# Compact operators on sequence spaces associated with the Copson matrix of order α

## Abstract

In this work, we study characterizations of some matrix classes $$(\mathcal{C}^{(\alpha )}(\ell ^{p}),\ell ^{\infty })$$, $$(\mathcal{C}^{(\alpha )}(\ell ^{p}),c)$$, and $$(\mathcal{C}^{(\alpha )}(\ell ^{p}),c^{0})$$, where $$\mathcal{C}^{(\alpha )}(\ell ^{p})$$ is the domain of Copson matrix of order α in the space $$\ell ^{p}$$ ($$0< p<1$$). Further, we apply the Hausdorff measures of noncompactness to characterize compact operators associated with these matrices.

## Introduction

By $$l^{\diamond }= \{ \zeta =(\zeta _{k}):\text{each }\xi _{k} \text{ is real} \}$$. The sequence space $$\ell ^{p}$$ is defined by

$$\ell ^{p}:= \Biggl\{ \zeta =(\zeta _{k})\in l^{\diamond }:\sum_{k=0}^{ \infty } \vert \zeta _{k} \vert ^{p}< \infty , p>0 \Biggr\} .$$

This is a Banach space with the norm

$$\Vert \zeta \Vert _{\ell ^{p}}= \Biggl( \sum _{k=0}^{\infty } \vert \zeta _{k} \vert ^{p} \Biggr) ^{1/p}< \infty \quad (1\leq p< \infty )$$

and complete p-normed space with the p-norm

$$\Vert \zeta \Vert _{\ell ^{p}}=\sum_{k=0}^{\infty } \vert \zeta _{k} \vert ^{p}< \infty\quad (0< p< 1).$$

Further,

\begin{aligned}& c^{0}:= \bigl\{ \zeta =(\zeta _{k})\in l^{\diamond }: \zeta _{k} \rightarrow 0 (k\rightarrow \infty ) \bigr\} , \\& c:= \Bigl\{ \zeta =(\zeta _{k})\in l^{\diamond }:\lim _{k\rightarrow \infty }\zeta _{k} \text{ exists} \Bigr\} , \\& \ell ^{\infty }:= \Bigl\{ \zeta =(\zeta _{k})\in l^{\diamond }: \sup_{k} \vert \zeta _{k} \vert < \infty \Bigr\} \end{aligned}

are Banach spaces with $$\Vert \zeta \Vert _{\ell ^{\infty }}=\sup_{k}| \zeta _{k}|$$.

The Copson matrix $$\mathcal{C}^{(1)}=(c_{j,k})_{j,k\in \mathbb{N}_{0}}$$ of order 1 is defined by

$$c_{j,k}=\textstyle\begin{cases} \frac{1}{k+1} & 0\leq j\leq k, \\ 0 & \text{otherwise.}\end{cases}$$

Note that $$\Vert \mathcal{C}^{(1)}\Vert _{\ell ^{p}}=p$$. The Copson matrix is the transpose of the Cesàro matrix

$$c_{j,k}^{t}=\textstyle\begin{cases} \frac{1}{k+1} & 0\leq k\leq j, \\ 0 & \text{otherwise.}\end{cases}$$

The Copson matrix of order $$\alpha >0$$, $$\mathcal{C}^{(\alpha )}=(c_{j,k}^{(\alpha )})$$ is defined by

$$c_{j,k}^{(\alpha )}=\textstyle\begin{cases} \frac{{\binom{n+k-j-1}{k-j}}}{{\binom{n+k}{k}}} & 0\leq j\leq k \\ 0 & \text{otherwise},\end{cases}$$

which is the transpose of Cesàro matrix of order α, and the $$\ell ^{p}$$-norm of $$\mathcal{C}^{(\alpha )}$$ is (see [18, 19])

$$\bigl\Vert \mathcal{C}^{(\alpha )} \bigr\Vert _{\ell ^{p}}= \frac{\Gamma (\alpha +1)\Gamma (1/p)}{\Gamma (\alpha +1/p)}.$$

For $$\alpha =0$$, $$\mathcal{C}^{(0)}=I$$, where I is the identity matrix, and for $$\alpha =1$$, it is $$\mathcal{C}^{(1)}$$.

Recently, these types of sequence spaces have been studied in [1822]. Most recently, Roopaei [19] studied the following spaces:

\begin{aligned}& \mathcal{C}^{(\alpha )}\bigl(c^{0}\bigr)= \Biggl\{ \zeta =(\zeta _{j})\in l^{ \diamond }:\lim_{j\rightarrow \infty }\sum _{k=j}^{\infty } \frac{{\binom{\alpha +k-j-1}{k-j}}}{{\binom{\alpha +k}{k}}}\zeta _{k}=0 \Biggr\} , \\& \mathcal{C}^{(\alpha )}(c)= \Biggl\{ \zeta =(\zeta _{j})\in l^{ \diamond }:\lim_{j\rightarrow \infty }\sum_{k=j}^{\infty } \frac{{\binom{\alpha +k-j-1}{k-j}}}{{\binom{\alpha +k}{k}}}\zeta _{k} \text{ exists} \Biggr\} , \end{aligned}

and

$$\mathcal{C}^{(\alpha )}\bigl(\ell ^{p}\bigr)= \Biggl\{ \zeta =(\zeta _{j})\in l^{ \diamond }:\sum_{j=0}^{\infty } \Biggl\vert \sum_{k=j}^{\infty } \frac{{\binom{\alpha +k-j-1}{k-j}}}{{\binom{\alpha +k}{k}}}\zeta _{k} \Biggr\vert ^{p}< \infty \Biggr\} \quad (0< p< 1).$$

In terms of matrix domains, these spaces are defined as follows:

$$\mathcal{C}^{(\alpha )}\bigl(c^{0}\bigr)=\bigl(c^{0} \bigr)_{\mathcal{C}^{(\alpha )}},\qquad \mathcal{C}^{(\alpha )}(c)=(c)_{\mathcal{C}^{(\alpha )}}, \quad \text{and}\quad \mathcal{C}^{(\alpha )}\bigl(\ell ^{p}\bigr)= \bigl(\ell ^{p}\bigr)_{\mathcal{C}^{( \alpha )}}.$$

Throughout the study, $$\eta =(\eta _{j})$$ will be the $$\mathcal{C}^{(\alpha )}$$-transform of a sequence $$\zeta =(\zeta _{j})$$; that is,

$$\eta _{j}=\bigl(\mathcal{C}^{(\alpha )}\zeta \bigr)_{j}= \sum_{k=j}^{\infty } \frac{{\binom{n+k-j-1}{k-j}}}{{\binom{n+k}{k}}}\zeta _{k}$$
(1.1)

for all $$j\in \mathbb{N}_{0}$$. Also, the relation

${\zeta }_{k}=\sum _{i=k}^{\mathrm{\infty }}{\left(-1\right)}^{i-k}\left(\begin{array}{c}n+k\\ k\end{array}\right)\left(\begin{array}{c}n\\ i-k\end{array}\right){\eta }_{i}$
(1.2)

holds for all $$k\in \mathbb{N}_{0}$$.

The spaces $$\mathcal{C}^{(\alpha )}(c^{0})$$ and $$\mathcal{C}^{(\alpha )}(c)$$ are Banach spaces with the norm $$\Vert \zeta \Vert _{\mathcal{C}^{(\alpha )}(c^{0})}=\Vert \zeta \Vert _{\mathcal{C}^{(\alpha )}(c)}=\Vert \mathcal{C}^{(\alpha )}\zeta \Vert _{\ell ^{\infty }}$$, and $$\mathcal{C}^{(\alpha )}(\ell ^{p})$$ ($$0< p<1$$) is a complete p-normed space with the p-norm $$\Vert \zeta \Vert _{\mathcal{C}^{(\alpha )}(\ell ^{p})}=\Vert \mathcal{C}^{(\alpha )}\zeta \Vert _{\ell ^{p}}$$. Furthermore, $$\mathcal{C}^{(\alpha )}(c^{0})\simeq c^{0}$$ and $$\mathcal{C}^{(\alpha )}(c)\simeq c$$, while $$\mathcal{C}^{(\alpha )}(\ell ^{p})\simeq \ell ^{p}$$.

The main theme of this article is to characterize some matrix classes $$(\mathcal{C}^{(\alpha )}(\ell ^{p}),E)$$, where $$E=\ell ^{\infty },c,c^{0}$$. Furthermore, we apply the techniques of measures of noncompactness to characterize compact operators associated with these matrix classes.

## Matrix classes

Let $$c_{00}:= \{ \zeta =(\zeta _{j})\in l^{\diamond }:\zeta _{j} \neq 0\text{ for finite }j;\text{and }0\text{ elsewhere} \}$$. For a BK-space $$\mathfrak{U}\supset c_{00}$$ and $$\gamma =(\gamma _{k})\in l^{\diamond }$$, we define

$$\mathcal{ \Vert \gamma \Vert }_{\mathfrak{U}}^{\ast }=\sup _{ \zeta \in S_{\mathfrak{X}}} \Biggl\vert \sum_{k=0}^{\infty } \gamma _{k} \zeta _{k} \Biggr\vert$$
(2.1)

provided $$\gamma \in \mathfrak{U}^{\beta }= \{ \gamma =(\gamma _{k})\in l^{ \diamond }:\sum_{k=0}^{\infty }\gamma _{k}\zeta _{k} \text{ converges for all }\zeta =(\zeta _{k})\in \mathfrak{U} \}$$.

For FK-, BK-, AK-spaces and the relevant literature, we refer to [1, 2, 11], and [12].

We need the following lemmas.

### Lemma 2.1

([23])

We have the following:

1. (i)

$$D=(d_{jk})\in (c_{0},c_{0})\Leftrightarrow$$

\begin{aligned}& \sup_{j\in \mathbb{N}_{0}}\sum_{k=0}^{\infty } \vert d_{jk} \vert < \infty \end{aligned}
(2.2)
\begin{aligned}& \lim_{j\rightarrow \infty }d_{jk}=0\quad \textit{for each } k\in \mathbb{N}_{0}. \end{aligned}
(2.3)
2. (ii)

$$D=(d_{jk})\in (c_{0},c) \Leftrightarrow$$ (2.2) holds, and

$$\exists \alpha _{k}\in \mathbb{R}\ni \lim_{j\rightarrow \infty }d_{jk}= \alpha _{k} \quad \textit{for each } k\in \mathbb{N}_{0}.$$
(2.4)
3. (iii)

$$D=(d_{jk})\in (c:c_{0})\Leftrightarrow$$ (2.2), (2.3) hold, and

$$\lim_{j\rightarrow \infty }\sum_{k=0}^{\infty }d_{jk}=0.$$
(2.5)
4. (iv)

$$D=(d_{jk})\in (c,c)\Leftrightarrow$$ (2.2) and (2.4) hold, and

$$\lim_{j\rightarrow \infty }\sum_{k=0}^{\infty }d_{jk} \quad \textit{exists.}$$
(2.6)
5. (v)

$$D=(d_{jk})\in (c_{0},\ell _{\infty })=(c,\ell _{\infty }) \Leftrightarrow$$ (2.2) holds.

### Lemma 2.2

We have the following:

1. (i)

[8, Theorem 1(i) with $$p_{k}=p$$ for all k] $$D=(d_{jk})\in (\ell _{p},\ell _{\infty })\Leftrightarrow$$

$$\sup_{j,k\in \mathbb{N}_{0}} \vert d_{jk} \vert ^{p}< \infty .$$
(2.7)
2. (ii)

[8, Corollary for Theorem 1 with $$p_{k}=p$$ for all k] $$D=(d_{jk})\in (\ell _{p},c)\Leftrightarrow$$ (2.4) and (2.7) hold.

The following results give the relation between $$(\mathfrak{U,V})$$ and $$\mathcal{B}(\mathfrak{U,V})$$ [1].

### Lemma 2.3

Let $$\mathfrak{U}\supset c^{00}$$ and $$\mathfrak{V}$$ be BK-spaces. Then,

1. (a)

$$(\mathfrak{U,V})\subset \mathcal{B}(\mathfrak{U,V})$$, i.e., every matrix $$\mathfrak{A}\in (\mathfrak{U,V})$$ is associated with an operator $$L_{\mathfrak{A}}\in \mathcal{B}(\mathfrak{U,V})$$ by $$L_{\mathfrak{A}}(\zeta )=\mathfrak{A}\xi$$ for all $$\zeta \in \mathfrak{U}$$.

2. (b)

If $$\mathfrak{U}$$ has AK, then the reverse inclusion also holds.

### Lemma 2.4

Let $$\mathfrak{U}\supset c^{00}$$ be a BK-space and $$\mathfrak{V\in }\{c^{0},c,\ell ^{\infty }\}$$. Then

$$\Vert L_{\mathfrak{A}} \Vert = \mathcal{ \Vert \mathfrak{A} \Vert }_{(\mathfrak{U},\ell ^{\infty })}= \sup_{n}\Vert \mathfrak{A}_{n}\Vert _{\mathfrak{U}}^{\ast }< \infty\quad \textit{for } \mathfrak{A}\in (\mathfrak{U,V}).$$

Next, we characterize the matrix classes $$(\mathcal{C}^{(\alpha )}(\ell ^{p}),\ell ^{\infty })$$, $$(\mathcal{C}^{(\alpha )}(\ell ^{p}),c)$$, and $$(\mathcal{C}^{(\alpha )}(\ell ^{p}),c^{0})$$. Hereafter, we write $$\mathfrak{A}=(a_{jk})_{j,k\in \mathbb{N}_{0}}$$ for an infinite matrix.

The β-dual of a sequence space $$\mathfrak{U}$$, i.e., $$\mathfrak{U}^{\beta }= \{ a=(a_{k})\in l^{\diamond }:\sum_{k=0}^{ \infty }a_{k}\zeta _{k}\text{ converges for}\text{ }\text{all }\zeta =(\zeta _{k}) \in \mathfrak{U} \}$$ plays an important role in matrix transformations. The β-dual of $$\mathcal{C}^{(\alpha )}(\ell ^{p})$$ ($$0< p<1$$) is

$$\bigl( \mathcal{C}^{(\alpha )}\bigl(\ell ^{p}\bigr) \bigr) ^{\beta }:= \Biggl\{ b=(b_{k})\in l^{\diamond }: \sup _{j} \Biggl\vert \sum_{i=0}^{j}(-1)^{j-i}{\binom{n+i}{i}} {\binom{n}{j-i}}b_{i} \Biggr\vert ^{p}< \infty \Biggr\} .$$

### Theorem 2.5

$$\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),\ell ^{\infty }) \Leftrightarrow$$

$$\sup_{j,k\in \mathbb{N}_{0}} \Biggl\vert \sum_{i=0}^{k}(-1)^{k-i}{ \binom{\alpha +i}{i}} {\binom{\alpha }{k-i}}a_{ji} \Biggr\vert ^{p}< \infty .$$
(2.8)

### Proof

Necessity. Suppose $$\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),\ell _{\infty })$$ and $$\xi =(\xi _{k})\in \mathcal{C}^{(\alpha )}(\ell ^{p})$$. Then $$\mathfrak{A}\xi$$ exists and $$\mathfrak{A}\xi \in \ell ^{\infty }$$. Then $$\mathfrak{A}_{j}=(a_{jk})_{k\in \mathbb{N}_{0}}\in (\mathcal{C}^{( \alpha )}(\ell ^{p}))^{\beta }$$ for each $$j\in \mathbb{N}_{0}$$, and hence (2.8) holds.

Sufficiency. Let (2.8) hold and that $$\zeta =(\zeta _{k})\in \mathcal{C}^{(\alpha )}(\ell ^{p})$$. Then $$\mathfrak{A}_{j}=(a_{jk})_{k\in \mathbb{N}_{0}}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}))^{ \beta }$$ for each $$j\in \mathbb{N}_{0}$$, which guarantees the existence of $$\mathfrak{A}\zeta$$. Fix $$j\in \mathbb{N}$$, then by (1.2), for $$r\in \mathbb{N}_{0}$$,

\begin{aligned} \sum_{k=0}^{r}a_{jk}\zeta _{k} =&\sum_{k=0}^{r}\sum _{i=k}^{ \infty }(-1)^{i-k}{ \binom{\alpha +k}{k}} {\binom{\alpha }{i-k}}a_{jk}y_{i} \\ =&\sum_{k=0}^{r} \Biggl( \sum _{i=0}^{k}(-1)^{k-i}{ \binom{\alpha +i}{i}} {\binom{\alpha }{k-i}}a_{ji} \Biggr) y_{k} \\ &{}+\sum_{k=r+1}^{\infty } \Biggl( \sum _{i=0}^{r}(-1)^{r-i}{ \binom{\alpha +i}{i}} {\binom{\alpha }{r-i}}a_{ji} \Biggr) y_{k} \end{aligned}

for all $$j,r\in \mathbb{N}_{0}$$. Now, by letting $$r\rightarrow \infty$$, we have

$$(A\zeta )_{j}=\sum_{k=0}^{\infty }a_{jk} \zeta _{k}=\sum_{k=0}^{ \infty }b_{jk}y_{k}=(By)_{j}$$
(2.9)

for all $$j\in \mathbb{N}_{0}$$, where

$$b_{jk}=\sum_{i=0}^{k}(-1)^{k-i}{ \binom{\alpha +i}{i}} { \binom{\alpha }{k-i}}a_{ji}$$
(2.10)

for all $$j,r\in \mathbb{N}_{0}$$. Therefore, condition (2.7) of Lemma 2.2 is satisfied by the matrix $$B=(b_{jk})$$. Hence $$By=\mathfrak{A}\zeta \in \ell ^{\infty }$$, i.e., $$\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),\ell _{\infty })$$. □

### Theorem 2.6

$$\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),c)$$ (2.8) holds and there exists $$\beta _{k}\in \mathbb{R}$$ such that

$$\lim_{j\rightarrow \infty }\sum_{i=0}^{k}(-1)^{k-i}{ \binom{\alpha +i}{i}} {\binom{\alpha }{k-i}}a_{ji}=\beta _{k}$$
(2.11)

for each $$k\in \mathbb{N}_{0}$$.

### Proof

Necessity. Let $$\mathfrak{A}=(a_{nk})\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),c)$$. Then $$\mathfrak{A\zeta }$$ exists and $$\mathfrak{A\zeta }\in c$$ for all $$\mathfrak{\zeta }=(\zeta _{k})\in \mathcal{C}^{(\alpha )}(\ell ^{p})$$. Since $$c\subset \ell ^{\infty }$$, condition (2.8) follows from Theorem 2.5. Condition (2.11) immediate follows by taking the sequence $$\zeta ^{(i)}= \{ \zeta _{k}^{(i)} \} \in \mathcal{C}^{( \alpha )}(\ell ^{p})$$ defined by

$$\zeta _{k}^{(i)}:=\textstyle\begin{cases} (-1)^{k-i}{\binom{\alpha +i}{i}}{\binom{\alpha }{k-i}} , & k\geq i, \\ 0 , & 0\leq k\leq i-1,\end{cases}$$

for all $$i,k\in \mathbb{N}_{0}$$ that $$\mathfrak{A}\zeta ^{(k)}= \{ \sum_{i=0}^{k}(-1)^{k-i}{ \binom{\alpha +i}{i}}{\binom{\alpha }{k-i}}a_{ji} \} \in c$$ for each $$k\in \mathbb{N}_{0}$$.

Sufficiency. Suppose that conditions (2.8) and (2.11) hold, and that $$\zeta =(\zeta _{k})\in \mathcal{C}^{(\alpha )}(\ell ^{p})$$. Existence of $$\mathfrak{A}\zeta$$ follows from the fact that $$\mathfrak{A}_{j}=(a_{jk})_{k\in \mathbb{N}_{0}}\in (\mathcal{C}^{( \alpha )}(\ell ^{p}))^{\beta }$$ for each $$j\in \mathbb{N}_{0}$$. Therefore, it follows from (2.9) that conditions (2.8) and (2.11) correspond to (2.7) and (2.4) with $$b_{jk}$$ instead of $$d_{jk}$$, respectively, where $$b_{jk}$$ is given by (2.10). Thus, $$By\in c$$, and we get by (2.9) that $$\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),c)$$. □

### Corollary 2.7

$$\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),c^{0}) \Leftrightarrow$$ (2.8) holds and (2.11) also holds with $$\beta _{k}=0$$ for all $$k\in \mathbb{N}_{0}$$.

### Corollary 2.8

For $$\mathfrak{A}=(a_{nk})$$, write $$c(j,k)=\sum_{i=0}^{j}a_{ik}$$ for all $$k,n\in \mathbb{N}_{0}$$. Then, from Theorem 2.5, Theorem 2.6, and Corollary 2.7, we get:

1. (i)

$$\mathfrak{A}=(a_{nk})\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),bs) \Leftrightarrow$$ (2.8) holds with $$a_{jk}$$ is replaced by $$c(j,k)$$.

2. (ii)

$$\mathfrak{A}=(a_{nk})\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),cs) \Leftrightarrow$$ (2.8) and (2.11) hold with $$a_{jk}$$ is replaced by $$c(j,k)$$.

3. (iii)

$$\mathfrak{A}=(a_{nk})\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),cs_{0})\Leftrightarrow$$ (2.8) and (2.11) hold with $$a_{jk}$$ is replaced by $$c(j,k)$$, with $$\beta _{k}=0$$ for all $$k\in \mathbb{N}_{0}$$, where bs, cs, and $$s_{0}$$ are the space of bounded, convergent, and null series, respectively.

## Compactness of matrix operators

We apply the techniques of [37, 9, 10], and [1317].

Let $$\mathcal{M}_{\mathfrak{U}}:=\{\mathfrak{B\subset U}:\mathfrak{B}\text{ is bounded} \}$$. The Hausdorff measure of noncompactness (HMNC) of $$\mathfrak{B\in }\mathcal{M}_{\mathfrak{U}}$$ is defined by

$$\chi (\mathfrak{B})=\inf \{ \varepsilon >0:\mathfrak{B} \text{ has finite } \varepsilon \text{-net} \} .$$

Let $$\mathfrak{U}$$ and $$\mathfrak{V}$$ be Banach spaces and $$\mathfrak{D}\in \mathcal{B}(\mathfrak{U},\mathfrak{V})$$. Then the HMNC of $$\mathfrak{D}$$ is defined by

$${\mathcal{ \Vert \mathfrak{D} \Vert }}_{\chi }=\chi \bigl( \mathfrak{D}({S}_{\mathfrak{U}})\bigr)=\chi \bigl(\mathfrak{D}( \bar{B}_{\mathfrak{U}})\bigr),$$
(3.1)

and we have

$$\mathfrak{D} \quad \text{is compact if and only if}\quad { \mathcal{ \Vert \mathfrak{D} \Vert }}_{\chi }=0.$$
(3.2)

In what follows, we denote the set of all compact operators from $$\mathfrak{U}$$ into $$\mathfrak{V}$$ by $$\mathfrak{C}(\mathfrak{U,V})$$.

### Theorem 3.1

Let $$\mathfrak{U}$$ be a Banach space with a Schauder basis $$(b_{k})_{k=0}^{\infty }$$, $${\mathcal{\mathfrak{D}}}\in \mathcal{M}_{\mathfrak{U}}$$ and $$\mathfrak{P}_{n}:\mathfrak{U}\rightarrow \mathfrak{U}$$ ($$n\in \mathbb{N}$$) be the projector onto the linear span of $$\{b_{0},b_{1},\ldots ,b_{n}\}$$. Then we have

\begin{aligned}& \frac{1}{\limsup_{n\rightarrow \infty } \Vert I-\mathfrak{P}_{n} \Vert } \cdot \limsup_{n\rightarrow \infty }{ \Bigl(}\sup _{\zeta \in { \mathcal{\mathfrak{D}}}} \bigl\Vert (I-\mathfrak{P}_{n}) (\zeta ) \bigr\Vert { \Bigr)} \\& \quad \leq \chi ({ \mathcal{\mathfrak{D}}})\leq \limsup _{n\rightarrow \infty }{ \Bigl(}\sup_{x \in {\mathcal{\mathfrak{D}}}} \bigl\Vert (I- \mathfrak{P}_{n}) (\zeta ) \bigr\Vert { \Bigr)}. \end{aligned}

### Theorem 3.2

Let $${\mathcal{\mathfrak{D}}}\in \mathcal{M}_{\mathfrak{U}}$$, where $$\mathfrak{U}=\ell _{p}$$ ($$1\leq p<\infty$$) or $$c^{0}$$. If $$\mathfrak{P}_{n}:\mathfrak{U}\rightarrow \mathfrak{U}$$ ($$n\in \mathbb{N}$$) is the operator defined by $$\mathfrak{P}_{n}(\zeta )=\zeta ^{{}[ n]}=(\zeta _{0},\zeta _{1}, \ldots ,\zeta _{n},0 ,0,\ldots )$$ for all $$\zeta =(\zeta _{k})_{k=0}^{\infty }\in \mathfrak{U}$$, then

$$\chi ({\mathcal{\mathfrak{D}}})=\lim_{n\rightarrow \infty }{ \Bigl(}\sup_{\zeta \in {\mathcal{\mathfrak{D}}}} \bigl\Vert (I-\mathfrak{P}_{n}) ( \zeta ) \bigr\Vert { \Bigr)}.$$

### Lemma 3.3

([13])

Let $$\mathfrak{U}\supset c^{00}$$ be a BK-space with AK or $$\mathfrak{U}=\ell _{\infty }$$. If $$\mathfrak{A}\in (\mathfrak{U},c)$$, then

\begin{aligned}& \alpha _{k}=\lim_{j\rightarrow \infty }a_{jk} \quad \textit{exists for every } k\in \mathbb{N}, \end{aligned}
(3.3)
\begin{aligned}& \alpha =(\alpha _{k})\in \mathfrak{U}^{\beta }, \end{aligned}
(3.4)
\begin{aligned}& \sup_{j}\Vert \mathfrak{A}_{j}-\alpha \Vert _{\mathfrak{U}}^{\ast }< \infty , \end{aligned}
(3.5)
\begin{aligned}& \lim_{j\rightarrow \infty }\mathfrak{A}_{j}(x)=\sum _{k=0}^{\infty } \alpha _{jk}x_{k}\quad \textit{for all } x=(x_{k})\in \mathfrak{U}. \end{aligned}
(3.6)

### Theorem 3.4

([13])

Let $$\mathfrak{U}\supset c^{00}$$ be a BK-space. Then we have

(a)

$$\Vert L_{\mathfrak{A}}\Vert _{\chi }= \limsup _{n\rightarrow \infty }\Vert \mathfrak{A}_{n} \Vert _{\mathfrak{U}}^{\ast }\quad \textit{for }\mathfrak{A} \in \bigl( \mathfrak{U},c^{0}\bigr)$$

and

$$L_{\mathfrak{A}}\in \mathfrak{C} \bigl(\mathfrak{U},c^{0}\bigr) \quad \Leftrightarrow \quad \lim_{n\rightarrow \infty } \Vert \mathfrak{A}_{n}\Vert _{\mathfrak{U}}^{\ast }=0.$$

(b) If $$\mathfrak{U}$$ has AK or $$\mathfrak{U}=\ell ^{\infty }$$, then

$$\frac{1}{2}\cdot \limsup_{n\rightarrow \infty }\Vert \mathfrak{A}_{n}-\alpha \Vert _{\mathfrak{U}}^{\ast } \leq \Vert L_{\mathfrak{A}}\Vert _{\chi } \leq \limsup_{n \rightarrow \infty }\Vert \mathfrak{A}_{n}- \alpha \Vert _{\mathfrak{U}}^{\ast }\quad \textit{for }\mathfrak{A} \in (\mathfrak{U},c)$$

and

$$L_{\mathfrak{A}}\in \mathfrak{C} (\mathfrak{U},c) \quad \Leftrightarrow \quad \lim_{n\rightarrow \infty }\Vert \mathfrak{A}_{n}-\alpha \Vert _{\mathfrak{U}}^{\ast }=0,$$

where $$\alpha =(\alpha _{k})=(\lim_{n\rightarrow \infty }a_{nk})$$ for all $$k\in \mathbb{N}$$.

(c)

$$0\leq \Vert L_{\mathfrak{A}}\Vert _{\chi } \leq \limsup_{n\rightarrow \infty }\Vert \mathfrak{A}_{n} \Vert _{\mathfrak{U}}^{\ast }\quad \textit{for }\mathfrak{A} \in \bigl( \mathfrak{U},\ell ^{\infty }\bigr)$$

and

$$L_{\mathfrak{A}}\in \mathfrak{C} \bigl(\mathfrak{U},\ell ^{\infty }\bigr) \quad \textit{if } \lim_{n\rightarrow \infty }\Vert \mathfrak{A}_{n} \Vert _{\mathfrak{U}}^{\ast }=0.$$
(3.7)

We now state and prove the following.

### Theorem 3.5

Let $$1\leq p<\infty$$. Then we have

(a)

$$\Vert L_{\mathfrak{A}}\Vert _{\chi }= \lim _{r\rightarrow \infty }\sup_{j} \Biggl( \sum _{k=r+1}^{\infty } \vert a_{jk} \vert ^{p} \Biggr) ^{1/p}\quad \textit{for }\mathfrak{A}\in \bigl( \mathcal{C}^{(\alpha )}\bigl( \ell ^{p}\bigr),c^{0}\bigr).$$
(3.8)

(b)

\begin{aligned}& \frac{1}{2}\cdot \lim_{r\rightarrow \infty }\sup_{j} \Biggl( \sum_{k=r+1}^{ \infty } \vert a_{jk}-\beta _{k} \vert ^{p} \Biggr) ^{1/p} \\& \quad \leq \Vert L_{\mathfrak{A}}\Vert _{\chi }\leq \lim_{r \rightarrow \infty }\sup_{j} \Biggl( \sum_{k=r+1}^{\infty } \vert a_{jk}- \beta _{k} \vert ^{p} \Biggr) ^{1/p}\quad \textit{for }\mathfrak{A}\in \bigl( \mathcal{C}^{(\alpha )} \bigl(\ell ^{p}\bigr),c\bigr), \end{aligned}
(3.9)

where $$\beta =(\beta _{k})=(\lim_{j\rightarrow \infty }b_{jk})$$ for all $$k\in \mathbb{N}$$.

(c)

$$0\leq \Vert L_{\mathfrak{A}}\Vert _{\chi } \leq \lim _{r\rightarrow \infty }\sup_{j} \Biggl( \sum _{k=r+1}^{ \infty } \vert a_{jk} \vert ^{p} \Biggr) ^{1/p}\quad \textit{for }\mathfrak{A}\in \bigl( \mathcal{C}^{(\alpha )}\bigl(\ell ^{p}\bigr),\ell ^{\infty } \bigr).$$
(3.10)

### Proof

(a) Note that the limits in (3.8), (3.9), and (3.10) exist by Lemmas 2.4 and 3.3. Let $$\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),c^{0})$$. Then $$\mathfrak{A}_{j}=(a_{jk})_{k\in \mathbb{N}_{0}}\in {}[ \mathcal{C}^{( \alpha )}(\ell ^{p})]^{\beta }$$ for each $$j\in \mathbb{N}_{0}$$, and we have

$$\Vert \mathfrak{A}\Vert _{\mathcal{C}^{( \alpha )}(\ell ^{p})}^{\ast }= \Vert B_{j} \Vert _{\ell ^{p}}= \Biggl( \sum _{k=0}^{\infty } \vert a_{jk} \vert ^{p} \Biggr) ^{1/p}.$$
(3.11)

Write $$S=S_{\mathcal{C}^{(\alpha )}(\ell ^{p})}$$ for short. Then we have $$\mathfrak{A}S\in \mathcal{M}_{c^{0}}$$. From Theorem 3.2, we get

\begin{aligned}& \Vert L_{\mathfrak{A}}\Vert _{\chi }= \chi (\mathfrak{A}S)=\lim_{r\rightarrow \infty }\sup_{\zeta \in S} \bigl\Vert (I-\mathfrak{P}_{r}) (\mathfrak{A}\zeta ) \bigr\Vert _{\ell ^{p}}. \end{aligned}
(3.12)
\begin{aligned}& \lim_{r\rightarrow \infty }\sup_{y\in S_{\ell ^{p}}} \bigl\Vert (I- \mathfrak{P}_{r}) (By) \bigr\Vert _{\ell _{p}}{=}\lim _{r\rightarrow \infty }\sup_{j} \Biggl( \sum _{k=r+1}^{\infty } \vert a_{jk} \vert ^{p} \Biggr) ^{1/p}. \end{aligned}
(3.13)

We get (3.8) by (3.13).

(b) We have $$\mathfrak{A}S\in \mathcal{M}_{c}$$. Suppose that $$\mathfrak{P}_{r}:c\rightarrow c$$ ($$r\in \mathbb{N}$$) are the projectors defined by (2.3).

Now, since $$\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),c)$$, we have $$B\in (\ell ^{p},c)$$ and $$\mathfrak{A}\xi =By$$. Thus, it follows from Lemma 3.3 that the limits $$\beta _{k}=\lim_{j\rightarrow \infty }a_{jk}$$ exist for all k, $$\beta =(\beta _{k})\in {\ell }^{1}={c}^{\beta }$$ and $$\lim_{j\rightarrow \infty }B_{j}(y)=\sum_{k=0}^{\infty }a_{jk}y_{k}$$. Therefore, we get

\begin{aligned} \Vert (I-\mathfrak{P}_{r}) ( \mathfrak{A} \zeta )\Vert _{\ell ^{p}}& =\Vert (I-\mathfrak{P}_{r}) (By)\Vert _{\ell ^{p}} \\ & =\sup_{j} \Biggl( \sum_{k=r+1}^{\infty } \vert a_{jk}-\beta _{k} \vert ^{p} \Biggr) ^{1/p} \end{aligned}

for all $$\zeta =(\zeta _{k})\in \mathcal{C}^{(\alpha )}(\ell ^{p})$$. Now, (3.12) and (3.1) imply that

$$\frac{1}{2}\cdot \lim \sup_{r\rightarrow \infty } \Vert B_{j}-\beta \Vert _{\ell ^{p}}\leq \mathcal{ \Vert }L_{\mathfrak{A}}\Vert _{\chi }\leq \lim \sup _{r \rightarrow \infty }\Vert B_{j}-\beta \Vert _{\ell ^{p}}.$$
(3.14)

Hence, we get (3.9) from (3.14), since the limit in (3.9) exists.

(c) Define $$\mathfrak{P}_{r}:\ell ^{\infty }\rightarrow \ell ^{\infty }$$ ($$r\in \mathbb{N}$$) as in (a) for all $$\zeta =(\zeta _{k})\in \ell ^{\infty }$$. Then

$$\mathfrak{A}S\subset \mathfrak{P}_{r}(\mathfrak{A}S)+(I- \mathfrak{P}_{r}) (\mathfrak{A}S);\quad (r\in \mathbb{N}).$$

Therefore

\begin{aligned} 0&\leq \chi (\mathfrak{A}S) \\ & \leq \chi \bigl( \mathfrak{P}_{r}( \mathfrak{A}S)\bigr)+\chi \bigl((I- \mathfrak{P}_{r}) (\mathfrak{A}S)\bigr) \\ & = \chi \bigl((I-\mathfrak{P}_{r}) (\mathfrak{A}S)\bigr) \\ & \leq \sup_{\xi \in S}\Vert (I-\mathfrak{P}_{r}) ( \mathfrak{A}\xi )\Vert _{\ell ^{p}} \\ & = \lim_{r\rightarrow \infty }\sup_{j} \Biggl( \sum _{k=r+1}^{ \infty } \vert a_{jk} \vert ^{p} \Biggr) ^{1/p}. \end{aligned}

From this and (3.12), we get (3.10), which concludes the proof. □

### Corollary 3.6

We have the following:

(a) For $$\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),c_{0})$$,

$$L_{\mathfrak{A}}\in \mathfrak{C}\bigl(\mathcal{C}^{(\alpha )}\bigl(\ell ^{p}\bigr),c^{0}\bigr)\quad \Leftrightarrow \quad \lim _{r\rightarrow \infty }\sup_{j} \Biggl( \sum _{k=r+1}^{ \infty } \vert a_{jk} \vert ^{p} \Biggr) ^{1/p}=0.$$

(b) For $$\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell _{p}),c)$$,

$$L_{\mathfrak{A}}\in \mathfrak{C}\bigl(\mathcal{C}^{(\alpha )}\bigl(\ell ^{p}\bigr),c\bigr) \quad \Leftrightarrow\quad \lim_{r\rightarrow \infty }\sup _{j} \Biggl( \sum_{k=r+1}^{\infty } \vert a_{jk}-\beta _{k} \vert ^{p} \Biggr) ^{1/p}=0,$$

where $$\beta =(\beta _{k})=(\lim_{j\rightarrow \infty }a_{jk})$$ for all $$k\in \mathbb{N}$$.

(c) For $$\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),\ell ^{\infty })$$, then

$$L_{\mathfrak{A}}\in \mathfrak{C}\bigl(\mathcal{C}^{(\alpha )}\bigl(\ell ^{p}\bigr), \ell ^{\infty }\bigr)\quad \textit{if } \lim _{r\rightarrow \infty }\sup_{j} \Biggl( \sum _{k=r+1}^{\infty } \vert b_{jk} \vert ^{p} \Biggr) ^{1/p}=0.$$
(3.15)

### Corollary 3.7

From Theorem 3.4and Corollary 2.11, we have the following:

(a) For $$\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),cs^{0})$$,

$$\Vert L_{\mathfrak{A}}\Vert _{\chi }= \lim _{r\rightarrow \infty }\sup_{j} \Biggl( \sum _{k=r+1}^{\infty } \bigl\vert c(j,k) \bigr\vert ^{p} \Biggr) ^{1/p}.$$
(3.16)

(b) For $$\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),cs)$$,

\begin{aligned}& \frac{1}{2}\cdot \lim_{r\rightarrow \infty }\sup_{j} \Biggl( \sum_{k=r+1}^{ \infty } \bigl\vert c(j,k)- \beta _{k} \bigr\vert ^{p} \Biggr) ^{1/p} \\& \quad \leq \Vert L_{\mathfrak{A}}\Vert _{\chi }\leq \lim_{r \rightarrow \infty }\sup_{j} \Biggl( \sum _{k=r+1}^{\infty } \bigl\vert c(j,k)- \beta _{k} \bigr\vert ^{p} \Biggr) ^{1/p}, \end{aligned}
(3.17)

where $$\beta =(\beta _{k})=(\lim_{j\rightarrow \infty }b_{jk})$$ for all $$k\in \mathbb{N}$$.

(c) For $$\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),bs)$$,

$$0\leq \Vert L_{\mathfrak{A}}\Vert _{\chi } \leq \lim _{r\rightarrow \infty }\sup_{j} \Biggl( \sum _{k=r+1}^{ \infty } \bigl\vert c(j,k) \bigr\vert ^{p} \Biggr) ^{1/p}.$$
(3.18)

### Corollary 3.8

From Corollary 3.5and Corollary 2.11, we have the following:

(a) For $$\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),cs^{0})$$,

$$L_{\mathfrak{A}}\in \mathfrak{C}\bigl(\mathcal{C}^{(\alpha )}\bigl(\ell ^{p}\bigr),cs^{0}\bigr) \quad\Leftrightarrow \quad \lim _{r\rightarrow \infty }\sup_{j} \Biggl( \sum _{k=r+1}^{\infty } \bigl\vert c(j,k) \bigr\vert ^{p} \Biggr) ^{1/p}=0.$$

(b) For $$\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),cs)$$,

$$L_{\mathfrak{A}}\in \mathfrak{C}\bigl(\mathcal{C}^{(\alpha )}\bigl(\ell ^{p}\bigr),cs\bigr) \quad\Leftrightarrow \quad \lim _{r\rightarrow \infty }\sup_{j} \Biggl( \sum _{k=r+1}^{\infty } \bigl\vert c(j,k)-\beta _{k} \bigr\vert ^{p} \Biggr) ^{1/p}=0,$$

where $$\beta =(\beta _{k})=(\lim_{j\rightarrow \infty }c(j,k))$$ for all $$k\in \mathbb{N}$$.

(c) For $$\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),bs)$$,

$$L_{\mathfrak{A}}\in \mathfrak{C}\bigl(\mathcal{C}^{(\alpha )}\bigl(\ell ^{p}\bigr),bs\bigr) \quad\Leftrightarrow \quad \mathit{ if} \lim_{r\rightarrow \infty } \sup_{j} \Biggl( \sum _{k=r+1}^{\infty } \bigl\vert c(j,k) \bigr\vert ^{p} \Biggr) ^{1/p}=0.$$

Not applicable.

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## Acknowledgements

We are thankful to the learned reviewers whose suggestions led to the improvement of the presentation.

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Mursaleen, M., Edely, O.H.H. Compact operators on sequence spaces associated with the Copson matrix of order α. J Inequal Appl 2021, 178 (2021). https://doi.org/10.1186/s13660-021-02713-9

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• DOI: https://doi.org/10.1186/s13660-021-02713-9

• 26D15
• 40C05
• 40G05
• 47B37

### Keywords

• Sequence spaces
• Cesàro matrix
• Copson matrix
• Copson matrix of order α
• Matrix transformations
• Hausdorff measure of noncompactness
• Compact operators