For a Norlund sequence space E, the components of ideal $U_{E}^{\mathrm {app}}(X,Y)$ are complete.

Theorem 3.8

$\operatorname {ces}((p_{n}), (q_{n}))$ is a pre-modular (sss) if conditions (b1), (b2) and (b3) are satisfied.

Proof

We define a functional ρ on $\operatorname {ces}((p_{n}), (q_{n}))$ as $\rho (x)=\sum^{\infty}_{n=0} (\frac{\sum^{n}_{k=0}q_{k}\vert x_{k}\vert }{\sum_{k=0}^{n}q_{k}} )^{p_{n}}$.

(i) Evidently, $\rho(x)\geq0$ and $\rho(x)=0\Leftrightarrow x=\theta$.

(ii) There exists steady $L=\max \{1,\sup_{n}\vert \lambda \vert ^{p_{n}} \} \geq1$ such that $\rho(\lambda x)\leq L\lambda\rho( x)$ for all $x\in \operatorname {ces}((p_{n}), (q_{n}))$ and for each real λ.

(iii) For some numbers $K=\max\{1, 2^{h-1}\}\geq1$, we have the inequality $\rho(x+y)\leq K(\rho(x)+\rho(y))$ for all $x, y\in \operatorname {ces}((p_{n}), (q_{n}))$.

(iv) Clearly, (2) of Theorem 3.1 holds.

(v) It is obtained from (3) of Theorem 3.1 that $K_{0}\geq (2^{2h-1}+2^{h-1}+2^{h})\geq1$.

(vi) It is clear that $\overline{F}=\operatorname {ces}((p_{n}), (q_{n}))$.

(vii) There exists a steady $0<\xi\leq \vert \lambda \vert ^{p_{0}-1}$ such that $\rho(\lambda, 0, 0, 0,\ldots)\geq\xi \vert \lambda \vert \rho(1, 0, 0, 0,\ldots)$ for all $\lambda\in\mathbb{R}$. □

Theorem 3.9

$U_{E_{\rho}}^{\mathrm {app}}(X,Y)$ is complete if X and Y are Banach spaces and $E_{\rho}$ is a pre-modular (sss).

Proof

Let $(T_{m})$ be a Cauchy sequence in $U_{E_{\rho}}^{\mathrm {app}}(X,Y)$. Then, by utilizing part (vii) of Definition 2.11 and since $L(X, Y)\supseteq U_{E_{\rho}}^{\mathrm {app}}(X,Y)$, we get

$$ \begin{aligned} \rho \bigl(\bigl(\alpha_{n}(T_{i}-T_{j}) \bigr)_{n=0}^{\infty} \bigr)&\geq \rho \bigl(\alpha_{0}(T_{i}-T_{j}), 0, 0, 0,\ldots \bigr)\\ &= \rho \bigl( \Vert T_{i}-T_{j} \Vert , 0, 0, 0,\ldots \bigr)\geq\xi \Vert T_{i}-T_{j} \Vert \rho(1, 0, 0, 0,\ldots), \end{aligned} $$

then $(T_{m})_{m\in\mathbb{N}}$ is plus a Cauchy sequence in $L(X, Y)$. While the space $L(X, Y)$ is a Banach space, so there exists $T\in L(X, Y)$ such that $\lim_{m\rightarrow\infty} \Vert T_{m}-T \Vert =0$ and while $(\alpha_{n}(T_{m}))_{n=0}^{\infty}\in E$ for each $m\in\mathbb{N}$. Therefore, using parts (iii) and (iv) of Definition 2.11 and ρ is continuous at θ, we obtain

$$ \begin{aligned} \rho \bigl(\bigl(\alpha_{n}(T) \bigr)_{n=0}^{\infty} \bigr)&=\rho \bigl(\bigl(\alpha _{n}(T-T_{m}+T_{m})\bigr)_{n=0}^{\infty} \bigr) \\ &\leq K\rho \bigl(\bigl(\alpha_{[\frac{n}{2}]}(T-T_{m}) \bigr)_{n=0}^{\infty}\bigr)+K \rho \bigl(\bigl(\alpha_{[\frac{n}{2}]}(T_{m})_{n=0}^{\infty} \bigr) \bigr) \\ &\leq K\rho \bigl(\bigl( \Vert T_{m}-T \Vert \bigr)_{n=0}^{\infty} \bigr)+K\rho \bigl(\bigl(\alpha_{n}(T_{m})_{n=0}^{\infty} \bigr) \bigr)< \varepsilon, \end{aligned} $$

we have $(\alpha_{n}(T))_{n=0}^{\infty}\in E$, then $T\in U_{E_{\rho }}^{\mathrm {app}}(X,Y)$. □

Corollary 3.10

On the off chance that X and Y are Banach spaces and conditions (b1), (b2) and (b3) are fulfilled, then $U_{\operatorname {ces}((p_{n}), (q_{n}))}^{\mathrm {app}}(X,Y)$ is complete.

Corollary 3.11

If X and Y are Banach spaces and $(p_{n})$ is an increasing sequence, $\lim_{n\rightarrow\infty}\inf p_{n}>1$ and $\lim_{n\rightarrow\infty}\sup p_{n}<\infty$, then $U_{\operatorname {ces}((p_{n}))}^{\mathrm {app}}(X,Y)$ is complete.

Corollary 3.12

If X and Y are Banach spaces and $1< p<\infty$, then $U_{\operatorname {ces}_{p}}^{\mathrm {app}}(X,Y)$ is complete.