A weighted mean Hausdorff type operator and its summability matrix domain

In this research, we firstly introduce a criterion for factorizing an operator based on the gamma operator, and as a result we present two factorizations for the Cesàro and Hilbert operators. As another point of view, we study some properties of the matrix domain associated with the gamma matrix of order n and compute the duals of this space. Moreover, we compute the norm of well-known operators into, from, and on the gamma sequence space.


Introduction
By ω we denote the set of all real-valued sequences, and a sequence space is a subspace of ω. The following sets are some examples for the sequence spaces.
In this paper, the supremum is taken over all k ∈ N 0 = {0, 1, 2, 3, . . .}. Also, we use the notion N = {1, 2, 3, . . .}. For two sequence spaces X, Y and an infinite matrix A = (a j,k ), we define a matrix transformation from X into Y as Ax = ((Ax) j ) = ( ∞ k=0 a j,k x k ) provided that the series is convergent for each j ∈ N 0 . The class of all infinite matrices from X into Y is denoted by (X, Y ).
The matrix domain of an infinite matrix A in a sequence space X is defined as which is also a sequence space.
In the literature, there are many papers related to new sequence spaces constructed by means of the matrix domain of a special triangle. In order to give full knowledge on the domains of triangular matrices in classical sequence spaces and related topics, the articles [1][2][3][4][5][6][7][8][9][10][11] and the monographs [12] and [13] are recommended.
Let X and Y be subsets of ω. The set M(X, Y ) = {a ∈ ω : ax ∈ Y for all x ∈ X} is called the multiplier space of X and Y . In the special cases Y = 1 , Y = cs, and Y = bs, the multiplier spaces X α = M(X, 1 ), X β = M(X, cs), and X γ = M(X, bs) are called the α-, β-, and γduals of the space X, respectively, that is, Hausdorff matrices. 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 follows: • 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, dθ gives the Hölder matrix of order n, • The choice dμ(θ ) = point evaluation at θ = n gives the Euler matrix of order n. Hardy's formula ( [14], Theorem 216) states that the Hausdorff matrix is a bounded operator on p if and only if Cesàro matrix of order n. The measure dμ(θ ) = n(1θ ) n-1 dθ gives the Cesàro matrix C n = (c n j,k ) of order n, which is defined by Hence, according to Hardy's formula, C n has the p -norm C n p → p = (n + 1) (1/p * ) (n + 1/p * ) , (1.2) where p * is the conjugate of p i.e. 1 p + 1 p * = 1. Note that C 0 = I, where I is the identity matrix, and is the classical Cesàro matrix which has the p -norm C p → p = p p-1 .
Gamma matrix of order n. By letting dμ(θ ) = nθ n-1 dθ in the definition of the Hausdorff matrix, the gamma matrix n = (γ n j,k ) of order n is given by which according to relation (1.1) has the p -norm While 1 is the classical Cesàro matrix C, other values of n give birth to different matrices. For example, In this study, we firstly introduce a criterion for factorizing an operator based on the gamma operator. Secondly we investigate the matrix domain of the gamma matrix of order n in the spaces p (1 ≤ p < ∞) and ∞ and compute the duals of the resulting spaces.
We shall deal with the spaces p of absolutely p-summable real sequences with 1 < p < ∞ endowed with the norm Motivation. Let us recall the definition of Nörlund and weighted mean matrices. Suppose that a = (a j ) ∞ j=0 is a nonnegative sequence with a 0 > 0 and A j = a 0 +a 1 +· · ·+a j . The Nörlund and weighted mean matrices N a = (a j,k ) and M a = (a j,k ) are lower triangular matrices which are defined by respectively. The sequence (a j ) ∞ j=0 is called the "symbol" of Nörlund and weighted mean matrices.
It must be mentioned that the Cesàro and gamma matrices of order n are the Nörlund and weighted mean matrices with symbol a n j = n+j-1 j . For example, for n = 1 and n = 2, the sequences a 1 j = 1 and a 2 j = 1 + j are the symbols of the Cesàro and gamma matrices of order 1 (the well-known Cesàro matrix) and the Cesàro and gamma matrices of order 2, with the entries presented in (1.4). As we mentioned in the introduction, both these matrices are also in the classes of Hausdorff matrices with different choosing of their probability measures.
From both aspects of being a Hausdorff operator or a weighted mean matrix, the gamma operator has not been seen as it deserves. Many mathematicians have done and still publish numerous articles about the Cesàro matrix, Cesàro matrix domain, and Cesàro function spaces [15][16][17][18][19], while the importance of the gamma operator or its associated matrix domain has been ignored under the shadow of its rival Cesàro matrix. In this present study, the authors try to reveal some brilliant characteristics of this operator as an independent matrix.

Factorization based on gamma operator
In this section, we introduce the necessary and sufficient conditions for factorizing an operator based on the gamma matrix, then as the result we obtain two factorizations for the Cesàro and Hilbert matrices.
j,k=0 be a matrix and n = (γ n j,k ) be the gamma matrix of order n. There exists a factor S n = (s n j,k ) ∞ j,k=0 such that T = S n n if and only if (i) , which gives (ii).
This implies that S n n exists and T = S n n .
Recall the definition of the Hilbert matrix H = (h j,k ) = ( 1 j+k+1 ) of nonnegative integers j, k. The Hilbert operator has the matrix representation which is a bounded operator on p and H p = π csc(π/p), ( [20], Theorem 323). Bennett in [21] showed that the Hilbert matrix admits a factorization of the form H = BC, where C is the Cesàro matrix and B = (b j,k ) is defined by The matrix B is a bounded operator on p and B p = π p * csc(π/p). For obtaining our first result, we need the Hellinger-Toeplitz theorem.

Corollary 2.3
The Hilbert matrix has a factorization of the form H = S n n , where S n = (s n j,k ) with the entries is a bounded operator on p and S n p → p = π(1 -1 np ) csc(π/p). In particular, for n = 1, H = BC, where C is the Cesàro matrix and B is the matrix defined by relation (2.1).
Proof Let H = (h j,k ) be the Hilbert matrix. Since therefore H has a factorization H = S n n , where the factor S n = (s n j,k ) in Theorem 2.1 is

Now, our factorization and relation (1.3) result in
For proving the other side of the above inequality, by using the definition of the matrix B, as in relation (2.1), we can rewrite Hence, by applying the Hellinger-Toeplitz theorem, we have which completes the proof. In a special case n = 1, S 1 = B, S 1 p → p = B p → p = π p * csc(π/p), 1 = C and the Hilbert matrix has the factorization H = BC.

2)
where C n-1 is the Cesàro matrix of order n -1.
Proof In this case, for t n j,k = c n j,k = , we have Hence C n has the factorization of the form C n = S n n . Now, the factor S n = (S n j,k ) in Theorem 2.1 is Hence we have proved C n = C n-1 n . But, since every two Hausdorff matrices commute ( [14], Theorem 197), we have the desired result.
Roopaei in [22,23] introduced more factorizations for the Hilbert and Cesàro operators based on gamma matrices of order n.

Matrix domain of gamma matrix in p
The sequence space associated with n is the set {x = (x k ) ∈ ω : n x ∈ p }. That is, which is called the gamma space of order n. In a special case n = 1, we show the gamma sequence space G 1 (p) by the notation ces(p). Also, we define the following sequence space G n (∞): In the study, by y = (y j ), we mean the n -transform of a sequence x = (x j ), that is, for all j ∈ N 0 . The gamma matrix of order n, n , is invertible and its inverse -n = (γ -n j,k ) is defined by Proof We omit the proof which is a routine verification.

Theorem 3.3
The spaces G n (p) and G n (∞) are linearly isomorphic to p and ∞ , respectively.
Proof We only prove the first one and the other one can be proved in a similar way. Since n is invertible, the map x → n x defines a bijection between G n (p) and p . Also, since x G n (p) = n x p holds, the defined map preserves the norm, which completes the proof.
Remark 3.4 If n = 1, the Cesàro sequence spaces ces(p) and ces(∞) are linearly isomorphic to p and ∞ .
Remark 3.5 The space n 2 is an inner product space with the inner product defined as x,x n 2 = n x, nx 2 , where ·, · 2 is the inner product on 2 .

Theorem 3.6 The space G n (p) is not an inner product space for p = 2. Then the space G n (p)
is not a Hilbert space for p = 2.
Remark 3.7 If n = 1, the Cesàro sequence space ces(p) is not a Hilbert space for p = 2.
Proof Given any x ∈ G n (p), we have n x ∈ p . Since the inclusion p ⊂ q holds for 1 ≤ p < q < ∞, we have n x ∈ q . This implies that x ∈ G n (q). Hence, we conclude that the inclusion G n (p) ⊂ G n (q) holds.
Also, since the inclusion p ⊂ q is strict, we can choose y = (y j ) ∈ q \ p . If we define a sequence x = (x j ) as then we have n x j = y j for every j ∈ N 0 . This means n x = y, and so n x ∈ q \ p . Since we have x ∈ G n (q)\G n (p), we conclude that the inclusion G n (p) ⊂ G n (q) strictly holds.
Proof Choose any x ∈ G n (p). Then we have n x ∈ p . Since the inclusion p ⊂ ∞ holds for 1 ≤ p < ∞, we have n x ∈ ∞ . This implies that x ∈ G n (∞). Hence, we conclude that the inclusion G n (p) ⊂ G n (∞) holds. Consider the sequence x = (x j ) with

It follows that
Since n x = ((-1) j ) ∈ ∞ \ p holds, we have x ∈ G n (∞)\G n (p). This means that the inclusion G n (p) ⊂ G n (∞) is strict.

Theorem 3.12
The space G n (p) (1 ≤ p < ∞) has a basis (c k ) defined as Further, every x ∈ G n (p) is written in the form uniquely.
Proof From (3.2), we have that n (c k ) = e k ∈ p , and so c k ∈ G n (p) for each k ∈ N. Now, let x ∈ G n (p). For every m ∈ N 0 , write Then we have that and it follows that Given any ε > 0, there exists M ∈ N 0 such that We have that for every m ≥ M, which implies that lim m→∞ xx m G n (p) = 0. This implies that x = k ( n x) k c k . Finally, we show that the representation (3.3) of x ∈ G n (p) is unique. On the contrary, we suppose that x = k ( x) k c k . By using the continuity of the mapping L : G n (p) → p defined in the proof of Theorem 3.3, we have Hence, the representation (3.3) of x ∈ G n (p) is unique, which completes the proof.
The following lemma is essential to determine the dual spaces. Throughout the paper, N is the collection of all finite subsets of N. Firstly, we list some conditions: lim j→∞ a j,k exists for each k ∈ N, Table 1. Theorem 3.14 The α-dual, β-dual, and γ -dual of the space G n (1) are as follows: Proof Firstly, we compute the α-dual. Given any a = (a k ) ∈ ω, we define a matrix B = (b j,k ) as

Lemma 3.13 ([25]) The necessary and sufficient conditions for
If we choose any x = (x j ) ∈ G n (1), we obtain that a j x j = 1 + j n a j y j -j n a j y j-1 = (By) j for all j ∈ N. Hence, it follows that ax ∈ 1 for x ∈ G n (1) if and only if By ∈ 1 for y ∈ 1 . Thus, we have a ∈ (G n (1)) α if and only if B ∈ ( 1 , 1 ). By using Lemma 3.13, we conclude that holds. Now, we compute the β-dual. a = (a k ) ∈ (G n (1)) β if and only if the series ∞ k=0 a k x k is convergent for all x = (x k ) ∈ G n (1). Then, from the equality  Finally, the γ -dual can be computed by utilizing the same technique with the β-dual. From equality (3.12), we deduce that a = (a k ) ∈ (G n (1)) γ if and only if the matrix C = (c j,k ) is in the class ( 1 , ∞ ). Hence, the proof follows from Lemma 3.13.

Theorem 3.15
The α-dual, β-dual, and γ -dual of the space G n (p) for 1 < p < ∞ are as follows: Proof Consider the matrices B = (b j,k ) and C = (c j,k ) defined in the proof of Theorem 3.14. Let 1 < p < ∞. The α-, β-, and γ -dual of the space G n (p) can be computed from the facts that B ∈ ( p , 1 ), C ∈ ( p , c), and C ∈ ( p , ∞ ), respectively.
Proof Consider the matrices B = (b j,k ) and C = (c j,k ) defined in the proof of Theorem 3.14.

Norm of operators on a gamma sequence space
In this section, we investigate the norm of well-known operators from/into the gamma matrix domain, and we firstly start with the Hausdorff operators. For computing the norm