A study on Copson operator and its associated sequence space II

In this paper, we investigate some properties of the domains c(Cn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c(C^{n})$\end{document}, c0(Cn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c_{0}(C^{n})$\end{document}, and ℓp(Cn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell _{p}(C^{n})$\end{document}(0<p<1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(0< p<1)$\end{document} of the Copson matrix of order n, where c, c0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$c_{0}$\end{document}, and ℓp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell _{p}$\end{document} are the spaces of all convergent, convergent to zero, and p-summable real sequences, respectively. Moreover, we compute the Köthe duals of these spaces and the lower bound of well-known operators on these sequence spaces. The domain ℓp(Cn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell _{p}(C^{n})$\end{document} of Copson matrix Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C^{n}$\end{document} of order n in the sequence space ℓp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell _{p}$\end{document}, the norm of operators on this space, and the norm of Copson operator on several matrix domains have been investigated recently in (Roopaei in J. Inequal. Appl. 2020:120, 2020), and the present study is a complement of our previous research.


Introduction
Let ω denote the set of all real-valued sequences. Any linear subspace of ω is called a sequence space. For 0 < p < 1, the complete p-normed space p is the set of all real sequences x = (x k ) ∞ k=0 ∈ ω such that By c and c 0 , we denote the spaces of all convergent and convergent to zero real sequences, respectively. These spaces are Banach spaces with the norm x ∞ = sup k |x k |. Here and in the rest of the paper, the supremum is taken over all k ∈ N 0 = {0, 1, 2, 3, . . . }. Also, we use the notion N = {1, 2, 3, . . . }. One can consider an infinite matrix as a linear operator from a sequence space to another one. Given any two arbitrary sequence spaces X, Y and an infinite matrix T = (t i,j ), we define a matrix transformation from X into Y as Tx = ((Tx) i ) = ( ∞ j=0 t i,j x j ) provided that the series is convergent for each i ∈ N 0 . By (X, Y ), we denote the family of all infinite matrices from X into Y .
The domain X T of an infinite matrix T in a sequence space X is defined as X T = {x ∈ ω : Tx ∈ X}, (1.1) which is also a sequence space. By using matrix domains of special triangle matrices in classical spaces, many authors have introduced and studied new Banach spaces. For the relevant literature, we refer to the papers [1,2,4,9,13,16,22,23,[26][27][28][29] and textbooks [3,20,21]. The Köthe duals (α-, β-, γ -duals) of a sequence space X are defined by |a k x k | < ∞ for all x = (x k ) ∈ X , a k x k ∈ c for all x = (x k ) ∈ X , respectively. Copson matrix. The Copson matrix is an upper-triangular matrix which is defined by , and has the p -norm C p = p. This matrix is the transpose of the well-known Cesàro matrix.
Copson matrix of order n. Consider the Hausdorff matrix H μ = (h j,k ) ∞ j,k=0 , with entries of the form: where μ is a probability measure on [0, 1]. The Hausdorff matrix contains the famous classes of matrices. For positive integer n, these classes are as follow: • The choice dμ(θ ) = n(1θ ) n-1 dθ gives the Cesàro matrix of order n, • The choice dμ(θ ) = nθ n-1 dθ gives the Gamma matrix of order n, • The choice dμ(θ ) = | log θ| n-1 (n) dθ gives the Hölder matrix of order n, • The choice dμ(θ ) = point evaluation at θ = r (0 < r < 1) gives the Euler matrix of order r.
We use the notation hau(p) as the set of all sequences whose H μ -transforms are in the space p , that is, where μ is a fixed probability measure on [0, 1].
Hardy's formula [11,Theorem 216] states that the Hausdorff matrix is a bounded operator on p if and only if Hausdorff operator has the following norm property.
The following theorem is an analog of Hardy's formula. In order to define and know the Copson matrix details, we need the following theorem also known as Hellinger-Toeplitz theorem.
For a nonnegative real number n, and by choosing dμ(θ ) = n(1-θ ) n-1 dθ in the definition of Hausdorff matrix, we gain the Cesàro matrix of order n, which, according to Hardy's formula, has the p -norm (n + 1) (1/p * ) (n + 1/p * ) . Now, the Copson matrix of order n, C n = (c n j,k ), which is defined as the transpose of Cesàro matrix of order n, has the entries 4) and, according to Hellinger-Toeplitz theorem, the p -norm C n p = (n + 1) (1/p) (n + 1/p) .
Note that C 0 = I, where I is the identity matrix and C 1 = C is the well-known Copson matrix. For more examples, In this study, after introducing the domains of the Copson matrix of order n in the spaces c 0 , c, and p , we study some properties of the Copson spaces. We also compute α-, β-, γ -duals of these spaces and determine Schauder basis. We seek lower bounds of the form valid for every x ∈ p with x 0 > x 1 > · · · > 0. Here T is a matrix with nonnegative entries, assumed to map p into itself, and L is a constant not depending on x. The lower bound of T is the greatest possible value of L, which we denote by L(T). Throughout this paper, we use the notations L(·) for the lower bound of operators on p and L(·) X,Y for the lower bound of operators from the sequence space X into the sequence space Y .
Motivation. Many mathematicians have and still publish numerous articles about the Cesàro matrix, Cesàro matrix domain, and Cesàro function spaces [8,9,14,19,29], while the importance of the Copson operator and its associated matrix domains have been ignored under the shadow of its transpose Cesàro matrix. Recently, the author have investigated the sequence space p (C n ) for 1 ≤ p < ∞, as well as found the norm of well-known operators on this matrix domain. In this research, as a complement of [24], the matrix domains c 0 (C n ), c(C n ), and p (C n ), 0 < p < 1, are investigated, while the lower bound of well-known operators on the Copson sequence space and the lower bound of the Copson operator on some matrix domains are computed as well, which has never been done before.

Copson Banach spaces c 0 (C n ), c(C n ), and p (C n )
In this section, the sequence spaces c 0 (C n ), c(C n ), and p (C n ) are introduced and the inclusion relations as well as dual spaces of these new spaces are determined.

Lemma 2.1
The Copson matrix of order n, C n , is invertible and its inverse, C -n = (c -n j,k ), is defined by Proof Let us recall the forward difference matrix of order n, n = (δ n j,k ), which is a lowertriangular matrix with entries This matrix has the inverse -n = (δ -n j,k ) with the following entries: From the relation (1.4), one can see that the Copson matrix of order n and its inverse can be rewritten based on the forward difference operator and its inverse. For j ≤ k, we have Now, by a simple calculation, we deduce that which completes the proof. Now, we introduce the sequence spaces c 0 (C n ), c(C n ), and p (C n ) as the set of all sequences whose C n -transforms are in the spaces c 0 , c, and p , respectively, that is, With the notation of (1.1), the spaces c 0 (C n ), c(C n ), and p (C n ) can be redefined as follows: Throughout the study, y = (y j ) will be the C n -transform of a sequence x = (x j ), that is, for all j ∈ N 0 . Also, the relation

Theorem 2.2 The following statements hold:
• The spaces c 0 (C n ) and c(C n ) are Banach spaces with the norm We omit the proof which is a routine verification.

Theorem 2.3 The following statements hold:
• The spaces c 0 (C n ) and c(C n ) are linearly norm-isomorphic to c 0 and c, respectively.
Proof The proof follows from the fact that the mapping L : X(C n ) → X defined by x → Lx = y = C n x is a norm-preserving linear bijection, where X ∈ {c 0 , c, p } and y = (y j ) is given by (2.1).
Proof Choose any x ∈ q (C n ). Then, C n x ∈ q . Since the inclusion q ⊂ p holds for 0 < p < q < 1, we have C n x ∈ p . This implies that x ∈ p (C n ). Hence, we conclude that the inclusion q (C n ) ⊂ p (C n ) holds. Now, we show that the inclusion is strict. Since the inclusion q ⊂ p is strict, we can choose y = (y j ) ∈ p \ q . Define a sequence x = (x j ) as Then, we have C n x j = y j for every j ∈ N 0 , which means C n x = y, and so C n x ∈ p \ q . Hence, we conclude that x ∈ p (C n )\ q (C n ), and so the inclusion q (C n ) ⊂ p (C n ) is strict.
It is known from Theorem 2.3 of Jarrah and Malkowsky [15] that if T is triangular then the domain X T of T in a normed sequence space X has a basis if and only if X has a basis. As a direct consequence of this fact, we have Then, the sequence (b (k) ) is a basis for the spaces c 0 (C n ) and p (C n ), and every sequence x ∈ c 0 (C n ) or x ∈ p (C n ) has a unique representation of the form The following lemma is essential to determine the dual spaces. Throughout the paper, N is the collection of all finite subsets of N.

2)
and lim j→∞ t j,k exists for each k ∈ N.

Lemma 2.8
The following statements hold: c) if and only if (2.6) holds and ∃α k ∈ C lim j→∞ t j,k = α k for each k ∈ N. (2.7)

Theorem 2.9
The α-duals of the spaces c 0 (C n ) and c(C n ) are as follows: otherwise.
Given any x = (x j ) ∈ X(C n ), we have b j x j = (Ay) j for all j ∈ N, where X ∈ {c 0 , c}. This implies that bx ∈ 1 with x ∈ X(C n ) if and only if Ay ∈ 1 with y ∈ X. Hence, we conclude that b ∈ (X(C n )) α if and only if A ∈ (X, 1 ). This completes the proof by part (i) of Lemma 2.7.

Theorem 2.10
Let define the following sets: We deduce that b = (b k ) ∈ (c 0 (C n )) β if and only if the matrix B = (b j,k ) is in the class (c 0 , c). Hence, we deduce from part (ii) of Lemma 2.7 that which means b = (b k ) ∈ A 1 , and so we have (c 0 (C n )) β = A 1 . The other results can be proved similarly.

Theorem 2.11
The γ -duals of the spaces c 0 (C n ), c(C n ), and p (C n ) (0 < p < 1) are as follows: Proof This follows by applying the same technique used in the proof of Theorem 2.10.

Lower bound of operators on the Copson matrix domain for (0 < p < 1)
In this section, we assume 0 < p < 1 and intend to compute the lower bound of operators from p into p (C n ), from p (C n ) into p , and from p (C n ) into itself. In so doing, we need the following lemma.
We emphasize again that we use the notations L(·) for the lower bound of operators on p and L(·) X,Y for the lower bound of operators from the sequence space X into the sequence space Y .

Lemma 3.1 ([25, Lemma 2.1]) Let U is a bounded operator on p and A p and B p be two matrix domains such that A p
p . Then, the following statements hold: • If BT is a bounded operator on p , then T is a bounded operator from p into B p and T p ,B p = T p and L(T) p ,B p = L(BT).

• If T has a factorization of the form T = UA, then T is a bounded operator from the matrix domain A p into p and T A p , p = U p and L(T) A p , p = L(U).
• If BT = UA, then T is a bounded operator from the matrix domain A p into B p and T A p ,B p = U p and L(T) A p ,B p = L(U).

Lower bound of operators from p into p (C n )
In this part of study we intend to compute the lower bound of transposed Hausdorff operators on the Copson matrix domain.

Lower bound of operators from p (C n ) into p
In this section, we intend to find the lower bound of the transposed Hausdorff operators from p (C n ) into p .  (m + 1) (n + 1/p * ) (n + 1) (m + 1/p * ) .

Lower bound of operators on p (C n )
In this part of the study, we try to find the lower bound of the transposed Hausdorff operators on the space p (C n ). In particular, the transposed Cesàro, Gamma, Hölder, and Euler matrices of order m are bounded operators on p (C n ) and L C mt p (C n ) = (m + 1) (1/p) (m + 1/p) (m > 0), Proof Since Hausdorff matrices commute, we have C n H μt = H μt C n . Thus, part (iii) of Lemma 3.1 and relation (1.3) complete the proof.
Throughout the next two sections we assume that 1 ≤ p < ∞.

Lower bound of operators on the Copson matrix domain for 1 ≤ p < ∞
In this section, we intend to compute the lower bound of operators from p into p (C n ), from p (C n ) into p , and from p (C n ) into itself.
Recall the Hilbert matrix H = (h j,k ), which was introduced by David Hilbert in 1894 to study a question in approximation theory: We know that for p ≥ 1, the Hilbert operator H is a bounded operator on p with H p = π csc(π/p) (see [12,Theorem 323]) and the lower bound ζ (p) 1/p (see [6,Theorem 5]).
For a positive integer n, we define the Hilbert matrix of order n, H n = (h n j,k ), by h n j,k = 1 j + k + n + 1 (j, k = 0, 1, . . . ).
We say that Q = (q n,k ) is a quasisummability matrix if it is an upper-triangular matrix, i.e., q n,k = 0 for n < k, and k n=0 q n,k = 1 for all k. The product of two quasisummability matrices is also a quasisummability matrix and all these matrices have the lower bound 1 on p , according to the following theorem.

Lower bound of operators from p into p (C n )
In this part of study we compute the lower bound of some well-known operators like Hilbert and transposed Hausdorff operators on the domain of Copson matrix.  and p -norm S α,n p = (α + 1) (n + 1/p) (n + 1) (α + 1/p) .

Corollary 4.5
Let α, n be two nonnegative integers that α > n ≥ 0. The matrix S α,n defined in relation (4.2) has a lower bound from p into p (C n ) and L S α,n p , p (C n ) = 1.
Proof According to Lemmas 3.1 and 4.4 and Theorem 4.2, we have L S α,n p , p (C n ) = L C n S α,n = L C α = 1.

Corollary 4.6
For every quasisummability matrix Q, we have L(Q) p , p (C n ) = 1.
In particular, for every Hausdorff matrix H μ , we have L(H μt ) p , p (C n ) = 1.

Lower bound of operators from p (C n ) into p
In this part of study we compute the lower bound of transposed Hausdorff operators on the domain of Copson matrix. Proof Since the factor H ϕt in the factorization H μt = C n H ϕt is a quasisummability matrix, the proof is similar to that of Theorem 2.1, so

Lower bound of operators on p (C n )
In this part of study we compute the lower bound of Hilbert and transposed Hausdorff operators on the domain of Copson matrix.

Lower bound of Copson operator on some sequence spaces
In this section, we investigate the problem of finding the lower bound of Copson operator on some sequence spaces. Through out this section we assume that 1 ≤ p < ∞.

Lower bound of Copson operator on the difference sequence spaces
In this part of study, we investigate the lower bound of the Copson matrix of order n on the difference sequence spaces. In so doing we need the following preliminaries. Let n ∈ N and n F = (δ n F j,k ) be the forward difference operator of order n with entries We define the sequence space p ( n F ) as the set {x = (x k ) : with seminorm, · p ( n F ) , which is defined by Note that this function will not be a norm, since if x = (1, 1, 1, . . .) then x p ( n F ) = 0 while x = 0. The definition of the backward difference sequence space p ( n B ) is similar to p ( n F ), except that · p ( n B ) is a norm. For the special case n = 1, we use the notations B and F to indicate the backward and forward difference matrices of order 1, respectively. These matrices are defined by respectively. The domains c 0 ( F ), c( F ), and ∞ ( F ) of the forward difference matrix F in the spaces c 0 , c, and ∞ were introduced by Kizmaz [17]. Moreover, the domain bv p of the backward difference matrix B in the space p has been recently investigated for 0 < p < 1 by Altay and Başar [1], and for 1 ≤ p ≤ ∞ by Başar and Altay [5].
The constant in (5.1) is the best possible, and there is equality only when x = 0 or p = 1, or when dμ(θ ) is the point mass at 1.
For example, by choosing dμ(θ ) = n(1θ ) n-1 dθ , the lower bound of the Cesàro matrix of order n is In particular, for n = 1, the well-known Cesàro operator has the lower bound L(C t ) = ζ (p) 1/p .

Theorem 5.2
The Copson matrix of order n, C n , is a bounded operator from p ( n B ) into p ( n F ) and L C n p ( n B ), p ( n F ) = ∞ k=0 n n + k p 1/p . In particular, the Copson matrix is a bounded operator from p ( B ) into p ( F ) and L(C) p ( B ), p ( F ) = ζ (p) 1/p . Proof Let n F C n = D n . It has proved by Theorem 4.1 in [24] that the matrix D n = (d n i,j ) = I i,j / n+j j is a diagonal matrix, where I is the identity matrix. The facts that n B is the transpose of n F and n F C n is a diagonal matrix result in the identity n F C n = C nt n B . Now, by applying Lemma 3.1 and relation (5.2), we have L C n p ( n B ), p ( n F ) = L C nt = ∞ k=0 n n + k p 1/p , which completes the proof.

Lower bound of Copson operator on the domain of Hilbert matrix
Let n be a nonnegative integer and hil(n, p) be the sequence space associated with the Hilbert matrix of order n, H n , which is hil(n, p) = x = (x k ) ∈ ω : Note that, by letting n = 0 in the above definition, we obtain the domain of Hilbert matrix hil(p).

Corollary 5.3
The Copson operator of order n, C n , is a bounded operator from hil(n, p) into hil(p) and L C n hil(n,p),hil(p) = 1.
Proof According to Lemma 4.1, we have HC n = C n H n . Now, Lemma 3.1 and Theorem 4.2 complete the proof.

Lower bound of Copson operator on the domain of the transposed Hausdorff matrix
Let hau t (p) be the domain of the transposed Hausdorff matrix in the space p , that is, {x ∈ p : H μt x ∈ p }. Then

Corollary 5.4
The Copson operator of order n, C n , is a bounded operator from • p into hau t (p) and L(C n ) p ,hau t (p) = 1, • hau t (p) into p and L(C n ) hau t (p), p = 1, • hau t (p) into itself and L(C n ) hau t (p) = 1.
Proof According to Lemma 3.1, L(C n ) p ,hau t (p) = L(H μt C n ). Now, since the product of any two quasi-Hausdorff matrices is also a quasi-Hausdorff matrix, Theorem 4.2 completes the proof.
Let ν be the quotient measure in the factorization of the Copson matrix C n = H ν H μt , where H ν is a quasi Hausdorff matrix. Now, by applying Lemma 3.1 and Theorem 4.2, L(C n ) hau t (p), p = L(H ν ) = 1.
The fact that the Hausdorff matrices commute is also valid for their transposes H μt C n = C n H μt . Hence the proof is obvious by applying Lemma 3.1 and Theorem 4.2.