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\((r_{1},r_{2})\)-Cesàro summable sequence space of non-absolute type and the involved pre-quasi ideal

Abstract

We suggest a sufficient setting on any linear space of sequences \(\mathcal{V}\) such that the class \(\mathbb{B}^{s}_{\mathcal{V}}\) of all bounded linear mappings between two arbitrary Banach spaces with the sequence of s-numbers in \(\mathcal{V}\) constructs a map ideal. We define a new sequence space \((\mathit{ces}_{r_{1},r_{2}}^{t} )_{\upsilon }\) for definite functional υ by the domain of \((r_{1},r_{2})\)-Cesàro matrix in \(\ell _{t}\), where \(r_{1},r_{2}\in (0,\infty )\) and \(1\leq t<\infty \). We examine some geometric and topological properties of the multiplication mappings on \((\mathit{ces}_{r_{1},r_{2}}^{t} )_{\upsilon }\) and the pre-quasi ideal \(\mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t} )_{\upsilon }}\).

Introduction

Finding out about \((r_{1},r_{2})\)-mathematics or \((r_{1},r_{2})\)-analogues of recognized consequences dates back to the time of Euler. It has several functions in the discipline of arithmetic specifically in the area of dynamical systems, combinatorics, special functions, quantum groups, learning about fractals and multi-fractal measures, and so forth. By \((r_{1},r_{2})\)-analogue of a recognized expression, we suggest the generalization of that expression, the use of new parameters \((r_{1},r_{2})\), which returns again to the authentic expression as \((r_{1},r_{2})\rightarrow (1,1)\). In functional analysis, the multiplication mappings, and mapping ideals have an important role in spectrum theorem, fixed point theorem, the topological and geometric structure of Banach spaces, etc. We use the following conventions throughout the article; if others are used, we will state them.

Conventions 1.1

([1, 2])

\(\mathrm{N}=\{0, 1, 2,\ldots\}\). \(\mathcal{C}\)::

The complex numbers.

\(\mathfrak{F}\)::

The space of all sets with a finite number of elements.

\(\mathcal{C}^{\mathrm{N}}\)::

The space of all sequences of complex numbers.

\(\ell _{\infty }\)::

The space of bounded sequences of complex numbers.

\(\ell _{r}\)::

The space of r-absolutely summable sequences of complex numbers.

\(c_{0}\)::

The space of null sequences of complex numbers.

\(e_{l}=(0, 0,\ldots,1,0,0,\ldots)\),:

as 1 lies at the \(l^{th}\) coordinate for all \(l\in \mathrm{N}\).

\(\mathcal{F}\)::

The space of all sequences with infinite zero coordinates.

::

The space of all increasing sequences of real numbers.

\(\mathbb{B}(\mathcal{P}, \mathcal{Q})\)::

The space of all bounded linear mappings from a Banach space \(\mathcal{P}\) into a Banach space \(\mathcal{Q}\).

\(\mathbb{B}(\mathcal{P})\)::

The space of all bounded linear mappings from a Banach space \(\mathcal{P}\) into itself.

\(\mathbb{F}(\mathcal{P}, \mathcal{Q})\)::

The space of finite rank mappings from a Banach space \(\mathcal{P}\) into a Banach space \(\mathcal{Q}\).

\(\mathbb{F}(\mathcal{P})\)::

The space of finite rank mappings from a Banach space \(\mathcal{P}\) into itself.

\(\mathcal{A}(\mathcal{P}, \mathcal{Q})\)::

The space of approximable mappings from a Banach space \(\mathcal{P}\) into a Banach space \(\mathcal{Q}\).

\(\mathcal{A}(\mathcal{P})\)::

The space of approximable mappings from a Banach space \(\mathcal{P}\) into itself.

\(\mathcal{K}(\mathcal{P}, \mathcal{Q})\)::

The space of compact mappings from a Banach space \(\mathcal{P}\) into a Banach space \(\mathcal{Q}\).

\(\mathcal{K}(\mathcal{P})\)::

The space of compact mappings from a Banach space \(\mathcal{P}\) into itself.

Lemma 1.2

([2])

If \(U\in \mathbb{B}(\mathcal{P}, \mathcal{Q})\) and \(U\notin \mathcal{A}(\mathcal{P}, \mathcal{Q})\), then we have mappings \(X\in \mathbb{B}(\mathcal{P})\) and \(Y\in \mathbb{B}(\mathcal{Q})\) such that \(YUX e_{l}=e_{l}\) with \(l\in \mathrm{N}\).

Definition 1.3

([2])

A Banach space \(\mathcal{V}\) is known as simple if the algebra \(\mathbb{B}(\mathcal{V})\) contains one and only one nontrivial closed ideal.

Theorem 1.4

([2])

Assume that \(\mathcal{V}\) is an infinite dimensional Banach space, then

$$ \mathbb{F}(\mathcal{V})\varsubsetneqq \mathcal{A}(\mathcal{V}) \varsubsetneqq \mathcal{K}(\mathcal{V})\varsubsetneqq \mathbb{B}( \mathcal{V}). $$

Definition 1.5

([3])

A mapping \(U\in \mathbb{B}(\mathcal{V})\) is known as Fredholm if \(\dim (\operatorname{Range}(U))^{c}<\infty \), \(\dim (\ker (U))<\infty \), and \(\operatorname{Range}(U)\) is closed, where \((\operatorname{Range}(U))^{c}\) describes the complement of \(\operatorname{Range}(U)\).

Definition 1.6

([4])

A subclass \(\mathbb{W}\subseteq \mathbb{B}\) is known as an ideal if all elements \(\mathbb{W}(\mathcal{P}, \mathcal{Q})=\mathbb{W}\cap \mathbb{B}( \mathcal{P}, \mathcal{Q})\) satisfy the following conditions:

  1. (i)

    \(I_{\Omega }\in \mathbb{W}\) if Ω indicates a Banach space of one dimension;

  2. (ii)

    \(\mathbb{W}(\mathcal{P}, \mathcal{Q})\) is a linear space on \(\mathcal{C}\);

  3. (iii)

    If \(X\in \mathbb{B}(\mathcal{P}_{0}, \mathcal{P})\), \(Y\in \mathbb{W}(\mathcal{P}, \mathcal{Q})\), and \(Z\in \mathbb{B}(\mathcal{Q}, \mathcal{Q}_{0})\), then \(ZYX\in \mathbb{W}(\mathcal{P}_{0}, \mathcal{Q}_{0})\), where \(\mathcal{P}_{0}\) and \(\mathcal{Q}_{0}\) are normed spaces.

Definition 1.7

([5])

A function \(\Psi : \mathbb{W}\rightarrow [0, \infty )\) is known as a pre-quasi norm on the ideal \(\mathbb{W}\) if the following setting is confirmed:

  1. (1)

    For all \(X\in \mathbb{W}(\mathcal{P}, \mathcal{Q})\), \(\Psi (X)\geq 0\) and \(\Psi (X)=0\Longleftrightarrow X=0\);

  2. (2)

    One has \(E_{0}\geq 1\) such that \(\Psi (\kappa X)\leq E_{0} \vert \kappa \vert \Psi (X)\) for all \(X\in \mathbb{W}(\mathcal{P}, \mathcal{Q})\) and \(\kappa \in \mathcal{C}\);

  3. (3)

    One has \(G_{0}\geq 1\) such that \(\Psi (Z_{1}+Z_{2})\leq G_{0}[\Psi (Z_{1})+\Psi (Z_{2})]\) for every \(Z_{1},Z_{2}\in \mathbb{W}(\mathcal{P}, \mathcal{Q})\);

  4. (4)

    One has \(D_{0}\geq 1\) such that if \(X\in \mathbb{B}(\mathcal{P}_{0}, \mathcal{P})\), \(Y\in \mathbb{W}(\mathcal{P}, \mathcal{Q})\), and \(Z\in \mathbb{B}(\mathcal{Q}, \mathcal{Q}_{0})\), then \(\Psi (ZYX)\leq D_{0} \Vert Z \Vert \Psi (Y) \Vert X \Vert \).

Theorem 1.8

([5])

If Ψ is a quasi norm on the ideal \(\mathbb{W}\), then Ψ is a pre-quasi norm on the ideal \(\mathbb{W}\).

Definition 1.9

([6])

An s-number function is a map acting on \(\mathbb{B}(\mathcal{P}, \mathcal{Q})\), which gives to each map \(X\in \mathbb{B}(\mathcal{P}, \mathcal{Q})\) a nonnegative scaler sequence \((s_{l}(X))_{l=0}^{\infty }\) satisfying the following set-up:

  1. (a)

    \(\Vert X \Vert =s_{0}(X)\geq s_{1}(X)\geq s_{2}(X)\geq \cdots\geq 0\) for every \(X\in \mathbb{B}(\mathcal{P}, \mathcal{Q})\);

  2. (b)

    \(s_{l+a-1}(X_{1}+X_{2})\leq s_{l}(X_{1})+s_{a}(X_{2})\) for each \(X_{1}, X_{2}\in \mathbb{B}(\mathcal{P}, \mathcal{Q})\) and l, \(a\in \mathrm{N}\);

  3. (c)

    Ideal property: \(s_{a}(ZYX)\leq \Vert Z \Vert s_{a}(Y) \Vert X \Vert \) for all \(X\in \mathbb{B}(\mathcal{P}_{0}, \mathcal{P})\), \(Y\in \mathbb{B}(\mathcal{P}, \mathcal{Q})\), and \(Z\in \mathbb{B}(\mathcal{Q}, \mathcal{Q}_{0})\), where \(\mathcal{P}_{0}\) and \(\mathcal{Q}_{0}\) are any Banach spaces;

  4. (d)

    If \(G\in \mathbb{B}(\mathcal{P}, \mathcal{Q})\) and \(\gamma \in \mathcal{C}\), then \(s_{a}(\gamma G)= \vert \gamma \vert s_{a}(G)\);

  5. (e)

    Rank property: Suppose \(\operatorname{rank}(X)\leq a\), then \(s_{a}(X)=0\) for all \(X\in \mathbb{B}(\mathcal{P}, \mathcal{Q})\);

  6. (f)

    Norming property: \(s_{l\geq a}(I_{a})=0\) or \(s_{l< a}(I_{a})=1\), where \(I_{a}\) denotes the unit mapping on the a-dimensional Hilbert space \(\ell _{2}^{a}\).

We mention here some examples of s-numbers:

  1. (1)

    The ath Kolmogorov number, described by \(d_{a}(X)\), is marked by

    $$\begin{aligned}& d_{a}(X)=\inf_{\dim J\leq a}\sup_{ \Vert f \Vert \leq 1}\inf _{g\in J} \Vert Xf-g \Vert . \end{aligned}$$
  2. (2)

    The ath approximation number, described by \(\alpha _{a}(X)\), is marked by

    $$\begin{aligned}& \alpha _{a}(X)=\inf \bigl\{ \Vert X-Y \Vert :Y\in \mathbb{B}( \mathcal{P}, \mathcal{Q}) \text{ and } \operatorname{rank}(Y)\leq a \bigr\} . \end{aligned}$$

Notations 1.10

([5])

$$\begin{aligned} &\mathbb{B}^{s}_{\mathcal{V}}:= \bigl\{ \mathbb{B}^{s}_{ \mathcal{V}}( \mathcal{P}, \mathcal{Q}); \mathcal{P} \text{ and } \mathcal{Q} \text{ are Banach spaces} \bigr\} , \\ &\quad \text{where } \mathbb{B}^{s}_{ \mathcal{V}}( \mathcal{P}, \mathcal{Q}):= \bigl\{ X\in \mathbb{B}( \mathcal{P}, \mathcal{Q}):( \bigl(s_{a}(X) \bigr)_{a=0}^{\infty }\in \mathcal{V} \bigr\} , \\ & \mathbb{B}^{\alpha }_{\mathcal{V}}:= \bigl\{ \mathbb{B}^{\alpha }_{ \mathcal{V}}( \mathcal{P}, \mathcal{Q}); \mathcal{P} \text{ and } \mathcal{Q} \text{ are Banach spaces} \bigr\} , \\ &\quad \text{where } \mathbb{B}^{ \alpha }_{\mathcal{V}}( \mathcal{P}, \mathcal{Q}):= \bigl\{ X\in \mathbb{B}(\mathcal{P}, \mathcal{Q}):( \bigl(\alpha _{a}(X) \bigr)_{a=0}^{\infty } \in \mathcal{V} \bigr\} , \\ & \mathbb{B}^{d}_{\mathcal{V}}:= \bigl\{ \mathbb{B}^{d}_{\mathcal{V}}( \mathcal{P}, \mathcal{Q}); \mathcal{P} \text{ and } \mathcal{Q} \text{ are Banach spaces} \bigr\} , \\ &\quad \text{where } \mathbb{B}^{d}_{ \mathcal{V}}( \mathcal{P}, \mathcal{Q}):= \bigl\{ X\in \mathbb{B}( \mathcal{P}, \mathcal{Q}):( \bigl(d_{a}(X) \bigr)_{a=0}^{\infty }\in \mathcal{V} \bigr\} . \end{aligned}$$

Theorem 1.11

([7])

For s-type \(\mathcal{V}_{\upsilon }:= \{f=(s_{r}(X))\in \mathcal{C}^{ \mathrm{N}}: X\in \mathbb{B}(\mathcal{P}, \mathcal{Q}) \textit{ and } \upsilon (f)<\infty \}\). If \(\mathbb{B}^{s}_{\mathcal{V}_{\upsilon }}\) is a map ideal, then the following conditions are verified:

  1. 1.

    \(\mathcal{F}\subset s\textit{-type }\mathcal{V}_{\upsilon }\).

  2. 2.

    Assume \((s_{r}(X_{1}) )_{r=0}^{\infty }\in s\textit{-type } \mathcal{V}_{\upsilon }\) and \((s_{r}(X_{2}) )_{r=0}^{\infty }\in s\textit{-type } \mathcal{V}_{\upsilon }\), then \((s_{r}(X_{1}+X_{2}) )_{r=0}^{\infty }\in s\textit{-type } \mathcal{V}_{\upsilon }\).

  3. 3.

    If \(\lambda \in \mathcal{C}\) and \((s_{r}(X) )_{r=0}^{\infty }\in s\textit{-type } \mathcal{V}_{\upsilon }\), then \(\vert \lambda \vert (s_{r}(X) )_{r=0}^{\infty }\in s\textit{-type } \mathcal{V}_{\upsilon }\).

  4. 4.

    The sequence space \(\mathcal{V}_{\upsilon }\) is solid, i.e., if \((s_{r}(Y) )_{r=0}^{\infty }\in s\textit{-type } \mathcal{V}_{\upsilon }\) and \(s_{r}(X)\leq s_{r}(Y)\) for all \(r\in \mathrm{N}\) and \(X,Y\in \mathbb{B}(\mathcal{P}, \mathcal{Q})\), then \((s_{r}(X) )_{r=0}^{\infty }\in s\textit{-type } \mathcal{V}_{\upsilon }\).

Some mapping ideals in the class of Banach spaces or Hilbert spaces are generated by sequence spaces of numbers. As the ideal of compact mappings is generated by \(c_{0}\) and \(d_{a}(X)\) with \(X\in \mathbb{B}(\mathcal{P}, \mathcal{Q})\). Pietsch [2] discussed the quasi-ideals \(\mathbb{B}^{\alpha }_{\ell _{b}}\) when \(0< b<\infty \). He examined that the ideals of nuclear mappings and of Hilbert–Schmidt mappings between Hilbert spaces are constructed by \(\ell _{1}\) and \(\ell _{2}\), respectively. He showed that \(\mathbb{F}(\ell _{b})\) are dense in \(\mathbb{B}(\ell _{b})\), and the algebra \(\mathbb{B}(\ell _{b})\), where \((1\leq b<\infty )\), produced simple Banach space. Pietsch [8] showed that \(\mathbb{B}^{\alpha }_{\ell _{b}}\), with \(0< b<\infty \), is small. Makarov and Faried [9] proved that, for every infinite dimensional Banach space \(\mathcal{P}\), \(\mathcal{Q}\) and \(r>b>0\), then \(\mathbb{B}^{\alpha }_{\ell _{b}}(\mathcal{P}, \mathcal{Q}) \varsubsetneqq \mathbb{B}^{\alpha }_{\ell _{r}}(\mathcal{P}, \mathcal{Q})\subsetneqq \mathbb{B}(\mathcal{P}, \mathcal{Q})\). Yaying et al. [10] introduced the sequence space \(\chi _{r}^{t}\) whose r-Cesàro matrix in \(\ell _{t}\) with \(r\in (0,1]\) and \(1\leq t\leq \infty \). They investigated the quasi Banach ideal of type \(\chi _{r}^{t}\) for \(r\in (0,1]\) and \(1< t<\infty \). They found its Schauder basis, α-, β-, and γ-duals, and determined certain matrix classes related to this sequence space. Başarir and Kara suggested the compact mappings on some Euler \(B(m)\)-difference sequence spaces [11], some difference sequence spaces of weighted means [12], the Riesz \(B(m)\)-difference sequence space [13], the B-difference sequence space derived by weighted mean [14], and the mth order difference sequence space of generalized weighted mean [15]. Mursaleen and Noman [16, 17] introduced the compact mappings on some difference sequence spaces. The multiplication maps on Cesàro sequence spaces with the Luxemburg norm were examined by Komal et al. [18]. İlkhan et al. [19] considered the multiplication maps on Cesàro second order function spaces. In the near past, several authors in the literature investigated some non-absolute type sequence spaces and introduced recent high quality papers; for example, Mursaleen and Noman [20] defined the sequence spaces \(\ell _{p}^{\lambda }\) and \(\ell _{\infty }^{\lambda }\) of non-absolute type and showed that the spaces \(\ell _{p}^{\lambda }\) and \(\ell _{p}^{\lambda }\) are linearly isomorphic for \(0< p \leq \infty \), \(\ell _{p}^{\lambda }\) is a p-normed space and a BK-space in the cases for \(0< p<1\) and \(1\leq p \leq \infty \), and formed the basis for the space \(\ell _{p}^{\lambda }\) for \(1\leq p<\infty \). In [21], they studied the α-, β-, and γ-duals of \(\ell _{p}^{\lambda }\) and \(\ell _{\infty }^{\lambda }\) of non-absolute type for \(1\leq p <\infty \). They characterized some related matrix classes and derived the characterizations of some other classes by means of a given basic lemma. On Cesàro summable sequences, Mursaleen and Başar [22] defined some spaces of double sequences whose Cesàro transforms are bounded, convergent in the Pringsheim sense, null in the Pringsheim sense, both convergent in the Pringsheim sense and bounded, regularly convergent and absolutely q-summable, respectively, and examined some topological properties of those sequence spaces. The next inequality will be used in the sequel [23]: Suppose \(1\leq t<\infty \) and \(x_{a},z_{a}\in \mathcal{C}\), then

$$ \vert x_{a}+z_{a} \vert ^{t}\leq 2^{t-1} \bigl( \vert x_{a} \vert ^{t}+ \vert z_{a} \vert ^{t} \bigr). $$
(1)

The design of this article is arranged as follows: In Sect. 2, we investigate sufficient conditions on any linear space of sequences \(\mathcal{V}\) so that \(\mathbb{B}^{s}_{\mathcal{V}}\) describes a mapping ideal. We apply this theorem on \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) for definite functional υ. We examine the sufficient conditions on it to generate a pre-quasi Banach sequence space. In Sect. 3, we define a multiplication map on \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) and introduce the necessity and sufficient conditions on this sequence space in order for the multiplication mapping to be bounded, approximable, invertible, Fredholm, and closed range. In Sect. 4, firstly, we give the sufficient conditions (not necessary) on \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) so that \(\overline{\mathbb{F}}=\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{ \upsilon }}\). This gives a counter example of Rhoades [24] open problem about the linearity of s-type \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) spaces. Secondly, we explore the set-up on \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) so that \(\mathbb{B}^{s}_{\mathit{ces}_{r_{1},r_{2}}^{t}}\) is Banach and closed. Thirdly, we offer the sufficient set-up on \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) in order for \(\mathbb{B}^{\alpha }_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}\) to be strictly confined for distinct powers. We advance the conditions so that \(\mathbb{B}^{\alpha }_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}\) is minimum. Fourthly, we make known the conditions in order that the \(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}\) is a simple Banach space. Fifthly, we declare the sufficient set-up on \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) such that the class of all bounded linear mappings whose sequence of eigenvalues in \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) is strictly contained in \(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}\). In Sect. 5, we give our conclusion.

The sequence space \((\mathit{ces}_{r_{1},r_{2}}^{t} )_{\upsilon }\)

We introduce in this section the definition of the sequence space \((\mathit{ces}_{r_{1},r_{2}}^{t} )_{\upsilon }\) under the functional υ. We suggest a subspace of any linear space of sequences \(\mathcal{V}\) (private sequence space (\(\mathfrak{pss}\))) such that the class \(\mathbb{B}^{s}_{\mathcal{V}}\) generates an ideal. We apply these conditions on \((\mathit{ces}_{r_{1},r_{2}}^{t} )_{\upsilon }\) equipped with definite functional υ to create a pre-modular \(\mathfrak{pss}\) and a pre-quasi Banach \(\mathfrak{pss}\).

Definition 2.1

For all \(r_{1},r_{2}\in (0,\infty )\) and \(1\leq t<\infty \), the sequence space \((\mathit{ces}_{r_{1},r_{2}}^{t} )_{\upsilon }\) under the functional υ is defined as follows:

$$\begin{aligned} & \bigl(\mathit{ces}_{r_{1},r_{2}}^{t} \bigr)_{\upsilon }= \bigl\{ f=(f_{k}) \in \mathcal{C}^{\mathrm{N}}:\upsilon (\rho f)< \infty \text{ for every } \rho >0 \bigr\} , \\&\quad \text{as } \upsilon (f)= \sum ^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z}f_{z} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \quad \text{and} \\ & [l]_{r_{1},r_{2}}=\sum^{l-1}_{z=0}r_{1}^{z}r_{2}^{l-1-z}= \textstyle\begin{cases} \frac{r_{1}^{l}-r_{2}^{l}}{r_{1}-r_{2}}, & r_{1}\neq r_{2}\neq 1, \\ lr_{1}^{l-1}, & r_{1}=r_{2}\neq 1, \\ [l]_{r_{2}}, & r_{1}=1, \\ [l]_{r_{1}}, & r_{2}=1, \\ l, & r_{1}=r_{2}= 1. \end{cases}\displaystyle \end{aligned}$$

Remark 2.2

  1. (1)

    Assume \(r_{1}=r\) and \(r_{2}= 1\), the sequence space \(\mathit{ces}_{r_{1},r_{2}}^{t}=\chi _{r}^{t}\) was investigated by Yaying et al. [10].

  2. (2)

    If \(r_{1}=r_{2}= 1\), hence \(\mathit{ces}_{r_{1},r_{2}}^{t}=\mathit{ces}^{t}\), was made current and considered by Ng and Lee [25]. Distinctive classification of \(\mathit{ces}^{t}\) has been examined by many authors [21, 2629].

Theorem 2.3

If \(r_{1},r_{2}\in (0,\infty )\) and \(1\leq t<\infty \), then \((\mathit{ces}_{r_{1},r_{2}}^{t} )_{\upsilon }\) is of non-absolute type.

Proof

By taking \(f=(-1,1,0,0,0,\ldots )\), then \(\vert f \vert =(1,1,0,0,0,\ldots )\). We have

$$\begin{aligned} \upsilon (f)={}&1+ \biggl(\frac{ \vert -r_{2}+r_{1} \vert }{[2]_{r_{1},r_{2}}} \biggr)^{t}+ \biggl( \frac{ \vert -r_{2}^{2}+r_{1}r_{2} \vert }{[3]_{r_{1},r_{2}}} \biggr)^{t}+ \ldots \\ \neq{}& 1+ \biggl(\frac{ \vert r_{2}+r_{1} \vert }{[2]_{r_{1},r_{2}}} \biggr)^{t}+ \biggl(\frac{ \vert r_{2}^{2}+r_{1}r_{2} \vert }{[3]_{r_{1},r_{2}}} \biggr)^{t}+ \ldots = \upsilon \bigl( \vert f \vert \bigr). \end{aligned}$$

Therefore, the sequence space \((\mathit{ces}_{r_{1},r_{2}}^{t} )_{\upsilon }\) is of non-absolute type.

We name the sequence space \((\mathit{ces}_{r_{1},r_{2}}^{t} )_{\upsilon }\) as \((r_{1},r_{2})\)-Cesàro summable sequence space of non-absolute type since it is constructed by the domain of \((r_{1},r_{2})\)-Cesàro matrix in \(\ell _{t}\), where the \((r_{1},r_{2})\)-Cesàro matrix \(\Lambda (r_{1},r_{2})=(\lambda _{lz}(r_{1},r_{2}))\) is defined as

$$ \lambda _{lz}(r_{1},r_{2})= \textstyle\begin{cases} \frac{r_{1}^{z}r_{2}^{l-z}}{[l+1]_{r_{1},r_{2}}}, & 0\leq z\leq l, \\ 0, &z>l. \end{cases} $$

 □

Definition 2.4

Pick up a linear space of sequences \(\mathcal{V}\). The subspace \(\mathcal{V}\) is known as a \(\mathfrak{pss}\) if it supports the next set-up:

  1. (1)

    \(e_{b}\in \mathcal{V}\) for each \(b\in \mathrm{N}\);

  2. (2)

    If \(f=(f_{b})\in \mathcal{C}^{\mathrm{N}}\), \(\vert g \vert =( \vert g_{b} \vert )\in \mathcal{V}\), and \(\vert f_{b} \vert \leq \vert g_{b} \vert \) for \(b\in \mathrm{N}\), then \(\vert f \vert \in \mathcal{V}\), i.e., \(\mathcal{V}\) is solid;

  3. (3)

    For \(( \vert f_{b} \vert )_{b=0}^{\infty }\in \mathcal{V}\), we have \(( \vert f_{[\frac{b}{2}]} \vert )_{b=0}^{\infty }\in \mathcal{V}\), where \([\frac{b}{2}]\) indicates the integral part of \(\frac{b}{2}\).

Theorem 2.5

Assume that the linear sequence space \(\mathcal{V}\) is a \(\mathfrak{pss}\), then \(\mathbb{B}^{s}_{\mathcal{V}}\) is an ideal.

Proof

Similar to the proof of Theorem 3.2 in [5]. □

Theorem 2.6

\(\mathit{ces}_{r_{1},r_{2}}^{t}\) is a \(\mathfrak{pss}\) whenever \(1< t<\infty \) and \(r_{1}\leq r_{2}\).

Proof

(1-i) Let \(f, g\in \mathit{ces}_{r_{1},r_{2}}^{t}\). Since \(1< t<\infty \), we obtain

$$\begin{aligned}& \sum^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z}(f_{z}+g_{z}) \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t}\\& \quad \leq 2^{t-1} \Biggl( \sum ^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z}f_{z} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t}+ \sum^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z}g_{z} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \Biggr)< \infty , \end{aligned}$$

hence \(f+g\in \mathit{ces}_{r_{1},r_{2}}^{t}\).

(1-ii) Assume \(\rho \in \mathcal{C}\), \(f\in \mathit{ces}_{r_{1},r_{2}}^{t}\), and since \(1< t<\infty \), one has

$$ \sum^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z}\rho f_{z} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} = \vert \rho \vert ^{t} \sum ^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z}f_{z} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t}< \infty . $$

So \(\rho f\in \mathit{ces}_{r_{1},r_{2}}^{t}\). By using (1-i) and (1-ii), one has \(\mathit{ces}_{r_{1},r_{2}}^{t}\) is a linear space.

Besides, when \(1< t<\infty \), we have

$$ \sum^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z}(e_{b})_{z} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} = r_{1}^{bt}r_{2}^{(l-b)t} \sum^{\infty }_{l=b} \biggl( \frac{1}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}< \infty . $$

Hence, \(e_{b}\in \mathit{ces}_{r_{1},r_{2}}^{t}\) for each \(b\in \mathrm{N}\).

(2) Suppose \(\vert f_{b} \vert \leq \vert g_{b} \vert \) for all \(b\in \mathrm{N}\) and \(\vert g \vert \in \mathit{ces}_{r_{1},r_{2}}^{t}\). We have

$$ \sum^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z} \vert f_{z} \vert \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \leq \sum^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z} \vert g_{z} \vert \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t}< \infty , $$

so \(\vert f \vert \in \mathit{ces}_{r_{1},r_{2}}^{t}\).

(3) If \(( \vert f_{z} \vert )\in \mathit{ces}_{r_{1},r_{2}}^{t}\), \(1< t<\infty \), and \(r_{1}\leq r_{2}\), one has

$$\begin{aligned} & \sum^{\infty }_{l=0} \biggl( \frac{\sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z} \vert f_{[\frac{z}{2}]} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &\quad = \sum^{\infty }_{l=0} \biggl( \frac{\sum^{2l}_{z=0}r_{1}^{z}r_{2}^{l-z} \vert f_{[\frac{z}{2}]} \vert }{[2l+1]_{r_{1},r_{2}}} \biggr)^{t}+ \sum ^{\infty }_{l=0} \biggl( \frac{\sum^{2l+1}_{z=0}r_{1}^{z}r_{2}^{l-z} \vert f_{[\frac{z}{2}]} \vert }{[2l+2]_{r_{1},r_{2}}} \biggr)^{t} \\ &\quad \leq \sum^{\infty }_{l=0} \Biggl( \frac{1}{[2l+1]_{r_{1},r_{2}}} \Biggl(r_{1}^{2l}r_{2}^{-l} \vert f_{l} \vert + \sum^{l}_{z=0} \bigl(r_{1}^{2z}r_{2}^{l-2z}+r_{1}^{2z+1}r_{2}^{l-2z-1} \bigr) \vert f_{z} \vert \Biggr) \Biggr)^{t} \\ &\quad \quad{} + \sum^{\infty }_{l=0} \Biggl( \frac{1}{[2l+2]_{r_{1},r_{2}}}\sum^{l}_{z=0} \bigl(r_{1}^{2z}r_{2}^{l-2z}+r_{1}^{2z+1}r_{2}^{l-2z-1} \bigr) \vert f_{z} \vert \Biggr)^{t} \\ &\quad \leq 2^{t-1} \Biggl( \sum^{\infty }_{l=0} \Biggl( \frac{1}{[l+1]_{r_{1},r_{2}}}\sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z} \vert f_{z} \vert \Biggr)^{t}+ \sum ^{\infty }_{l=0} \Biggl( \frac{2}{[l+1]_{r_{1},r_{2}}}\sum ^{l}_{z=0}r_{1}^{z}r_{2}^{l-z} \vert f_{z} \vert \Biggr)^{t} \Biggr) \\ &\quad\quad{} + \sum ^{\infty }_{l=0} \Biggl( \frac{2}{[l+1]_{r_{1},r_{2}}}\sum ^{l}_{z=0}r_{1}^{z}r_{2}^{l-z} \vert f_{z} \vert \Biggr)^{t} \\ &\quad \leq \bigl(2^{2t-1}+3\times 2^{t-1} \bigr) \sum ^{\infty }_{l=0} \Biggl(\frac{1}{[l+1]_{r_{1},r_{2}}}\sum ^{l}_{z=0}r_{1}^{z}r_{2}^{l-z} \vert f_{z} \vert \Biggr)^{t}< \infty , \end{aligned}$$

so \(( \vert f_{[\frac{z}{2}]} \vert )\in \mathit{ces}_{r_{1},r_{2}}^{t}\). □

From Theorem 2.5, we conclude the following theorem.

Theorem 2.7

Assume \(1< t<\infty \) and \(r_{1}\leq r_{2}\), then \(\mathbb{B}^{s}_{\mathit{ces}_{r_{1},r_{2}}^{t}}\) is an ideal.

Definition 2.8

A subclass of the \(\mathfrak{pss}\) is called a pre-modular \(\mathfrak{pss}\) if the functional \(\upsilon : \mathcal{V}\rightarrow [0,\infty )\) satisfies the next conditions:

  1. (i)

    For \(f\in \mathcal{V}\), \(f=\theta \Longleftrightarrow \upsilon ( \vert f \vert )=0\) for all \(\upsilon (f)\geq 0\), with θ being the zero vector of \(\mathcal{V}\);

  2. (ii)

    For \(f\in \mathcal{V}\) and \(\rho \in \mathcal{C}\), one has \(E_{0}\geq 1\) so that \(\upsilon (\rho f)\leq \vert \rho \vert E_{0}\upsilon (f)\);

  3. (iii)

    \(\upsilon (f+g)\leq G_{0}(\upsilon (f)+\upsilon (g))\) verifies for some \(G_{0}\geq 1\), so that \(f, g\in \mathcal{V}\);

  4. (iv)

    If \(b\in \mathrm{N}\), \(\vert f_{b} \vert \leq \vert g_{b} \vert \), we have \(\upsilon (( \vert f_{b} \vert ))\leq \upsilon (( \vert g_{b} \vert ))\);

  5. (v)

    The inequality \(\upsilon (( \vert f_{b} \vert ))\leq \upsilon (( \vert f_{[\frac{b}{2}]} \vert ))\leq D_{0} \upsilon (( \vert f_{b} \vert ))\) is satisfied for some \(D_{0}\geq 1\);

  6. (vi)

    \(\overline{\mathcal{F}}=\mathcal{V}_{\upsilon }\);

  7. (vii)

    There is \(\varpi >0\) with \(\upsilon (\rho , 0, 0, 0,\ldots)\geq \varpi \vert \rho \vert \upsilon (1, 0, 0, 0,\ldots)\) for all \(\rho \in \mathcal{C}\).

Definition 2.9

The \(\mathfrak{pss}\) \(\mathcal{V}_{\upsilon }\) is called a pre-quasi normed \(\mathfrak{pss}\) if υ satisfies conditions (i)–(iii) of Definition 2.8. When \(\mathcal{V}\) is complete under υ, then \(\mathcal{V}_{\upsilon }\) is called a pre-quasi Banach \(\mathfrak{pss}\).

Theorem 2.10

Every pre-modular \(\mathfrak{pss}\) is a pre-quasi normed \(\mathfrak{pss}\) \(\mathcal{V}_{\upsilon }\).

Theorem 2.11

\((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) is a pre-modular \(\mathfrak{pss}\), whenever \(1< t<\infty \) and \(r_{1}\leq r_{2}\).

Proof

(i) We have \(\upsilon (f)\geq 0\) and \(\upsilon ( \vert f \vert )=0\Leftrightarrow f=\theta \).

(ii) One has \(E_{0}=\max \{1, \vert \rho \vert ^{t-1} \}\geq 1\) with \(\upsilon (\rho f)\leq E_{0} \vert \rho \vert \upsilon ( f)\) for all \(f\in \mathit{ces}_{r_{1},r_{2}}^{t}\) and \(\rho \in \mathcal{C}\).

(iii) We have \(\upsilon (f+g)\leq 2^{t-1}(\upsilon (f)+\upsilon (g))\) for all \(f, g\in \mathit{ces}_{r_{1},r_{2}}^{t}\).

(iv) Obviously, from the proof part (2) of Theorem 2.6.

(v) Obviously, from the proof part (3) of Theorem 2.6 that \(D_{0}\geq 2^{2t-1}+3\times 2^{t-1}\geq 1\).

(vi) Definitely, \(\overline{\mathcal{F}}=\mathit{ces}_{r_{1},r_{2}}^{t}\).

(vii) One has \(0<\varpi \leq \vert \rho \vert ^{t-1}\) for \(\upsilon (\rho , 0, 0, 0,\ldots)\geq \varpi \vert \rho \vert \upsilon (1, 0, 0, 0,\ldots)\) when \(\rho \neq 0\) and \(\varpi >0\) when \(\rho =0\). □

Theorem 2.12

If \(1< t<\infty \) and \(r_{1}\leq r_{2}\), then \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) is a pre-quasi Banach \(\mathfrak{pss}\).

Proof

Let the set-up be satisfied, then from Theorem 2.11 the space \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) is a pre-modular \(\mathfrak{pss}\). By using Theorem 2.10, the space \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) is a pre-quasi normed \(\mathfrak{pss}\). To show that \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) is a pre-quasi Banach \(\mathfrak{pss}\), assume \(f^{a}=(f_{z}^{a})_{z=0}^{\infty }\) to be a Cauchy sequence in \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\), then for all \(\varepsilon \in (0, 1)\), there is \(a_{0}\in \mathrm{N}\) so that, for all \(a,b\geq a_{0}\), one has

$$ \upsilon \bigl(f^{a}-f^{b} \bigr)= \sum ^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z}(f^{a}_{z}-f^{b}_{z}) \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t}< \varepsilon ^{t}. $$

Hence, for \(a,b\geq a_{0}\) and \(z\in \mathrm{N}\), we get \(\vert f^{a}_{z}-f^{b}_{z} \vert <\varepsilon \). So \((f^{b}_{z})\) is a Cauchy sequence in \(\mathcal{C}\) for fixed \(z\in \mathrm{N}\), this gives \(\lim_{b\rightarrow \infty }f^{b}_{z}=f^{0}_{z}\) for fixed \(z\in \mathrm{N}\). Hence \(\upsilon (f^{a}-f^{0})<\varepsilon ^{t}\) for all \(a\geq a_{0}\). Finally, to show that \(f^{0}\in (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\), one has \(\upsilon (f^{0})\leq 2^{t-1}(\upsilon (f^{a}-f^{0})+\upsilon (f^{a}))< \infty \), so \(f^{0}\in (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\). This means that \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) is a pre-quasi Banach \(\mathfrak{pss}\). □

Corollary 2.13

If \(1< t<\infty \), then \((\chi _{r}^{t})_{\upsilon }\) is a normed Banach \(\mathfrak{pss}\), where \(\upsilon (f)= [\sum^{\infty }_{l=0} ( \frac{ \vert \sum^{l}_{z=0}r^{z}f_{z} \vert }{[l+1]_{r}} )^{t} ]^{\frac{1}{t}}\) for all \(f\in \chi _{r}^{t}\).

By using Theorem 1.11, we conclude the following properties of the s-type \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\).

Theorem 2.14

For s-type \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }:= \{f=(s_{n}(X))\in \mathcal{C}^{\mathrm{N}}: X\in \mathbb{B}(\mathcal{P}, \mathcal{Q}) \textit{ and } \upsilon (f)<\infty \}\). The following settings are verified:

  1. 1.

    We have s-type \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\supset \mathcal{F}\).

  2. 2.

    If \((s_{r}(X_{1}) )_{r=0}^{\infty }\in s\textit{-type } (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) and \((s_{r}(X_{2}) )_{r=0}^{\infty }\in s\textit{-type } (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\), then \((s_{r}(X_{1}+X_{2}) )_{r=0}^{\infty }\in s\textit{-type } (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\).

  3. 3.

    For all \(\lambda \in \mathcal{C}\) and \((s_{r}(X) )_{r=0}^{\infty }\in s\textit{-type } (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\), then \(\vert \lambda \vert (s_{r}(X) )_{r=0}^{\infty }\in s\textit{-type } (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\).

  4. 4.

    The s-type \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) is solid.

Multiplication mappings on \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\)

We introduce in this section a multiplication mapping on \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\). We examine the necessity and sufficient conditions on \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) such that the multiplication mapping is invertible, bounded, Fredholm, approximable, and closed range.

Definition 3.1

Pick up \(\omega =(\omega _{k})\in \mathcal{C}^{\mathrm{N}}\) and \(\mathcal{V}_{\upsilon }\) is a pre-quasi normed \(\mathfrak{pss}\). The mapping \(H_{\omega }:\mathcal{V}_{\upsilon }\rightarrow \mathcal{V}_{\upsilon }\) is named a multiplication mapping on \(\mathcal{V}_{\upsilon }\) if \(H_{\omega }f= (\omega _{b}f_{b} )\in \mathcal{V}_{\upsilon }\), so that \(f\in \mathcal{V}_{\upsilon }\). The multiplication mapping is called produced by ω when \(H_{\omega }\in \mathbb{B}(\mathcal{V}_{\upsilon })\).

Theorem 3.2

Assume \(\omega \in \mathcal{C}^{\mathrm{N}}\), \(1< t<\infty \), and \(r_{1}\leq r_{2}\), then \(\omega \in \ell _{\infty }\) if and only if \(H_{\omega }\in \mathbb{B}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\).

Proof

Let the conditions be verified for \(\omega \in \ell _{\infty }\). So there is \(\nu >0\) such that \(\vert \omega _{b} \vert \leq \nu \) for all \(b\in \mathrm{N}\). If \(f\in (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\), we have

$$ \upsilon (H_{\omega }f)=\upsilon (\omega f)= \sum ^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z}\omega _{z}f_{z} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \leq \sum^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{z=0}\nu r_{1}^{z}r_{2}^{l-z}f_{z} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} =\nu ^{t}\upsilon (f). $$

Hence, \(H_{\omega }\in \mathbb{B}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\). However, suppose \(H_{\omega }\in \mathbb{B}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\) and \(\omega \notin \ell _{\infty }\). So, for each \(b\in \mathrm{N}\), one has \(x_{b}\in \mathrm{N}\) such that \(\omega _{x_{b}}> b\). We obtain

$$\begin{aligned} \upsilon (H_{\omega }e_{x_{b}})&=\upsilon (\omega e_{x_{b}})= \sum^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z}\omega _{z}(e_{x_{b}})_{z} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} = \sum^{\infty }_{l=x_{b}} \biggl( \frac{r_{1}^{x_{b}}r_{2}^{l-x_{b}} \vert \omega _{x_{b}} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &> \sum ^{\infty }_{l=x_{b}} \biggl( \frac{r_{1}^{x_{b}}r_{2}^{l-x_{b}}b}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}=b^{t} \upsilon (e_{x_{b}}). \end{aligned}$$

Therefore, \(H_{\omega }\notin \mathbb{B}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\). Hence \(\omega \in \ell _{\infty }\). □

Theorem 3.3

If \(\omega \in \mathcal{C}^{\mathrm{N}}\), \(1< t<\infty \), and \(r_{1}\leq r_{2}\). Then \(\omega _{b}=g\) for all \(b\in \mathrm{N}\) and \(g\in \mathcal{C}\) so that \(\vert g \vert =1\) if and only if \(H_{\omega }\) is an isometry.

Proof

Assume that the sufficient set-up is confirmed. We have

$$ \upsilon (H_{\omega }f)=\upsilon (\omega f)= \sum ^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{k=0}r_{1}^{k}r_{2}^{l-k}\omega _{k}f_{k} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} = \sum^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{k=0} \vert g \vert r_{1}^{k}r_{2}^{l-k}f_{k} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t}=\upsilon (f) $$

for all \(f\in (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\). Hence \(H_{\omega }\) is an isometry.

Suppose that the necessity set-up is verified and \(\vert \omega _{b} \vert <1\) for some \(b = b_{0}\). One can see

$$\begin{aligned} \upsilon (H_{\omega }e_{b_{0}})&=\upsilon (\omega e_{b_{0}})= \sum^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{k=0}r_{1}^{k}r_{2}^{l-k}\omega _{k}(e_{b_{0}})_{k} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} = \sum^{\infty }_{l=b_{0}} \biggl( \frac{r_{1}^{b_{0}}r_{2}^{l-b_{0}} \vert \omega _{b_{0}} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &< \sum ^{\infty }_{l=b_{0}} \biggl( \frac{r_{1}^{b_{0}}r_{2}^{l-b_{0}}}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}= \upsilon (e_{b_{0}}). \end{aligned}$$

Besides, if \(\vert \omega _{b_{0}} \vert >1\), one has \(\upsilon (H_{\omega }e_{b_{0}})>\upsilon (e_{b_{0}})\). For the two cases, we have a contradiction. So \(\vert \omega _{b} \vert =1\) with \(b\in \mathrm{N}\). □

Theorem 3.4

If \(\omega \in \mathcal{C}^{\mathrm{N}}\), \(1< t<\infty \), and \(r_{1}\leq r_{2}\). Then \(H_{\omega }\in \mathcal{A}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\) if and only if \((\omega _{b})_{b=0}^{\infty }\in c_{0}\).

Proof

Suppose \(H_{\omega }\in \mathcal{A}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\), hence \(H_{\omega }\in \mathcal{K}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\). Assume \(\lim_{b\rightarrow \infty }\omega _{b}\neq 0\). Hence, one has \(\varrho > 0\) such that \(K_{\varrho }=\{b\in \mathrm{N}: \vert \omega _{b} \vert \geq \varrho \} \nsubseteqq \mathfrak{F}\) when \(\{\alpha _{b}\}_{b\in \mathrm{N}}\subset K_{\varrho }\). Therefore, \(\{e_{\alpha _{b}}:\alpha _{b}\in K_{\varrho }\}\in \ell _{\infty }\) is infinite in \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\). As

$$ \begin{aligned} \upsilon (H_{\omega }e_{\alpha _{a}}-H_{\omega }e_{\alpha _{b}})&= \upsilon (\omega e_{\alpha _{a}}-\omega e_{\alpha _{b}})= \sum ^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{k=0}r_{1}^{k}r_{2}^{l-k}\omega _{k}((e_{\alpha _{a}})_{k}-(e_{\alpha _{b}})_{k}) \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &\geq \sum^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{k=0}r_{1}^{k}r_{2}^{l-k}\varrho ((e_{\alpha _{a}})_{k}-(e_{\alpha _{b}})_{k}) \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} =\varrho ^{t}\upsilon (e_{\alpha _{a}}-e_{\alpha _{b}}) \end{aligned} $$

for all \(\alpha _{a}, \alpha _{b}\in K_{\varrho }\). Hence, \(\{e_{\alpha _{b}}: \alpha _{b}\in K_{\varrho }\}\in \ell _{\infty }\), which cannot have a convergent subsequence with \(H_{\omega }\). So \(H_{\omega }\notin \mathcal{K}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\). This gives \(H_{\omega }\notin \mathcal{A}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\), which is unreliability. Therefore, \(\lim_{b\rightarrow \infty }\omega _{b}=0\). Conversely, suppose \(\lim_{b\rightarrow \infty }\omega _{b}=0\). Therefore, for each \(\varrho > 0\), we have \(K_{\varrho }=\{b\in \mathrm{N}: \vert \omega _{b} \vert \geq \varrho \}\subset \mathfrak{F}\). So, for all \(\varrho > 0\), one has \(\dim ( ((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon } )_{K_{\varrho }} )=\dim (\mathcal{C}^{K_{\varrho }} )<\infty \). Hence \(H_{\omega }\in \mathbb{F} ( ((\mathit{ces}_{r_{1},r_{2}}^{t})_{ \upsilon } )_{K_{\varrho }} )\). Assume \(\omega _{a}\in \mathcal{C}^{\mathrm{N}}\), with \(a\in \mathrm{N}\), by

$$ (\omega _{a})_{b} = \textstyle\begin{cases} \omega _{b}, & b\in K_{\frac{1}{a+1}}, \\ 0, & \text{otherwise.} \end{cases} $$

Obviously, \(H_{\omega _{a}}\in \mathbb{F} ( ((\mathit{ces}_{r_{1},r_{2}}^{t})_{ \upsilon } )_{B_{\frac{1}{a+1}}} )\) since \(\dim ( ((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon } )_{B_{ \frac{1}{a+1}}} )<\infty \) with \(a\in \mathrm{N}\). From \(1< t<\infty \) and \(r_{1}\leq r_{2}\), we have

$$\begin{aligned} \upsilon \bigl((H_{\omega }-H_{\omega _{a}})f \bigr)&=\upsilon \bigl( \bigl( \bigl(\omega _{b}-(\omega _{a})_{b} \bigr)f_{b} \bigr)_{b=0}^{\infty } \bigr) = \sum ^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{b=0}r_{1}^{b}r_{2}^{l-b}(\omega _{b}-(\omega _{a})_{b})f_{b} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &= \sum^{\infty }_{l=0, l\in K_{ \frac{1}{a+1}}} \biggl( \frac{ \vert \sum^{l}_{b=0}r_{1}^{b}r_{2}^{l-b}(\omega _{b}-(\omega _{a})_{b})f_{b} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &\quad{} + \sum^{\infty }_{l=0,l\notin K_{ \frac{1}{a+1}}} \biggl( \frac{ \vert \sum^{l}_{b=0}r_{1}^{b}r_{2}^{l-b}(\omega _{b}-(\omega _{a})_{b})f_{b} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &= \sum^{\infty }_{l=0,l\notin K_{\frac{1}{a+1}}} \biggl( \frac{ \vert \sum^{l}_{b=0}r_{1}^{b}r_{2}^{l-b}\omega _{b}f_{b} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &< \frac{1}{(a+1)^{t}} \sum ^{\infty }_{l=0,l \notin K_{\frac{1}{a+1}}} \biggl( \frac{ \vert \sum^{l}_{b=0}r_{1}^{b}r_{2}^{l-b}f_{b} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &< \frac{1}{(a+1)^{t}} \sum^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{b=0}r_{1}^{b}r_{2}^{l-b}f_{b} \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} =\frac{1}{(a+1)^{t}}\upsilon (f). \end{aligned}$$

Therefore, \(\Vert H_{\omega }-H_{\omega _{a}} \Vert \leq \frac{1}{(a+1)^{t}}\). Then \(H_{\omega }=\lim_{a\rightarrow \infty }H_{\omega _{a}}\) and hence \(H_{\omega }\in \mathcal{A}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\). □

Theorem 3.5

If \(\omega \in \mathcal{C}^{\mathrm{N}}\), \(1< t<\infty \), and \(r_{1}\leq r_{2}\), then \(H_{\omega }\in \mathcal{K}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\) if and only if \((\omega _{b})_{b=0}^{\infty }\in c_{0}\).

Proof

Evidently, as \(\mathcal{A}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\varsubsetneqq \mathcal{K}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\). □

Corollary 3.6

If \(1< t<\infty \) and \(r_{1}\leq r_{2}\), then \(\mathcal{K}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\varsubsetneqq \mathbb{B}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\).

Proof

Since the sequence \(\omega =(1, 1,\ldots )\) produces the multiplication mapping I on \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\). So, \(I\in \mathbb{B}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\) and \(I\notin \mathcal{K}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\). □

Theorem 3.7

If \(\omega \in \mathcal{C}^{\mathrm{N}}\), \(1< t<\infty \), \(r_{1}\leq r_{2}\), and \(H_{\omega }\in \mathbb{B}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\). Then there exist \(\alpha >0\) and \(\eta >0\) such that \(\alpha < \vert \omega _{b} \vert <\eta \) for all \(b\in (\ker (\omega ) )^{c}\) if and only if \(\operatorname{Range}(H_{\omega })\) is closed.

Proof

Let the sufficient setting be verified. So there is \(\varrho >0\) such that \(\vert \omega _{b} \vert \geq \varrho \) for all \(b\in (\ker (\omega ) )^{c}\). To prove that \(\operatorname{Range}(H_{\omega })\) is closed, suppose that g is a limit point of \(\operatorname{Range}(H_{\omega })\). One has \(H_{\omega }f_{b}\in (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) for each \(b\in \mathrm{N}\) so as to \(\lim_{b\rightarrow \infty }H_{\omega }f_{b}=g\). Clearly, the sequence \(H_{\omega }f_{b}\) is a Cauchy sequence. We have

$$\begin{aligned} \upsilon (H_{\omega }f_{a}-H_{\omega }f_{b})&= \sum^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{k=0}r_{1}^{k}r_{2}^{l-k}(\omega _{k}(f_{a})_{k}-\omega _{k}(f_{b})_{k}) \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &= \sum^{\infty }_{l=0,l\in (\ker (\omega ) )^{c}} \biggl( \frac{ \vert \sum^{l}_{k=0}r_{1}^{k}r_{2}^{l-k}(\omega _{k}(f_{a})_{k}-\omega _{k}(f_{b})_{k}) \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &\quad{}+ \sum^{\infty }_{l=0,l\notin (\ker (\omega ) )^{c}} \biggl( \frac{ \vert \sum^{l}_{k=0}r_{1}^{k}r_{2}^{l-k}(\omega _{k}(f_{a})_{k}-\omega _{k}(f_{b})_{k}) \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &\geq \sum^{\infty }_{l=0,l\in (\ker (\omega ) )^{c}} \biggl( \frac{ \vert \sum^{l}_{k=0}r_{1}^{k}r_{2}^{l-k}(\omega _{k}(f_{a})_{k}-\omega _{k}(f_{b})_{k}) \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &= \sum^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{k=0}r_{1}^{k}r_{2}^{l-k}(\omega _{k}(u_{a})_{k}-\omega _{k}(u_{b})_{k}) \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &> \sum^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{k=0}r_{1}^{k}r_{2}^{l-k}\varrho ((u_{a})_{k}-(u_{b})_{k}) \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} =\varrho ^{t}\upsilon (u_{a}-u_{b} ), \end{aligned}$$

where

$$ (u_{a})_{k}= \textstyle\begin{cases} (f_{a})_{k}, & k\in (\ker (\omega ) )^{c}, \\ 0, & k\notin (\ker (\omega ) )^{c}. \end{cases} $$

Then \((u_{a})\) is a Cauchy sequence in \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\). Since \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) is complete, there is \(f\in (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) so as to \(\lim_{b\rightarrow \infty }u_{b}=f\). As \(H_{\omega }\in \mathbb{B}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\), one has \(\lim_{b\rightarrow \infty }H_{\omega }u_{b}=H_{\omega }f\). From \(\lim_{b\rightarrow \infty }H_{\omega }u_{b}=\lim_{b\rightarrow \infty }H_{\omega }f_{b}=g\). Therefore, \(H_{\omega }f=g\). So \(g\in \operatorname{Range}(H_{\omega })\). Hence, \(\operatorname{Range}(H_{\omega })\) is closed. Then, let the necessity conditions be verified. Hence, there is \(\varrho >0\) such that \(\upsilon (H_{\omega }f)\geq \varrho \upsilon (f)\) for all \(f\in ((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon } )_{ (\ker ( \omega ) )^{c}}\). Suppose \(K= \{b\in (\ker (\omega ) )^{c}: \vert \omega _{b} \vert <\varrho \}\neq \emptyset \), hence if \(a_{0}\in K\), we have

$$\begin{aligned} \upsilon (H_{\omega }e_{a_{0}})&=\upsilon \bigl( \bigl( \omega _{b}(e_{a_{0}})_{b} \bigr) \bigr)_{b=0}^{\infty } )= \sum^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{b=0}r_{1}^{b}r_{2}^{l-b}\omega _{b}(e_{a_{0}})_{b}) \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\&< \sum ^{\infty }_{l=0} \biggl( \frac{ \vert \sum^{l}_{b=0}r_{1}^{b}r_{2}^{l-b}(e_{a_{0}})_{b}\varrho \vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} =\varrho ^{t}\upsilon (e_{a_{0}}), \end{aligned}$$

this implies unreliability. Hence, \(K=\phi \), one can see \(\vert \omega _{b} \vert \geq \varrho \) for every \(b\in (\ker (\omega ) )^{c}\). This shows the theorem. □

Theorem 3.8

If \(\omega \in \mathcal{C}^{\mathrm{N}}\), \(1< t<\infty \), and \(r_{1}\leq r_{2}\). Then there are \(\alpha >0\) and \(\eta >0\) such that \(\alpha < \vert \omega _{b} \vert <\eta \) for all \(b\in \mathrm{N}\) if and only if \(H_{\omega }\in \mathbb{B}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\) is invertible.

Proof

Let the sufficient setting be confirmed. Assume \(\kappa \in \mathcal{C}^{\mathrm{N}}\) with \(\kappa _{b}=\frac{1}{\omega _{b}}\). By Theorem 3.2, one has \(H_{\omega }\in \mathbb{B}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\) and \(H_{\kappa }\in \mathbb{B}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\). Therefore, \(H_{\omega }.H_{\kappa }=H_{\kappa }.H_{\omega }=I\). Therefore, \(H_{\kappa }=H^{-1}_{\omega }\). Then, let \(H_{\omega }\) be invertible. So \(\operatorname{Range}(H_{\omega })= ((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon } )_{ \mathrm{N}}\). Hence \(\operatorname{Range}(H_{\omega })\) is closed. By Theorem 3.7, one has \(\alpha >0\) such that \(\vert \omega _{b} \vert \geq \alpha \) with \(b\in (\ker (\omega ) )^{c}\). We have \(\ker (\omega )=\emptyset \), if \(\omega _{b_{0}}=0\) for \(b_{0}\in \mathrm{N}\), this gives \(e_{b_{0}}\in \ker (H_{\omega })\), which is unreliability, as \(\ker (H_{\omega })\) is trivial. Hence, \(\vert \omega _{b} \vert \geq \alpha \) with \(b\in \mathrm{N}\). As \(H_{\omega }\in \ell _{\infty }\). From Theorem 3.2, we have \(\eta >0\) such that \(\vert \omega _{b} \vert \leq \eta \) with \(b\in \mathrm{N}\). Hence, we have \(\alpha \leq \vert \omega _{b} \vert \leq \eta \) with \(b\in \mathrm{N}\). □

Theorem 3.9

If \(\omega \in \mathcal{C}^{\mathrm{N}}\), \(1< t<\infty \), \(r_{1}\leq r_{2}\), and \(H_{\omega }\in \mathbb{B}((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon })\). Then \(H_{\omega }\) is a Fredholm mapping if and only if (i) \(\ker (\omega )\subsetneqq \mathrm{N}\) is finite and (ii) \(\vert \omega _{b} \vert \geq \varrho \) with \(b\in (\ker (\omega ) )^{c}\).

Proof

Let the sufficient conditions be confirmed. Assume that \(\ker (\omega )\subsetneqq \mathrm{N}\) is infinite, so \(e_{b}\in \ker (H_{\omega })\) for \(b\in \ker (\omega )\). As \(e_{b}\)s are linearly independent, this implies that \(\dim (\ker (H_{\omega }))=\infty \), which is unreliability. As \(\ker (\omega )\subsetneqq \mathrm{N}\) must be finite. Setting (ii) follows from Theorem 3.7. Next, suppose that conditions (i) and (ii) are satisfied. By Theorem 3.7, one has that setting (ii) gives that \(\operatorname{Range}(H_{\omega })\) is closed. Condition (i) implies that \(\dim (\ker (H_{\omega }))<\infty \) and \(\dim ((\operatorname{Range}(H_{\omega }))^{c})<\infty \). This gives that \(H_{\omega }\) is Fredholm. □

Configuration of pre-quasi ideal

In this section, firstly, we examine the sufficient conditions (not necessary) on \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) so that \(\overline{\mathbb{F}}=\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{ \upsilon }}\), which implies a negative example of Rhoades [24] open problem about the linearity of s-type \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) spaces. Secondly, we give the set-up on \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) so as to \(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}\) is Banach and closed. Thirdly, we introduce the sufficient conditions on \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) such that \(\mathbb{B}^{\alpha }_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}\) is closely included for distinct t and minimum. Fourthly, we explain the conditions so that the Banach pre-quasi ideal \(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}\) is simple. Fifthly, we give the sufficient conditions on \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) such that the class \(\mathbb{B}\) has its sequence of eigenvalues in \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) closely included in \(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}\).

Denseness of finite rank mappings

Theorem 4.1

The settings \(1< t<\infty \) and \(r_{1}\leq r_{2}\) are sufficient only for \(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }} (\mathcal{P}, \mathcal{Q})=\overline{\mathbb{F}(\mathcal{P}, \mathcal{Q})}\).

Proof

Assume that the sufficient setting is confirmed. Since \(e_{l}\in (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) for every \(l\in \mathrm{N}\) and \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) is a linear space. Assume \(Z\in \mathbb{F}(\mathcal{P}, \mathcal{Q})\), we have \((s_{l}(Z))_{l=0}^{\infty }\in \mathcal{F}\). Therefore, \((s_{l}(Z))_{l=0}^{\infty }\in (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\), and this implies \(Z\in \mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }} ( \mathcal{P}, \mathcal{Q})\). So \(\overline{\mathbb{F}(\mathcal{P}, \mathcal{Q})}\subseteq \mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{ \upsilon }} (\mathcal{P}, \mathcal{Q})\). To prove that \(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}(\mathcal{P}, \mathcal{Q})\subseteq \overline{\mathbb{F}(\mathcal{P}, \mathcal{Q})}\). Since \(1< t<\infty \), we get \(\sum_{l=0}^{\infty } ( \frac{1}{[l+1]_{r_{1},r_{2}}} )^{t}<\infty \). Suppose \(Z\in \mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}(\mathcal{P}, \mathcal{Q})\), one has \((s_{l}(Z))_{l=0}^{\infty }\in (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\). Since \(\upsilon (s_{l}(Z))_{l=0}^{\infty }<\infty \), assume \(\rho \in (0, 1)\), then there exists \(l_{0}\in \mathrm{N}-\{0\}\) such that \(\upsilon ((s_{l}(Z))_{l=l_{0}}^{\infty })<\frac{\rho }{2^{t+3}\eta d}\) for some \(d\geq 1\), where \(\eta =\max \{1,\sum_{l=l_{0}}^{\infty } ( \frac{1}{[l+1]_{r_{1},r_{2}}} )^{t} \}\). Since \(s_{l}(Z)\) is decreasing, we have

$$\begin{aligned} \sum_{l=l_{0}+1}^{2l_{0}} \biggl( \frac{ \sum_{j=0}^{l}r_{1}^{j}r_{2}^{l-j}s_{2l_{0}}(Z)}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}&\leq \sum_{l=l_{0}+1}^{2l_{0}} \biggl( \frac{ \sum_{j=0}^{l}r_{1}^{j}r_{2}^{l-j}s_{j}(Z)}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}\leq \sum _{l=l_{0}}^{\infty } \biggl( \frac{ \sum_{j=0}^{l}r_{1}^{j}r_{2}^{l-j}s_{j}(Z)}{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ & < \frac{\rho }{2^{t+3}\eta d}. \end{aligned}$$
(2)

Hence, there exists \(Y\in \mathbb{F}_{2l_{0}}(\mathcal{P}, \mathcal{Q})\) with \(\operatorname{rank} (Y) \leq 2l_{0}\) and

$$ \sum_{l=2l_{0}+1}^{3l_{0}} \biggl( \frac{ \sum_{j=0}^{l}r_{1}^{j}r_{2}^{l-j} \Vert Z-Y \Vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \leq \sum_{l=l_{0}+1}^{2l_{0}} \biggl( \frac{ \sum_{j=0}^{l}r_{1}^{j}r_{2}^{l-j} \Vert Z-Y \Vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} < \frac{\rho }{2^{t+3}\eta d}, $$
(3)

as \(1< t<\infty \), one has

$$ \sup_{l=l_{0}}^{\infty } \Biggl( \sum _{j=0}^{l_{0}}r_{1}^{j}r_{2}^{l-j} \Vert Z-Y \Vert \Biggr)^{t}< \frac{\rho }{2^{2t+2}\eta }. $$
(4)

Hence, we have

$$ \sum_{l=0}^{l_{0}} \biggl( \frac{ \sum_{j=0}^{l}r_{1}^{j}r_{2}^{l-j} \Vert Z-Y \Vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t}< \frac{\rho }{2^{t+3}\eta d}. $$
(5)

Since \(1< t<\infty \) and by using inequalities (1)–(5), we obtain

$$\begin{aligned} d(Z, Y)&=\upsilon \bigl(s_{l}(Z-Y) \bigr)_{l=0}^{ \infty } = \sum_{l=0}^{3l_{0}-1} \biggl( \frac{ \sum_{j=0}^{l}r_{1}^{j}r_{2}^{l-j}s_{j}(Z-Y)}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}+ \sum_{l=3l_{0}}^{\infty } \biggl( \frac{ \sum_{j=0}^{l}r_{1}^{j}r_{2}^{l-j}s_{j}(Z-Y)}{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &\leq \sum_{l=0}^{3l_{0}} \biggl( \frac{ \sum_{j=0}^{l}r_{1}^{j}r_{2}^{l-j} \Vert Z-Y \Vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t}+ \sum_{l=l_{0}}^{\infty } \biggl( \frac{ \sum_{j=0}^{l+2l_{0}}r_{1}^{j}r_{2}^{l-j}s_{j}(Z-Y)}{[l+2l_{0}+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &\leq \sum_{l=0}^{3l_{0}} \biggl( \frac{ \sum_{j=0}^{l}r_{1}^{j}r_{2}^{l-j} \Vert Z-Y \Vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t}+ \sum_{l=l_{0}}^{\infty } \biggl( \frac{ \sum_{j=0}^{l+2l_{0}}r_{1}^{j}r_{2}^{l-j}s_{j}(Z-Y)}{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &\leq 3 \sum_{l=0}^{l_{0}} \biggl( \frac{ \sum_{j=0}^{i}r_{1}^{j}r_{2}^{l-j} \Vert Z-Y \Vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &\quad{} + \sum_{l=l_{0}}^{\infty } \biggl( \frac{ \sum_{j=0}^{2l_{0}-1}r_{1}^{j}r_{2}^{l-j}s_{j}(Z-Y)+ \sum_{j=2l_{0}}^{l+2l_{0}}r_{1}^{j}r_{2}^{l-j}s_{j}(Z-Y)}{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &\leq 3 \sum_{l=0}^{l_{0}} \biggl( \frac{ \sum_{j=0}^{l}r_{1}^{j}r_{2}^{l-j} \Vert Z-Y \Vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t}+ 2^{t-1} \Biggl[ \sum _{l=l_{0}}^{\infty } \biggl( \frac{ \sum_{j=0}^{2l_{0}-1}r_{1}^{j}r_{2}^{l-j}s_{j}(Z-Y)}{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &\quad{}+ \sum_{l=l_{0}}^{\infty } \biggl( \frac{ \sum_{j=2l_{0}}^{l+2l_{0}}r_{1}^{j}r_{2}^{l-j}s_{j}(Z-Y)}{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \Biggr] \\ &\leq 3 \sum_{l=0}^{l_{0}} \biggl( \frac{ \sum_{j=0}^{l}r_{1}^{j}r_{2}^{l-j} \Vert Z-Y \Vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t}+ 2^{t-1} \Biggl[ \sum _{l=l_{0}}^{\infty } \biggl( \frac{ \sum_{j=0}^{2l_{0}-1}r_{1}^{j}r_{2}^{l-j} \Vert Z-Y \Vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &\quad{}+ \sum_{l=l_{0}}^{\infty } \biggl( \frac{ \sum_{j=0}^{l}r_{1}^{j+2l_{0}}r_{2}^{l-j-2l_{0}}s_{j+2l_{0}}(Z-Y)}{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \Biggr] \\ &\leq 3 \sum_{l=0}^{l_{0}} \biggl( \frac{ \sum_{j=0}^{l}r_{1}^{j}r_{2}^{l-j} \Vert Z-Y \Vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &\quad{}+2^{t-1}\sup _{l=l_{0}}^{\infty } \Biggl( \sum_{j=0}^{2l_{0}-1}r_{1}^{j}r_{2}^{l-j} \Vert Z-Y \Vert \Biggr)^{t} \sum_{l=l_{0}}^{\infty } \bigl([l+1]_{r_{1},r_{ 2}} \bigr)^{-t} \\ &\quad{}+ 2^{t-1} \sum_{l=l_{0}}^{\infty } \biggl( \frac{ \sum_{j=0}^{l}r_{1}^{j}r_{2}^{l-j}s_{j}(Z)}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}< \rho . \end{aligned}$$

On the other hand, since \(I_{5}\in \mathbb{B}^{s}_{(\mathit{ces}_{0,1}^{1})_{ \upsilon }}(\mathcal{P}, \mathcal{Q})\) but \(t>1\) is not confirmed. This completes the proof. □

Corollary 4.2

Pick up \(1< t<\infty \) and \(0< r\leq 1\), then \(\mathbb{B}^{s}_{(\chi _{r}^{t})_{\upsilon }}(\mathcal{P}, \mathcal{Q})= \overline{\mathbb{F}(\mathcal{P}, \mathcal{Q})}\).

Banach and closed pre-quasi ideal

Theorem 4.3

Assume that \((\mathcal{V})_{\upsilon }\) is a pre-modular \(\mathfrak{pss}\), then the functional Ψ is a pre-quasi norm on \(\mathbb{B}^{s}_{(\mathcal{V})_{\upsilon }}\) with \(\Psi (Z)=\upsilon (s_{b}(Z))_{b=0}^{\infty }\) and \(Z\in \mathbb{B}^{s}_{(\mathcal{V})_{\upsilon }}(\mathcal{P}, \mathcal{Q})\).

Proof

Suppose that \((\mathcal{V})_{\upsilon }\) is a pre-modular \(\mathfrak{pss}\), so Ψ verifies the next set-up:

  1. (1)

    When \(X\in \mathbb{B}^{s}_{(\mathcal{V})_{\upsilon }}(\mathcal{P}, \mathcal{Q})\), \(\Psi (X)=\upsilon (s_{b}(X))_{b=0}^{\infty }\geq 0\) and \(\Psi (X)=\upsilon (s_{b}(X))_{b=0}^{\infty }=0\) if and only if \(s_{b}(X)=0\) for all \(b\in \mathrm{N}\) if and only if \(X=0\);

  2. (2)

    We have \(E_{0}\geq 1\) with \(\Psi (\rho X)=\upsilon (s_{b}(\rho X))_{b=0}^{\infty }\leq E_{0} \vert \rho \vert \Psi (X)\) for every \(X\in \mathbb{B}^{s}_{(\mathcal{V})_{\upsilon }}(\mathcal{P}, \mathcal{Q})\) and \(\rho \in \mathcal{C}\);

  3. (3)

    One has \(D\geq 1\) so that, for \(X_{1},X_{2}\in \mathbb{B}^{s}_{(\mathcal{V})_{\upsilon }}(\mathcal{P}, \mathcal{Q})\), one can see

    $$ \begin{aligned} \Psi (X_{1}+X_{2})={}&\upsilon \bigl(s_{b}(X_{1}+X_{2}) \bigr)_{b=0}^{ \infty } \leq G_{0} \bigl(\upsilon \bigl(s_{[\frac{b}{2}]}(X_{1}) \bigr)_{b=0}^{ \infty }+\upsilon \bigl(s_{[\frac{b}{2}]}(X_{2}) \bigr)_{b=0}^{\infty } \bigr) \\\leq {}&G_{0}D_{0} \bigl(\upsilon \bigl(s_{b}(X_{1}) \bigr)_{b=0}^{\infty }+ \upsilon \bigl(s_{b}(X_{2}) \bigr)_{b=0}^{\infty } \bigr) \\ \leq{}&D \bigl[\Psi (X_{1})+\Psi (X_{2}) \bigr]; \end{aligned} $$
  4. (4)

    We have \(\varrho \geq 1\) if \(X\in \mathbb{B}(\mathcal{P}_{0}, \mathcal{P})\), \(Y\in \mathbb{B}^{s}_{(\mathcal{V})_{\upsilon }}(\mathcal{P}, \mathcal{Q})\), and \(Z\in \mathbb{B}(\mathcal{Q}, \mathcal{Q}_{0})\), then \(\Psi (ZYX)=\upsilon (s_{b}(ZYX))_{b=0}^{\infty }\leq \upsilon ( \Vert X \Vert \Vert Z \Vert s_{b}(Y))_{b=0}^{\infty }\leq \varrho \Vert X \Vert \Psi (Y) \Vert Z \Vert \).

 □

Theorem 4.4

If \(1< t<\infty \) and \(r_{1}\leq r_{2}\), then \((\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }},\Psi )\) is a pre-quasi Banach ideal.

Proof

Since \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) is a pre-modular \(\mathfrak{pss}\), by using Theorem 4.3, Ψ is a pre-quasi norm on \(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}\). If \((X_{b})_{b\in \mathrm{N}}\) is a Cauchy sequence in \(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}(\mathcal{P}, \mathcal{Q})\). As \(\mathbb{B}(\mathcal{P}, \mathcal{Q})\supseteq \mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{ \upsilon }}(\mathcal{P}, \mathcal{Q})\), we have

$$ \Psi (X_{a}-X_{b})= \sum^{\infty }_{l=0} \biggl( \frac{\sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z} s_{z}(X_{a}-X_{b})}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}\geq \sum ^{\infty }_{l=0} \biggl( \frac{r_{2}^{l} \Vert X_{a}-X_{b} \Vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \geq \Vert X_{a}-X_{b} \Vert ^{t}, $$

hence \((X_{b})_{b\in \mathrm{N}}\) is a Cauchy sequence in \(\mathbb{B}(\mathcal{P}, \mathcal{Q})\). As \(\mathbb{B}(\mathcal{P}, \mathcal{Q})\) is a Banach space, there exists \(X\in \mathbb{B}(\mathcal{P}, \mathcal{Q})\) such that \(\lim_{b\rightarrow \infty } \Vert X_{b}-X \Vert =0\). As \((s_{l}(X_{b}))_{l=0}^{\infty }\in (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) with \(b\in \mathrm{N}\). Hence, by using Definition 2.8 parts (ii), (iii), and (v), we get

$$ \begin{aligned} \Psi (X)&= \sum^{\infty }_{l=0} \biggl( \frac{\sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z} s_{z}(X)}{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &\leq 2^{t-1} \sum ^{\infty }_{l=0} \biggl( \frac{\sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z} s_{[\frac{z}{2}]}(X-X_{b})}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}+ 2^{t-1} \sum^{\infty }_{l=0} \biggl( \frac{\sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z} s_{[\frac{z}{2}]}(X_{b})}{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &\leq 2^{t-1} \sum^{\infty }_{l=0} \biggl( \frac{\sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z} \Vert X-X_{b} \Vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t}+ 2^{t-1}D_{0} \sum^{\infty }_{l=0} \biggl( \frac{\sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z} s_{z}(X_{b})}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}< \infty . \end{aligned} $$

Therefore, \((s_{l}(X))_{l=0}^{\infty }\in (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\), hence \(X\in \mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}(\mathcal{P}, \mathcal{Q})\). □

Theorem 4.5

If \(\mathcal{P}\), \(\mathcal{Q}\) are normed spaces, \(1< t<\infty \), and \(r_{1}\leq r_{2}\), then \((\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }},\Psi )\) is a pre-quasi closed ideal.

Proof

Since \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) is a pre-modular \(\mathfrak{pss}\), by following Theorem 4.3, we have Ψ is a pre-quasi norm on \(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}\). Let \(X_{b}\in \mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}( \mathcal{P}, \mathcal{Q})\) with \(b\in \mathrm{N}\) and \(\lim_{b\rightarrow \infty }\Psi (X_{b}-X)=0\). Since \(\mathbb{B}(\mathcal{P}, \mathcal{Q})\supseteq \mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{ \upsilon }}(\mathcal{P}, \mathcal{Q})\), one has

$$ \Psi (X-X_{b})= \sum^{\infty }_{l=0} \biggl( \frac{\sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z} s_{z}(X-X_{b})}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}\geq \sum ^{\infty }_{l=0} \biggl( \frac{r_{2}^{l} \Vert X-X_{b} \Vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t}\geq \Vert X-X_{b} \Vert ^{t}, $$

so \((X_{b})_{b\in \mathrm{N}}\) is a convergent sequence in \(\mathbb{B}(\mathcal{P}, \mathcal{Q})\). As \((s_{l}(X_{b}))_{l=0}^{\infty }\in (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) with \(b\in \mathrm{N}\). From Definition 2.8 parts (ii), (iii), and (v), we have

$$ \begin{aligned} \Psi (X)&= \sum^{\infty }_{l=0} \biggl( \frac{\sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z} s_{z}(X)}{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\&\leq 2^{t-1} \sum ^{\infty }_{l=0} \biggl( \frac{\sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z} s_{[\frac{z}{2}]}(X-X_{b})}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}+ 2^{t-1} \sum^{\infty }_{l=0} \biggl( \frac{\sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z} s_{[\frac{z}{2}]}(X_{b})}{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &\leq 2^{t-1} \sum^{\infty }_{l=0} \biggl( \frac{\sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z} \Vert X-X_{b} \Vert }{[l+1]_{r_{1},r_{2}}} \biggr)^{t}+ 2^{t-1}D_{0} \sum^{\infty }_{l=0} \biggl( \frac{\sum^{l}_{z=0}r_{1}^{z}r_{2}^{l-z} s_{z}(X_{b})}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}< \infty . \end{aligned} $$

One can see \((s_{l}(X))_{l=0}^{\infty }\in (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\), hence \(X\in \mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}(\mathcal{P}, \mathcal{Q})\). □

Minimum pre-quasi ideal

Theorem 4.6

If \(\mathcal{P}\) and \(\mathcal{Q}\) are infinite dimensional Banach spaces, \(1< t_{1}< t_{2}<\infty \), and \(r_{1}\leq r_{2}\), then

$$ \mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{1}} )_{\upsilon }}( \mathcal{P}, \mathcal{Q}) \varsubsetneqq \mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{2}} )_{\upsilon }}(\mathcal{P}, \mathcal{Q}) \subsetneqq \mathbb{B}( \mathcal{P}, \mathcal{Q}). $$

Proof

Assume \(Z\in \mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{1}} )_{ \upsilon }}(\mathcal{P}, \mathcal{Q})\), then \((s_{l}(Z))\in (\mathit{ces}_{r_{1},r_{2}}^{t_{1}} )_{\upsilon }\). We have

$$ \sum_{l=0}^{\infty } \biggl( \frac{\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}s_{z}(Z)}{[l+1]_{r_{1},r_{2}}} \biggr)^{t_{2}}< \sum_{l=0}^{\infty } \biggl( \frac{\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}s_{z}(Z)}{[l+1]_{r_{1},r_{2}}} \biggr)^{t_{1}}< \infty , $$

then \(Z\in \mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{2}} )_{ \upsilon }}(\mathcal{P}, \mathcal{Q})\). Next, by taking \((s_{l}(Z))_{l=0}^{\infty }\) such that \(\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}s_{z}(Z)= \frac{[l+1]_{r_{1},r_{2}}}{\sqrt[t_{1}]{l+1}}\), one has \(Z\in \mathbb{B}(\mathcal{P}, \mathcal{Q})\) with

$$ \sum_{l=0}^{\infty } \biggl( \frac{\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}s_{z}(Z)}{[l+1]_{r_{1},r_{2}}} \biggr)^{t_{1}}=\sum_{l=0}^{\infty } \frac{1}{l+1}=\infty $$

and

$$ \sum_{l=0}^{\infty } \biggl( \frac{\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}s_{z}(Z)}{[l+1]_{r_{1},r_{2}}} \biggr)^{t_{2}}=\sum_{l=0}^{\infty } \biggl( \frac{1}{l+1} \biggr)^{{ \frac{t_{2}}{t_{1}}}}< \infty . $$

Therefore, \(Z\notin \mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{1}} )_{ \upsilon }}(\mathcal{P}, \mathcal{Q})\) and \(Z\in \mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{2}} )_{ \upsilon }}(\mathcal{P}, \mathcal{Q})\). Evidently, \(\mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{2}} )_{\upsilon }}( \mathcal{P}, \mathcal{Q})\subset \mathbb{B}(\mathcal{P}, \mathcal{Q})\). Then, by choosing \((s_{l}(Z))_{l=0}^{\infty }\) with \(\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}s_{z}(Z)= \frac{[l+1]_{r_{1},r_{2}}}{\sqrt[t_{2}]{l+1}}\), one can conclude \(Z\in \mathbb{B}(\mathcal{P}, \mathcal{Q})\) and \(Z\notin \mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{2}} )_{ \upsilon }}(\mathcal{P}, \mathcal{Q})\). □

Corollary 4.7

If \(\mathcal{P}\) and \(\mathcal{Q}\) are infinite dimensional Banach spaces, \(1< q< t<\infty \), and \(0< r\leq 1\), then

$$ \mathbb{B}^{s}_{ (\chi _{r}^{q} )_{\upsilon }}(\mathcal{P}, \mathcal{Q})\varsubsetneqq \mathbb{B}^{s}_{ (\chi _{r}^{t} )_{\upsilon }}(\mathcal{P}, \mathcal{Q})\subsetneqq \mathbb{B}( \mathcal{P}, \mathcal{Q}). $$

Theorem 4.8

\(\mathbb{B}^{\alpha }_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}\) is minimum, when \(\mathcal{P}\) and \(\mathcal{Q}\) are infinite dimensional Banach spaces, \(1< t<\infty \), and \(r_{1}\leq r_{2}\).

Proof

Suppose that the sufficient set-up is verified. Hence \((\mathbb{B}^{\alpha }_{\mathit{ces}_{r_{1},r_{2}}^{t}}, \Psi )\) is a pre-quasi Banach ideal with \(\Psi (Z)=\sum_{l=0}^{\infty } ( \frac{\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}\alpha _{z}(Z)}{[l+1]_{r_{1},r_{2}}} )^{t}\). Assume \(\mathbb{B}^{\alpha }_{\mathit{ces}_{r_{1},r_{2}}^{t}}(\mathcal{P}, \mathcal{Q})= \mathbb{B}(\mathcal{P}, \mathcal{Q})\), so we have \(\eta >0\) with \(\Psi (Z)\leq \eta \Vert Z \Vert \) for \(Z\in \mathbb{B}(\mathcal{P}, \mathcal{Q})\). From Dvoretzky’s theorem [30], for \(b\in \mathrm{N}\), we have quotient spaces \(\mathcal{P}/Y_{b}\) and subspaces \(M_{b}\) of \(\mathcal{Q}\) which can be mapped onto \(\ell _{2}^{b}\) by isomorphisms \(V_{b}\) and \(X_{b}\) with \(\Vert V_{b} \Vert \Vert V_{b}^{-1} \Vert \leq 2\) and \(\Vert X_{b} \Vert \Vert X_{b}^{-1} \Vert \leq 2\). Let \(I_{b}\) be the identity map on \(\ell _{2}^{b}\), \(T_{b}\) be the quotient map from \(\mathcal{P}\) onto \(\mathcal{P}/Y_{b}\), and \(J_{b}\) be the natural embedding map from \(M_{b}\) into \(\mathcal{Q}\). Suppose that \(m_{z}\) is the Bernstein numbers [31], we have

$$ \begin{aligned} 1&=m_{z}(I_{b})=m_{z} \bigl(X_{b}X_{b}^{-1}I_{b}V_{b}V_{b}^{-1} \bigr) \leq \Vert X_{b} \Vert m_{z} \bigl(X_{b}^{-1}I_{b}V_{b} \bigr) \bigl\Vert V_{b}^{-1} \bigr\Vert \\ &= \Vert X_{b} \Vert m_{z} \bigl(J_{b}X_{b}^{-1}I_{b}V_{b} \bigr) \bigl\Vert V_{b}^{-1} \bigr\Vert \leq \Vert X_{b} \Vert d_{z} \bigl(J_{b}X_{b}^{-1}I_{b}V_{b} \bigr) \bigl\Vert V_{b}^{-1} \bigr\Vert \\ &= \Vert X_{b} \Vert d_{z} \bigl(J_{b}X_{b}^{-1}I_{b}V_{b}T_{b} \bigr) \bigl\Vert V_{b}^{-1} \bigr\Vert \leq \Vert X_{b} \Vert \alpha _{z} \bigl(J_{b}X_{b}^{-1}I_{b}V_{b}T_{b} \bigr) \bigl\Vert V_{b}^{-1} \bigr\Vert \end{aligned} $$

if \(0\leq l\leq b\). One can conclude

$$ \begin{aligned} &\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z} \leq \sum_{z=0}^{l} \Vert X_{b} \Vert r_{1}^{z}r_{2}^{l-z} \alpha _{z} \bigl(J_{b}X_{b}^{-1}I_{b}V_{b}T_{b} \bigr) \bigl\Vert V_{b}^{-1} \bigr\Vert \quad \Rightarrow \\ & \biggl( \frac{\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}\leq \bigl( \Vert X_{b} \Vert \bigl\Vert V_{b}^{-1} \bigr\Vert \bigr)^{t} \biggl( \frac{\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}\alpha _{z}(J_{b}X_{b}^{-1}I_{b}V_{b}T_{b})}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}. \end{aligned} $$

Therefore, for some \(\varrho \geq 1\), we have

$$\begin{aligned} & \sum_{l=0}^{b} \biggl( \frac{\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}\leq \varrho \Vert X_{b} \Vert \bigl\Vert V_{b}^{-1} \bigr\Vert \sum _{l=0}^{b} \biggl( \frac{\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}\alpha _{z}(J_{b}X_{b}^{-1}I_{b}V_{b}T_{b})}{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \quad \Rightarrow \\ & \sum_{l=0}^{b} \biggl( \frac{\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}\leq \varrho \Vert X_{b} \Vert \bigl\Vert V_{b}^{-1} \bigr\Vert \Psi \bigl(J_{b}X_{b}^{-1}I_{b}V_{b}T_{b} \bigr) \quad \Rightarrow \\ & \sum_{l=0}^{b} \biggl( \frac{\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}\leq \varrho \eta \Vert X_{b} \Vert \bigl\Vert V_{b}^{-1} \bigr\Vert \bigl\Vert J_{b}X_{b}^{-1}I_{b}V_{b}T_{b} \bigr\Vert \quad \Rightarrow \\ & \sum_{l=0}^{b} \biggl( \frac{\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}}{[l+1]_{r_{1},r_{2}}} \biggr)^{t} \\ &\quad \leq \varrho \eta \Vert X_{b} \Vert \bigl\Vert V_{b}^{-1} \bigr\Vert \bigl\Vert J_{b}X_{b}^{-1} \bigr\Vert \Vert I_{b} \Vert \Vert V_{b}T_{b} \Vert = \varrho \eta \Vert X_{b} \Vert \bigl\Vert V_{b}^{-1} \bigr\Vert \bigl\Vert X_{b}^{-1} \bigr\Vert \Vert I_{b} \Vert \Vert V_{b} \Vert \quad \Rightarrow \\ & \sum_{l=0}^{b} \biggl( \frac{\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}\leq 4\varrho \eta . \end{aligned}$$

For \(b\rightarrow \infty \) and since \(\sum_{l=0}^{\infty } ( \frac{\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}}{[l+1]_{r_{1},r_{2}}} )^{t}= \sum_{l=0}^{\infty } ( \frac{r_{2}^{l}+r_{2}^{l-1}r_{1}+\ldots +r_{2}r_{1}^{l-1}+r_{1}^{l}}{[l+1]_{r_{1},r_{2}}} )^{t}=\sum_{l=0}^{\infty }1=\infty \). This implies unreliability. Hence \(\mathcal{P}\) and \(\mathcal{Q}\) both cannot be infinite dimensional if \(\mathbb{B}^{\alpha }_{\mathit{ces}_{r_{1},r_{2}}^{t}}(\mathcal{P}, \mathcal{Q})= \mathbb{B}(\mathcal{P}, \mathcal{Q})\). This confirms the proof. □

Theorem 4.9

If \(\mathcal{P}\) and \(\mathcal{Q}\) are infinite dimensional Banach spaces, \(1< t<\infty \), and \(r_{1}\leq r_{2}\), then \(\mathbb{B}^{d}_{\mathit{ces}_{r_{1},r_{2}}^{t}}\) is minimum.

Corollary 4.10

If \(\mathcal{P}\) and \(\mathcal{Q}\) are infinite dimensional Banach spaces, \(1< t<\infty \), and \(0< r\leq 1\), then \(\mathbb{B}^{\alpha }_{\chi _{r}^{t}}\) is minimum.

Corollary 4.11

If \(\mathcal{P}\) and \(\mathcal{Q}\) are infinite dimensional Banach spaces, \(1< t<\infty \), and \(0< r\leq 1\), then \(\mathbb{B}^{d}_{\chi _{r}^{t}}\) is minimum.

Non-trivial closed pre-quasi ideal

Theorem 4.12

If \(\mathcal{P}\) and \(\mathcal{Q}\) are infinite dimensional Banach spaces, \(r_{1}\leq r_{2}\), and \(1< t_{1}< t_{2}<\infty \), then

$$ \mathbb{B} \bigl(\mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{2}} )_{\upsilon }}(\mathcal{P}, \mathcal{Q}), \mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{1}} )_{\upsilon }}(\mathcal{P}, \mathcal{Q}) \bigr)=\mathcal{A} \bigl(\mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{2}} )_{\upsilon }}(\mathcal{P}, \mathcal{Q}), \mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{1}} )_{\upsilon }}(\mathcal{P}, \mathcal{Q}) \bigr). $$

Proof

Assume \(X\in \mathbb{B} (\mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{2}} )_{\upsilon }}(\mathcal{P}, \mathcal{Q}), \mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{1}} )_{\upsilon }}(\mathcal{P}, \mathcal{Q}) )\) and \(X\notin \mathcal{A} (\mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{2}} )_{\upsilon }}(\mathcal{P}, \mathcal{Q}), \mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{1}} )_{\upsilon }}(\mathcal{P}, \mathcal{Q}) )\). By using Lemma 1.2, we have \(Y\in \mathbb{B} (\mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{2}} )_{\upsilon }}(\mathcal{P}, \mathcal{Q}) )\) and \(Z\in \mathbb{B} (\mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{1}} )_{\upsilon }}(\mathcal{P}, \mathcal{Q}) )\) so that \(ZXY I_{b}=I_{b}\). Hence, for all \(b\in \mathrm{N}\), one has

$$ \begin{aligned} \Vert I_{b} \Vert _{\mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{1}} )_{\upsilon }}(\mathcal{P}, \mathcal{Q})}&=\sum _{l=0}^{\infty } \biggl( \frac{\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}s_{z}(I_{b})}{[l+1]_{r_{1},r_{2}}} \biggr)^{t_{1}} \\ &\leq \Vert ZXY \Vert \Vert I_{b} \Vert _{\mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{2}} )_{\upsilon }}(\mathcal{P}, \mathcal{Q})}\leq \sum_{l=0}^{ \infty } \biggl( \frac{\sum_{z=0}^{i}r_{1}^{z}r_{2}^{l-z}s_{z}(I_{b})}{[l+1]_{r_{1},r_{2}}} \biggr)^{t_{2}}. \end{aligned} $$

This contradicts Theorem 4.6. Therefore, \(X\in \mathcal{A} (\mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{2}} )_{\upsilon }}(\mathcal{P}, \mathcal{Q}), \mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{1}} )_{\upsilon }}(\mathcal{P}, \mathcal{Q}) )\), which completes the proof. □

Corollary 4.13

If \(\mathcal{P}\) and \(\mathcal{Q}\) are infinite dimensional Banach spaces, \(r_{1}\leq r_{2}\), and \(1< t_{1}< t_{2}<\infty \), then

$$ \mathbb{B} \bigl(\mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{2}} )_{\upsilon }}(\mathcal{P}, \mathcal{Q}), \mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{1}} )_{\upsilon }}(\mathcal{P}, \mathcal{Q}) \bigr)=\mathcal{K} \bigl(\mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{2}} )_{\upsilon }}(\mathcal{P}, \mathcal{Q}), \mathbb{B}^{s}_{ (\mathit{ces}_{r_{1},r_{2}}^{t_{1}} )_{\upsilon }}(\mathcal{P}, \mathcal{Q}) \bigr). $$

Proof

Definitely, since \(\mathcal{A}\subset \mathcal{K}\). □

Theorem 4.14

If \(\mathcal{P}\) and \(\mathcal{Q}\) are infinite dimensional Banach spaces, \(1< t<\infty \), and \(r_{1}\leq r_{2}\), then \(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}\) is simple.

Proof

Assume that the closed ideal \(\mathcal{K}(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}( \mathcal{P}, \mathcal{Q}))\) contains a mapping \(X\notin \mathcal{A}(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{ \upsilon }}(\mathcal{P}, \mathcal{Q}))\). By using Lemma 1.2, we have \(Y, Z\in \mathbb{B}(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}( \mathcal{P}, \mathcal{Q}))\) with \(ZXY I_{b}=I_{b}\). This implies that \(I_{\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}(\mathcal{P}, \mathcal{Q})}\in \mathcal{K}(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{ \upsilon }}(\mathcal{P}, \mathcal{Q}))\). Therefore, \(\mathbb{B}(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}( \mathcal{P}, \mathcal{Q}))=\mathcal{K}(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{ \upsilon }}(\mathcal{P}, \mathcal{Q}))\). So \(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}\) is simple. □

Spectrum of pre-quasi ideal

Notation 4.15

$$\begin{aligned}& \bigl(\mathbb{B}^{s}_{\mathcal{V}} \bigr)^{\rho }:= \bigl\{ \bigl(\mathbb{B}^{s}_{\mathcal{V}} \bigr)^{\rho }(\mathcal{P}, \mathcal{Q}); \mathcal{P} \text{ and } \mathcal{Q} \text{ are Banach spaces} \bigr\} , \quad \text{where } \\& \begin{aligned} \bigl(\mathbb{B}^{s}_{\mathcal{V}} \bigr)^{\rho }( \mathcal{P}, \mathcal{Q}):= {}&\bigl\{ X\in \mathbb{B}(\mathcal{P}, \mathcal{Q}):( \bigl(\rho _{l}(X) \bigr)_{l=0}^{ \infty }\in \mathcal{V} \text{ and } \bigl\Vert X-\rho _{l}(X)I \bigr\Vert ^{-1}\\ &{} \text{ does not exist for every } l\in \mathrm{N} \bigr\} . \end{aligned} \end{aligned}$$

Theorem 4.16

If \(\mathcal{P}\) and \(\mathcal{Q}\) are infinite dimensional Banach spaces, \(1< t<\infty \), and \(r_{1}\leq r_{2}\), then

$$ \bigl(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }} \bigr)^{ \rho }(\mathcal{P}, \mathcal{Q})\subsetneqq \mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{ \upsilon }}(\mathcal{P}, \mathcal{Q}). $$

Proof

Assume \(X\in (\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }} )^{\rho }(\mathcal{P}, \mathcal{Q})\), so \((\rho _{l}(X))_{l=0}^{\infty }\in (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\) and \(\Vert X-\rho _{l}(X)I \Vert =0\) for every \(l\in \mathrm{N}\). One has \(X=\rho _{l}(X)I\) for all \(l\in \mathrm{N}\), hence \(s_{l}(X)=s_{l}(\rho _{l}(X)I)= \vert \rho _{l}(X) \vert \) for each \(l\in \mathrm{N}\). Hence \((s_{l}(X))_{l=0}^{\infty }\in (\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\), which implies \(X\in \mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}(\mathcal{P}, \mathcal{Q})\). Next, by putting \((\rho _{l}(X))_{l=0}^{\infty }\) so that \(\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}\rho _{z}(X)= \frac{[l+1]_{r_{1},r_{2}}}{\sqrt[t]{l+1}}\), we have \(X\in \mathbb{B}(\mathcal{P}, \mathcal{Q})\) so that

$$ \sum_{l=0}^{\infty } \biggl( \frac{\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}\rho _{z}(X)}{[l+1]_{r_{1},r_{2}}} \biggr)^{t}=\sum_{l=0}^{\infty } \frac{1}{l+1}=\infty , $$

and by choosing \((s_{l}(X))_{l=0}^{\infty }\) with \(\sum_{z=0}^{l}r_{1}^{z}r_{2}^{l-z}s_{z}(X)= \frac{[l+1]_{r_{1},r_{2}}}{l+1}\). Therefore, \(X\notin (\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }} )^{\rho }(\mathcal{P}, \mathcal{Q})\) and \(X\in \mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}(\mathcal{P}, \mathcal{Q})\). This confirms the proof. □

Conclusion

Many authors in the near past investigated and studied the r-Cesàro matrix and the linked summability methods [3235]. In this paper, we explain some topological and geometric structure of the class \(\mathbb{B}^{s}_{(\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }}\) and the multiplication mappings defined on \((\mathit{ces}_{r_{1},r_{2}}^{t})_{\upsilon }\). When \(r_{1}=r\) and \(r_{2}=1\), we have \(\mathit{ces}_{r_{1},r_{2}}^{t}=\chi _{t}^{r}\). Some new properties to the sequence space \(\chi _{t}^{r}\) have been added. This article has many benefits for researchers such as studying the fixed points of any contraction maps on this pre-quasi normed sequence space, which is more general than the quasi normed sequence spaces, a new general space of solutions for many difference equations, the spectrum of any bounded linear operators between any two Banach spaces with s-numbers in this sequence space, and noting that the operator ideals are the prime structural components of a vector lattice; consequently, closed ideals are bound to play a positive role in the theory of Banach lattices. We open the way for many authors to generalize the results by a sequence \(t=(t_{l})_{l=0}^{\infty }\) and build \((\mathit{ces}_{r_{1},r_{2}}^{(t)})_{\upsilon }\) of non-absolute type.

Availability of data and materials

Not applicable.

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Acknowledgements

This work was funded by the University of Jeddah, Saudi Arabia, under grant No.(UJ-20-078-DR). The authors, therefore, acknowledge with thanks the University’s technical and financial support. Also, the authors are extremely grateful to the reviewers for their valuable suggestions and leading a crucial role for a better presentation of this manuscript.

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Correspondence to Awad A. Bakery.

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Bakery, A.A., Mohamed, O.K.S.K. \((r_{1},r_{2})\)-Cesàro summable sequence space of non-absolute type and the involved pre-quasi ideal. J Inequal Appl 2021, 43 (2021). https://doi.org/10.1186/s13660-021-02572-4

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MSC

  • 46-02
  • 47-02
  • 46C05
  • 46B10
  • 46B15
  • 46E05
  • 46E15
  • 46E30
  • 47B10

Keywords

  • \((r_{1},r_{2})\)-Cesàro matrix
  • s-numbers
  • Simple space
  • Minimum space
  • Multiplication map
  • Pre-quasi ideal
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