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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

ζ k = i = k ( 1 ) i k ( n + k k ) ( n i k ) η 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 \(\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),bs)\),

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

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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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MSC

  • 26D15
  • 40C05
  • 40G05
  • 47B37

Keywords

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