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Compact operators on sequence spaces associated with the Copson matrix of order α
Journal of Inequalities and Applications volume 2021, Article number: 178 (2021)
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
This is a Banach space with the norm
and complete p-normed space with the p-norm
Further,
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
Note that \(\Vert \mathcal{C}^{(1)}\Vert _{\ell ^{p}}=p\). The Copson matrix is the transpose of the Cesàro matrix
The Copson matrix of order \(\alpha >0\), \(\mathcal{C}^{(\alpha )}=(c_{j,k}^{(\alpha )})\) is defined by
which is the transpose of Cesàro matrix of order α, and the \(\ell ^{p}\)-norm of \(\mathcal{C}^{(\alpha )}\) is (see [18, 19])
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:
and
In terms of matrix domains, these spaces are defined as follows:
Throughout the study, \(\eta =(\eta _{j})\) will be the \(\mathcal{C}^{(\alpha )}\)-transform of a sequence \(\zeta =(\zeta _{j})\); that is,
for all \(j\in \mathbb{N}_{0}\). Also, the relation
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
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
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
Theorem 2.5
\(\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),\ell ^{\infty }) \Leftrightarrow \)
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 \(r\in \mathbb{N}_{0}\),
for all \(j,r\in \mathbb{N}_{0}\). Now, by letting \(r\rightarrow \infty \), we have
for all \(j\in \mathbb{N}_{0}\), where
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
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
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
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
and we have
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
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
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
Theorem 3.4
([13])
Let \(\mathfrak{U}\supset c^{00}\) be a BK-space. Then we have
(a)
and
(b) If \(\mathfrak{U}\) has AK or \(\mathfrak{U}=\ell ^{\infty }\), then
and
where \(\alpha =(\alpha _{k})=(\lim_{n\rightarrow \infty }a_{nk})\) for all \(k\in \mathbb{N}\).
(c)
and
We now state and prove the following.
Theorem 3.5
Let \(1\leq p<\infty \). Then we have
(a)
(b)
where \(\beta =(\beta _{k})=(\lim_{j\rightarrow \infty }b_{jk})\) for all \(k\in \mathbb{N}\).
(c)
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
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
(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
for all \(\zeta =(\zeta _{k})\in \mathcal{C}^{(\alpha )}(\ell ^{p})\). Now, (3.12) and (3.1) imply that
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
Therefore
From this and (3.12), we get (3.10), which concludes the proof. □
Corollary 3.6
We have the following:
(a) For \(\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),c_{0})\),
(b) For \(\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell _{p}),c)\),
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
Corollary 3.7
From Theorem 3.4and Corollary 2.11, we have the following:
(a) For \(\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),cs^{0})\),
(b) For \(\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),cs)\),
where \(\beta =(\beta _{k})=(\lim_{j\rightarrow \infty }b_{jk})\) for all \(k\in \mathbb{N}\).
(c) For \(\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),bs)\),
Corollary 3.8
From Corollary 3.5and Corollary 2.11, we have the following:
(a) For \(\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),cs^{0})\),
(b) For \(\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),cs)\),
where \(\beta =(\beta _{k})=(\lim_{j\rightarrow \infty }c(j,k))\) for all \(k\in \mathbb{N}\).
(c) For \(\mathfrak{A}\in (\mathcal{C}^{(\alpha )}(\ell ^{p}),bs)\),
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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