A New Highly Accurate Approach to Compact Operators on Some Generalized Fibonacci Difference Sequence Spaces

1. Elementary Classical Concepts

Our goal is to continue using the matrix domain while also reminding readers of the knowledge they will need to properly apply calculus in their work in following parts. To do this, we continued to modify some of the measurement theory techniques while maintaining the paper’s mathematical quality, the new sequence space’s orientation to the Hausdorff measure, its focus on earlier research, and the diversity of the theorems. The degree of rigor is essentially the same, despite the fact that many of the presentations in this new work are notably more generic than those in preceding pieces. It will be beneficial for novices to examine the five important books listed in References [1,2,3,4,5] with accessible content as part of the entire review strategy, without sacrificing the standards or breadth their consumers want to see. Let’s attempt to clarify some of the fundamentals first without going overboard with the obvious. The history of numbers dates back almost as far as humankind itself, and they were developed to satisfy the universal and scientific demand for mathematics. This still holds true today as it did when the topic first emerged. The sequences in the majority of our work will have sets of natural numbers $\mathbb{N}=\{0,1,2,...\}$ and real numbers $\mathbb{R}$ as their domains and ranges, respectively. Instead of writing ${lim}_{k\to \infty }$, ${sup}_{k\in \mathbb{N}}$, ${inf}_{k\in \mathbb{N}}$ and ${\sum }_{k=0}^{\infty }$, respectively, we will write ${lim}_k$, ${sup}_k$, ${inf}_k$ and ${\sum }_k$.

Infinite sequences and infinite series are two related issues that will be discussed in the following sections. A function whose domain is the set of natural numbers is one that produces an infinite series of numbers. The word "series" always suggests an infinite amount of terms that may be added together in a certain order. All real sequences’ vector spaces are symbolized by the symbol $\omega$. We are well aware that each subspace of $\omega$ is referred to as a sequence space. A few more notations regarding sequences are required for use in this work. $\phi$, ${\ell }_{\infty }$, c and $c_0$. should be used to indicate the sets of all finite sequences, bounded sequences, convergent sequences, and null sequences, respectively. The notation ${\ell }_p$ stands for the sequence space $\left\{x\in \omega :{\sum }_k{\left|x_k\right|}^p<\infty \right\}$ for any real integer p with $1\le p<\infty$. Additionally, the notations e and $e^{\left(n\right)}$, respectively, are used to represent the sequence $(1,1,...)$ and the sequence with 1 only in the $n^{\mathrm{th}}$ term and 0 in all other terms for each natural integer n. The n-section of any sequence x is the sum ${\sum }_{k=0}^n{x}_k{e}^{\left(k\right)}$ denoted by $x^{\left[n\right]}$ and is known as the n-section of any sequence x. Besides, $cs$ and $bs$ notations are used to display series whose partial summation sequence are converging and constrained, respectively. The term $B-space$ refers to a complete normed space. While all coordinate functionals ${\pi }_k$, denoted by ${\pi }_k\left(x\right)=x_k$, are continuous in a $K-space$, all coordinate functionals ${\pi }_k$ are continious in a topological sequence space. In essence, a $BK-space$ is a Banach space with continuous coordinates that satisfies the criteria for both a $K-space$ and a $B-space$. It is claimed that a $BK-space$ labeled as $X\supset \phi$ has $AK$ if all sequences $x=\left(x_k\right)\in X$ have the same representation, in which $x={\sum }_k{x}_k{e}^{\left(k\right)}$. As an illustration, the sequence space ${\ell }_p$$(1\le p<\infty )$ may be thought of as a $BK-space$ with the norm ${∥x∥}_p={\left({\sum }_k{\left|x_k\right|}^p\right)}^{1/p}$. Plus, because they have the norm ${∥x∥}_{\infty }={sup}_k\left|x_k\right|$, $c_0$, c, and ${\ell }_{\infty }$ likewise qualify as $BK-spaces$. Furthermore, the $BK-spaces$ $c_0$ and ${\ell }_p$ exhibit $AK$, in which $1\le p<\infty$.

If there exists a singular sequence $\left({\alpha }_n\right)$ consisting of scalars such that $x={\sum }_n{\alpha }_n{b}_n$, meaning that ${lim}_m∥x-{\sum }_{n=0}^m{\alpha }_n{b}_n∥=0$, then the sequence $\left(b_n\right)$ in a normed space X is referred to as a Schauder basis for all $x\in X$.

The following is the definition of the $\beta$-dual of a sequence space X:

$$X^{\beta }=\left\{a=\left(a_k\right)\in \omega :ax=\left(a_k{x}_k\right)\in cs\mathrm{for}\mathrm{all}x=\left(x_k\right)\in X\right\}.$$

An infinite matrix of real numbers, denoted by $A={\left(a_{nk}\right)}_{n,k=0}^{\infty }$, where $n,k\in \mathbb{N}$, can be represented as $A_n$, which denotes the sequence in the $n^{\mathrm{th}}$ row of A. Furthermore, if $x={\left(x_k\right)}_{k=0}^{\infty }\in \omega$, the A-transform of x is defined as the sequence $Ax={\left\{A_n\left(x\right)\right\}}_{n=0}^{\infty }$, where

$$A_n\left(x\right)=\sum _{k=0}^{\infty }{a}_{nk}{x}_k;\left(n\in \mathbb{N}\right)$$

(1)

provided that the series on the right-hand side converges for each $n\in \mathbb{N}$.

$\left(X,Y\right)$ refers to the class of all infinite matrices that map from X to Y, where X and Y are subsets of $\omega$. To put it another way, $A\in \left(X,Y\right)$ if and only if $A_n\in X^{\beta }$ for every $n\in N$ and $Ax\in Y$ for every $x\in X$.

The matrix domain is one method for creating a new sequence space, and a comprehensive understanding of it needs significant knowledge. Assume X is any sequence space

$$X_A=\left\{x=\left(x_k\right)\in \omega :Ax\in X\right\},$$

(2)

therefore defines the domain $X_A$ of an infinite matrix A in X. Let us also point out that $X_A$ is a sequence space. The reader is encouraged to consult the recent publications [6,7,8,9,10] on the domains of certain triangles in classical sequence spaces and related subjects.

The following outcomes are basic and frequently used [11,12].

Lemma 1. 

Let $X\supset \varphi$ and Y be a $BK$–space.

(a) Therefore, for any matrix $A\in (X,Y)$, we get $(X,Y)\subset B(X,Y)$, so indicating that for any $x\in X$, $L_A\left(x\right)=Ax$ describes an operator $L_A\in B(X,Y)$.

(b) If X has $AK$, and after that $B(X,Y)\subset (X,Y)$, meaning that there is a $A\in (X,Y)$ with any operator having $L\in B(X,Y)$ and $L\left(x\right)=Ax$ for every $x\in X$.

2. The Hausdorff measure of non-compactness

The goal of this section is to discuss the Hausdorff measure, which is used in theory and practice to characterize compact operators between Banach spaces. This part begins with unambiguous formulations of relative definitions, rules, and theorems, as well as explanatory and other demonstrative topic matter. It is organized around proved and additional theorems. The proven theorems help to demonstrate and amplify the theory, as well as to restate the key concepts required for effective learning. In several parts of mathematics, the idea of Hausdorff measure of non-compactness emerges. Under specific conditions, this idea has recently been utilized to describe compact matrix operators between $BK$–spaces.

The notion of the Hausdorff measure of non-compactness is essentially derived from the research of Goldenštein, Gohberg and Markus [13], and it was later picked up and investigated by Goldenštein and Markus [14]. Nonetheless, some of its concepts date back to the time of Kuratowski [15]. Afterwards, Darbo [16] expanded additional idea in addition to the conventional Schauder fixed point principle.

It is necessary to rephrase the notion of a compact operator in the context of infinite-dimensional Banach spaces X and Y. A linear operator L that maps from X to Y is thought to be compact if it covers the full domain of X, and furthermore, if the sequence $\left(L\left(x_n\right)\right)$ that represents the images of all bounded sequences $\left(x_n\right)$ in X under L has a convergent subsequence. In the discipline of functional analysis, $C(X,Y)$ denotes the collection of all compact operators in $B(X,Y)$.

Assume $(X,d)$ is a metric space. The open ball $B(x,t)$ is defined as the set $\{x\in X:d(x,x_0)<t\}$, where t represents the radius and $x_0$ stands for the center.

Moreover, let us describe $M\left(X\right)$ as the collection of all bounded subsets of X. In case of $Q\in M\left(X\right)$, the Hausdorff measure of the given set Q non-compactness, represented by $\chi \left(Q\right)$, is described in the following form:

$$\chi \left(Q\right)=inf\left\{ϵ>0:Q\subset \bigcup _{k=1}^{n}B(x_k,t_k),x_k\in X,t_k<ϵ(k=1,2,...),n\in \mathbb{N}\right\}.$$

The function $\chi :M_X\to [0,\infty )$ is the definition of the Hausdorff measure of non-compactness.

The applicability of the Hausdorff measure theorems to condensing operators, compact matrix operators on certain $BK$-spaces, and measures of non-compactness in Banach spaces can be found examined more thoroughly in the earlier works such as those and therein [11,17,18,19,20].

This paragraph aims to give a brief explanation of the Hausdorff measure of non-compactness operators between Banach spaces. The Hausdorff measures of non-compactness on X and Y, respectively, are ${\chi }_1$ and ${\chi }_2$ in the case of X and Y are being Banach spaces. If $L\left(Q\right)\in M\left(Y\right)$ for all $Q\in M\left(X\right)$, and if there exists $C\ge 0$ such that ${\chi }_2\left(L\left(Q\right)\right)\le C{\chi }_1\left(Q\right)$ for all $Q\in M\left(X\right)$, then the operator $L:X\to Y$ is called as $({\chi }_1$,${\chi }_2)$-bounded. The quantity

$$\left|L\right|({\chi }_1,{\chi }_2)=inf\left\{C\ge 0:{\chi }_2\left(L\left(Q\right)\right)\le C{\chi }_1\left(Q\right)\mathrm{for} \mathrm{all} Q\in \mathcal{M}X\right\}$$

is described as the $({\chi }_1,{\chi }_2)$–measure of non-compactness of L if the operator L is $({\chi }_1$, ${\chi }_2)$-bounded.

It is vital to note the fact that if both ${\chi }_1$ and ${\chi }_2$ are denoted as $\chi$, then $\left|L\right|({\chi }_1,{\chi }_2)={\left|L\right|}_{\chi }$.

In the present setting, our main goal is to give a through explanation of how the Hausdorff measure of non-compactness may be used to describe compact operators between Banach spaces. Let X and Y be Banach spaces, and let L be an element of $B(X,Y)$, illustrating that L is a bounded linear operator from X to Y. If L is non-compact, the Hausdorff measure of non-compactness of L, denoted as ${\parallel L\parallel }_{\chi }$, is described in the following form ([20], [Theorem 2.25])

$$\parallel L\parallel \chi =\chi \left(L\right(SX\left)\right).$$

(3)

Additionally, the Hausdorff measure of non-compactness ${\parallel L\parallel }_{\chi }$, which is denoted by the expression given in Ref. ([20], [Corollary 2.26]), characterizes L as a compact operator if and only if it equals zero. Furthermore, L is considered a compact operator if and only if the Hausdorff measure of non-compactness ${\parallel L\parallel }_{\chi }$ is equal to zero, as stated in ([20], [Corollary 2.26])

$${\parallel L\parallel }_{\chi }=0.$$

(4)

The identities stated in (3) and (4) form the basis for calculating the Hausdorff measure of non-compactness, abbreviated as $\chi \left(Q\right)$, for bounded sets Q in a Banach space X. The characterisation of compact operators $L\in B(X,Y)$ is made easier by these identities. When X has a Schauder basis, estimates or even identities for $\chi \left(Q\right)$ can be derived.

Theorem 1.

([13] or [20], [Theorem 2.23]) Let X be a Banach space with a Schauder basis ${\left(b_k\right)}_{k=0}^{\infty }$, $Q\in M_X$, $P_n:X\to X$ will be the projectors onto the linear span of $\{b_0,b_1,\cdots ,b_n\}$ and $R_n=I-P_n$ for $n=0,1,\dots$, in which I indicates the identity map on X. Under these conditions, the following inequality is satisfied

$${\displaystyle \frac{1}{a}}·\underset{n\to \infty }{lim\; sup}\left(\underset{x\in Q}{sup}∥{\mathcal{R}}_n\left(x\right)∥\right)\le \chi \left(Q\right)\le \underset{n\to \infty }{lim\; sup}\left(\underset{x\in Q}{sup}∥{\mathcal{R}}_n\left(x\right)∥\right),$$

in which $a={lim\; sup}_{n\to \infty }\parallel R_n\parallel$.

In particular, the following result shows how to derive the Hausdorff measure of non-compactness in the $BK$-spaces with $AK$, $c_0$, and $el{l}_p$$(1\le p<\infty )$.

Theorem 2.

([20], [Theorem 2.15]) A bounded subset of the normed space X, in which X is ${\ell }_p$ for $1\le p<\infty$ or $c_0$, is defined as Q. We can have

$$\chi \left(Q\right)=\underset{n\to \infty }{lim}\left(\underset{x\in Q}{sup}\parallel {\mathcal{R}}_n\left(x\right)\parallel \right).$$

(5)

if $P_n:X\to X$ is the operator described by $P_n\left(x\right)=x^{\left[n\right]}$ for every $x={\left(x_k\right)}_{k=0}^{\infty }\in X$ and $R_n=I-P_n$ for $n=0,1,\dots$.

The above results, together with the Hausdorff measure of non-compactness, make it quite plausible to derive both necessary and sufficient requirements for matrix operators between a Schauder basis and a $BK$-space. By presenting bounded linear operators between these Banach spaces as a result of matrix mappings over $BK$-spaces, $AK$ operators are transformed into compact operators. Currently, many academics have used this strategy in a variety of research articles (see, for example, [21,22,23,24,25,26,27,28,29,30,31]). The importance of these ideas will become clear in later conversations. In this paper, we are going to give an explanation of the matrix classes $({\ell }_1,{\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right))$ $(1\le p<\infty )$. Furthermore, by using the Hausdorff measure of non-compactness, we develop requirements for obtaining the pertinent subclasses of compact matrix operators. In addition, we develop an identity for the matrix transformation-determined norms of the bounded linear operators $L_A$.

3. The Fibonacci Difference Sequence Spaces ${\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right)$ and ${\ell }_{\infty }\left(\tilde{F}(\tilde{r},\tilde{s})\right)$

Infinite sequences have been employed in mathematics since antiquity, but they were particularly prevalent in the early history of the calculus. The mathematician Fibonacci utilized a series of integers $1,1,2,3,5,\dots$ in his work Liber Abaci (1202) during the middle times. Fibonacci sequences may already be familiar to you, but even if not, you will be able to easily comprehend the following formula. The stages in the series are often marked $1,1,2,3,5,dots$, and so on, for simplicity. The $Fibonacci$ $sequences$ $f=\left(f_n\right)$ provide a much better explanation. Using the recursion formula

$$f_n=f_{n-1}+f_{n-2};n\ge 2,$$

begins with the expression $f_0=f_1=1.$

The usage of Fibonacci sequences is widespread and provides opportunities for practical application. The Fibonacci numbers were discovered to be connected to the most dramatic distinctions in plants, certain living things in nature, and art and architecture. A number of Fibonacci sequence applications are outside the purview of this work, but the information in this part can help you get ready for more study and provide you knowledge you can use when necessary. There is a wealth of knowledge on Fibonacci numbers, especially the Golden ratio, in the reference number [32].

Let $1\le p\le \infty$ and q represent the conjugate of p throughout, that is, $q=p/(p-1)$ for $1<p<\infty$, that is, $q=p/(p-1)$ for $1<p<\infty$, $q=\infty$ for $p=1$ or $q=1$ for $p=\infty .$

In 2015, right after Kara [33], Candan and Kara [34] introduced the generalized Fibonacci difference sequence spaces ${\ell }_p\left(\widehat{F}(r,s)\right)$ and ${\ell }_{\infty }\left(\widehat{F}(r,s)\right)$. Very recently, Candan [35] has described the matrix class $({\ell }_1,{\ell }_p\left(\widehat{F}(r,s)\right))$$(1\le p<\infty )$. Now, we introduce the generalized Fibonacci difference sequence spaces ${\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right)$ and ${\ell }_{\infty }\left(\tilde{F}(\tilde{r},\tilde{s})\right)$, as follows;

$${\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right)=\left\{x=\left(x_n\right)\in \omega :\sum _n{\left|r_n\frac{f_n}{f_{n+1}}{x}_n+s_{n-1}\frac{f_{n+1}}{f_n}{x}_{n-1}\right|}^p<\infty \right\};1\le p<\infty$$

and

$${\ell }_{\infty }\left(\tilde{F}(\tilde{r},\tilde{s})\right)=\left\{x=\left(x_n\right)\in \omega :\underset{n\in \mathbb{N}}{sup}\left|r_n\frac{f_n}{f_{n+1}}{x}_n+s_{n-1}\frac{f_{n+1}}{f_n}{x}_{n-1}\right|<\infty \right\}.$$

We note here that, the generalization is made using two real number sequences that do not convergence to zero.

When we use the equivalent notation of (2) for the sequence spaces ${\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right)$ and ${\ell }_{\infty }\left(\tilde{F}(\tilde{r},\tilde{s})\right)$, related sequence spaces become

$${\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right)={\left({\ell }_p\right)}_{\tilde{F}(\tilde{r},\tilde{s})}(1\le p<\infty )\mathrm{and}\mathrm{also}{\ell }_{\infty }\left(\tilde{F}(\tilde{r},\tilde{s})\right)={\left({\ell }_{\infty }\right)}_{\tilde{F}(\tilde{r},\tilde{s})},$$

(6)

in which the matrix $\tilde{F}(\tilde{r},\tilde{s})=\left({\widehat{f}}_{nk}(\tilde{r},\tilde{s})\right)$ is described by

$${\widehat{f}}_{nk}(\tilde{r},\tilde{s})=\left\{\begin{array}{cc}{s}_n\frac{f_{n+1}}{f_n}& (k=n-1)\\ r_n\frac{f_n}{f_{n+1}}& (k=n)\\ 0& (0\le k<n-1)\mathrm{or}(k>n)\end{array}\right.;(n,k\in \mathbb{N}).$$

(7)

To signal the fact that the sequence spaces ${\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right)$ and ${\ell }_{\infty }\left(\tilde{F}(\tilde{r},\tilde{s})\right)$ are $BK$-spaces according to the

$${∥x∥}_{{\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right)}={\left(\sum _n{\left|y_n\left(x\right)\right|}^p\right)}^{1/p};(1\le p<\infty )\mathrm{and}{∥x∥}_{{\ell }_{\infty }\left(\tilde{F}(\tilde{r},\tilde{s})\right)}=\underset{n\in \mathbb{N}}{sup}\left|y_n\left(x\right)\right|,$$

(8)

norms defined on them, respectively, in which the sequence $y=\left(y_n\right)=\left(\tilde{F}{(\tilde{r},\tilde{s})}_n\left(x\right)\right)$ which is the $\tilde{F}(\tilde{r},\tilde{s})$-transform of any sequence $x=\left(x_n\right)$ is used. That is

$$y_n=\tilde{F}{(\tilde{r},\tilde{s})}_n\left(x\right)=\left\{\begin{array}{cc}{r}_0\frac{f_0}{f_1}{x}_0=r_0{x}_0& (n=0)\\ r_n\frac{f_n}{f_{n+1}}{x}_n+s_{n-1}\frac{f_{n+1}}{f_n}{x}_{n-1}& (n\ge 1)\end{array}\right.;(n\in \mathbb{N}).$$

(9)

The fact that the results of this present study are more through than those of Alotaibi et al. [36] in 2015 and Candan [35] in 2024 should be noted.

4. Main results

Although this article cannot cover all possible uses of compact operators, the information in this part can help you grasp the topic and help you retain knowledge that you can need later on. From a historical standpoint, the current understanding of the Hausdorff measure is the result of the combined work of many people. But Goldenštein, Gohberg, and Markus proposed the idea of non-compactness’ Hausdorff measure in 1957, and Goldenštein and Markus went on to further develop it. In the study [37], the sequence spaces Y, ${\ell }_{\infty }$, $c_0$, and c are considered, letting the characterization of the classes $({\ell }_p\left(\widehat{F}\right),Y)$, $({\ell }_{\infty }\left(\widehat{F}\right),Y)$, $({\ell }_1\left(\widehat{F}\right),Y)$, as well as the compact operators $({\ell }_p\left(\widehat{F}\right),{\ell }_1)$ and $({\ell }_1\left(\widehat{F}\right),{\ell }_p)$. In this study, we introduce the classes $B({\ell }_1,{\ell }_p^{\lambda })$ for $1\le p<\infty$ and compute the operator norms in $B({\ell }_1,{\ell }_p^{\lambda })$. Furthermore, leveraging the findings from the earlier parts of the present study, we describe the classes $C({\ell }_1,{\ell }_p)$ for $1\le p<\infty$ and determine the Hausdorff measure of non-compactness for operators in $B({\ell }_1,{\ell }_p^{\lambda })$.

Again it is assumed that $1\le p<\infty$. Now a characterization of $B({\ell }_1,{\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right))$ is provided together with the computation of the operator norms in $B({\ell }_1,{\ell }_p\left(\widehat{F}(\tilde{r},\tilde{s})\right))$. Additionally, one can use those results given in the earlier sections to both specify the Hausdorff measure of non-compactness for operators in $B({\ell }_1,{\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right))$ and characterize the classes $C({\ell }_1,{\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right))$ for $1\le p<\infty$.

It is important to note that the following result has a particular advantage in specific proofs.

Lemma 2.

([20], [Theorem 3.8]) T is a triangular matrix and with it X and Y being any two sequence spaces; for the matrix A to be an element of the $(X,Y_T)$ class, the necessary and sufficient condition is that $C=T·A$ and the matrix C belongs to the class $(X,Y)$. In addition, if the X and Y are $BK$-spaces, and also if the matrix A is an element of the class $(X,Y_T)$, then

$$\parallel L_A\parallel =\parallel L_C\parallel .$$

(10)

Next, the identities for those operator norm and the characterizations of the classes $B({\ell }_1,{\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right))$ for $(1\le p<\infty )$ are described.

Theorem 3.

Let $1\le p<\infty$.

(a) We have $L\in B({\ell }_1,{\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right))$ if and only if there exists an infinite matrix $A\in ({\ell }_1,{\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right))$ such that

$$\parallel A\parallel =\underset{k}{sup}{\left(\sum _n{\left|r_n\frac{f_n}{f_{n+1}}{a}_{nk}+s_{n-1}\frac{f_{n+1}}{f_n}{a}_{n-1,k}\right|}^p\right)}^{1/p}<\infty$$

(11)

and

$$L\left(x\right)=Axforallx\in {\ell }_1.$$

(12)

(b) If $L\in B({\ell }_1,{\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right))$ then

$$\parallel L\parallel =\parallel A\parallel .$$

(13)

Proof. 

For (a), if one keeps in mind the fact that ${\ell }_1$ is a $BK$-space with $AK$, for $L\in B({\ell }_1,{\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right))$ from Lemma 1 when the condition $1\le p<\infty$ hypothesis condition; the necessary and sufficient condition is that there is an infinite matrix A such that $A\in ({\ell }_1,{\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right))$ provided that the condition (12) is satisfied. If we denote the product of the matrices $\tilde{F}(\tilde{r},\tilde{s})=\left({\tilde{f}}_{nk}(\tilde{r},\tilde{s})\right)$ and $A=\left(a_{nk}\right)$ by $C=\left(c_{nk}\right)$, one can express it neatly in the following form

$$c_{nk}=r_n\frac{f_n}{f_{n+1}}{a}_{nk}+s_{n-1}\frac{f_{n+1}}{f_n}{a}_{n-1,k}.$$

At the moment, it is quiet easy to tell the fact that using Lemma 2(a) that the necessary and sufficient condition $A\in ({\ell }_1,{\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right))$ is $C\in ({\ell }_1,{\ell }_p)$. If the [Example 8.4.1D] in the reference [12] is utilized at this level of the proof, one can obviously see that the necessary and sufficient condition for $C\in ({\ell }_1,{\ell }_p)$ is

$$\parallel C\parallel =\underset{k}{sup}{\left(\sum _{n=0}^{\infty }{\left|c_{nk}\right|}^p\right)}^{1/p}<\infty ,$$

which proves the claim. □

(b) First of all, we are going to illustrate that $\parallel L\parallel \le \parallel A\parallel$. Let $L\in B({\ell }_1,{\ell }_p^{\lambda })$. Again, one can see from (10) that $\parallel L\parallel =\parallel L_C\parallel$ for $L_C\in B({\ell }_1,{\ell }_p)$ is given by the equation $L_C\left(x\right)=Cx$ for every $x\in {\ell }_1$. At the moment, one can state using the Minkowsky’s inequality that we can write the following expressions

$$\begin{array}{cc}\hfill \parallel L_C{\left(x\right)\parallel }_p& ={\left(\sum _{n=0}^{\infty }{\left|\sum _{k=0}^{\infty }{c}_{nk}{x}_k\right|}^p\right)}^{1/p}\hfill \\ \hfill & \le \sum _{k=0}^{\infty }\left|x_k\right|{\left(\sum _{n=0}^{\infty }{\left|c_{nk}\right|}^p\right)}^{1/p}\hfill \\ \hfill & \le \parallel C\parallel ·\parallel x\parallel \hfill \\ \hfill & =\parallel A\parallel ·\parallel x\parallel ,\hfill \end{array}$$

and from here one can state the following inequality

$$\parallel L\parallel \le \parallel A\parallel$$

(14)

for the norms of L and A. Now, it is time to prove the other side of this inequality. For this purpose, if the condition $e^{\left(k\right)}\in S_{{\ell }_1}$$(k\in N)$ is considered, one sees that

$$\parallel L\parallel \ge \parallel A\parallel$$

(15)

from the equation below

$$∥L_C\left(e^{\left(k\right)}\right)∥={\left(\sum _{n=0}^{\infty }{\left|c_{nk}\right|}^p\right)}^p.$$

When (14) and (15) are considered together, it is proved that (13).

The Hausdorff measure of the non-compactness of operators in $B({\ell }_1,{\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right))$ are going to be given as stated in the following expression. Another closely related result which is going to be utilized in the first proof will be given below.

Lemma 3.

([38], [Theorem 4.2]) Let T be a triangle and χ and ${\chi }_T$ be the Hausdorff measures of non-compactness on $M_X$ and $M_{X_T}$, respectively. Assume that X is a linear metric space with a translation invariant metric. If $Q\in M_{X_T}$, then ${\chi }_T\left(Q\right)=\chi \left(TQ\right)$.

Theorem 4.

Let $L\in B({\ell }_1,{\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right))$ with $(1\le p<\infty )$ and A demonstrate the matrix which stands for L. In that case we get

$${\parallel L\parallel }_{{\chi }_{{\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right)}}=\underset{m\to \infty }{lim}{\left(\underset{k}{sup}\sum _{n=m}^{\infty }{\left|r_n\frac{f_n}{f_{n+1}}{a}_{jk}+s_{n-1}\frac{f_{n+1}}{f_n}{a}_{j-1,k}\right|}^p\right)}^{1/p}.$$

Proof. 

Firstly, one can write in a brief manner $S=S_{{\ell }_1}$, also $C^{\left[m\right]}$$(m\in \mathbb{N})$ for the matrix having the rows $C_n^{\left[m\right]}=0$ for $0\le n\le m$ and $C_n^{\left[m\right]}=C_n$ for $n\ge m+1$. Under these conditions, when both Lemma 3 and together with (3), (5), (11) and (13) are used, the following equations may easily be computed as

$$\begin{array}{cc}\hfill {\parallel L\parallel }_{{\chi }_{{\ell }_p\left(\widehat{F}(\tilde{r},\tilde{s})\right)}}& ={\chi }_{{\ell }_p\left(\widehat{F}(\tilde{r},\tilde{s})\right)}\left(L\left(S\right)\right)\hfill \\ \hfill & ={\chi }_{{\ell }_p}\left(L_C\left(S\right)\right)\hfill \\ \hfill & =\underset{m\to \infty }{lim}\left(\underset{x\in S}{sup}{∥{\mathcal{R}}_m\left(Cx\right)∥}_p\right)\hfill \\ \hfill & =\underset{m\to \infty }{lim}\left(\underset{x\in S}{sup}{\parallel C^{\left[m\right]}x\parallel }_p\right)\hfill \\ \hfill & =\underset{m\to \infty }{lim}∥C^{\left[m\right]}∥\hfill \\ \hfill & =\underset{m\to \infty }{lim}{\left(\underset{k}{sup}\sum _{n=m}^{\infty }{\left|r_n\frac{f_n}{f_{n+1}}{a}_{jk}+s_{n-1}\frac{f_{n+1}}{f_n}{a}_{j-1,k}\right|}^p\right)}^{1/p}.\hfill \end{array}$$

□

One can see that in fact this is the required result.

Now it is high time to present the following theorem, which gives the characterization of $C({\ell }_1,{\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right))$ by coordinating the condition presented in (4) and Theorem 4.

Theorem 5.

$(1\le p<\infty )$, if $L\in B({\ell }_1,{\ell }_p\left(\tilde{F}(\tilde{r},\tilde{s})\right))$ and at the same time the matrix A is the matrix representing L, a necessary and sufficient condition for L to be compact is that the following limit is equal to zero, that is

$$\underset{m\to \infty }{lim}\left(\underset{k}{sup}\sum _{n=m}^{\infty }{\left|r_n\frac{f_n}{f_{n+1}}{a}_{jk}+s_{n-1}\frac{f_{n+1}}{f_n}{a}_{j-1,k}\right|}^p\right)=0.$$