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.

1 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 [18–22]. 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

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

(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.

2 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:

• (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)

• (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)

• (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)

• (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)

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

Lemma 2.2

We have the following:

• (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)

• (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,

• (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 $\zeta \in \mathfrak{U}$.

• (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:

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

• (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 $c(j,k)$.

• (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.

3 Compactness of matrix operators

We apply the techniques of [3–7, 9, 10], and [13–17].

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

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