Fractional Cesàro Matrix and its Associated Sequence Space


 In this research, we introduce a new fractional Cesàro matrix and investigate the topological properties of the sequence space associated with this matrix.We also introduce a fractional Gamma matrix aswell and obtain some factorizations for the Hilbert operator based on Cesàro and Gamma matrices. The results of these factorizations are two new inequalities one ofwhich is a generalized version of thewell-known Hilbert’s inequality. There are also some challenging problems that authors share at the end of the manuscript and invite the researcher for trying to solve them.


Introduction
Let ω be the space of all real-valued sequences. The space p consists all real sequences u = (u k ) ∞ k= ∈ ω such that ∞ k= |u k | p < ∞ which a Banach space with the norm The spaces c, c and ∞ consist of all convergent, null and bounded real sequences, respectively. These spaces are all Banach spaces with the norm u ∞ = sup k |u k |. The supremum is taken over all k ∈ N = { , , , , ...}. Note that N is the set of all natural numbers , , , ....
A subspace of ω is called a sequence space. An in nite matrix is considered as a linear operator from a sequence space to another sequence space. Let U and V be sequence spaces and A = (a i,j ) be an in nite matrix. The mapping A : U → V de ned as Au = ((Au) i ) = ∞ j= a i,j u j is a matrix transformation if the series is convergent for each i ∈ N . By (U, V), we denote the family of all in nite matrices from U into V.
The sequence space is called the matrix domain of in nite matrix A in the sequence space U. 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, 3, 4, 6, 16, 18-20, 22, 27, 28, 30] and textbooks [2] and [21].
Recall the Hilbert matrix H = (h j,k ), which was introduced (1894) by David Hilbert to study a question in approximation theory: h j,k = j + k + (j, k = , , . . .).
Throughout this paper we suppose that n is a non-negative integer and we use the notation T n to denote the matrix T of order n which does not mean the n-th power of T.
Consider the Hausdor matrix H µ = (h j,k ), with entries of the form: where µ is a probability measure on [ , ]. The Hausdor matrix contains some famous classes of matrices. Two of these classes are as follow: (i) The choice dµ(θ) = n( − θ) n− dθ gives the Cesàro matrix of order n; (ii) The choice dµ(θ) = nθ n− dθ gives the Gamma matrix of order n.
By letting dµ(θ) = n( − θ) n− dθ and dµ(θ) = nθ n− dθ in the de nition of the Hausdor matrix, the Cesàro matrix of order n, C n = (c n j,k ), and the Gamma matrix of order n, Γ n = (γ n j,k ), are respectively, which according to the Hardy's formula have the norms where p * is the conjugate of p i.e. p + p * = .
Note that, for n = , C = I, where I is the identity matrix and for n = , C = Γ = C is the well-known Cesàro matrix, which is de ned by In 1986 Bennett [8], proved that the Hilbert matrix H, admits a factorization of the form H = BC, where C is the Cesàro matrix and the matrix B = (b j,k ) is de ned by and is a bounded operator on p with B p = π p * csc(π/p), ([8], Proposition 2). Later in 2019, Roopaei [24] generalized the Bennett's factorization of the forms H = B n C n and H = S n Γ n , where C n and Γ n are the Cesàro and Gamma matrices of order n respectively. The result of these factorizations are the inequalities and where for n = the rst one is the Hilbert's inequality and for n = the rst and second inequalities become In this study, we de ne fractional Cesàro and Gamma matrices and study the sequence space associated with fractional Cesàro matrix. Also, we tried to obtain several factorizations for the in nite Hilbert operator based on fractional Cesàro and Gamma matrices of the forms H = B n/m C n/m and H = S n/m Γ n/m . For another purpose of this study we obtain the upper bounds for the factors presented in Hilbert factorizations. Note that we use the notation . p for the norm of operators from p into itself.

Motivation.
The in nite Cesàro matrix was de ned by Ernesto Cesàro (1859 -1906) who was an Italian mathematician and worked in the eld of di erential geometry. After that, many mathematicians have done research on Cesàro sequence spaces and Cesàro function spaces, but all the researches have been around the Cesàro matrix of order one. Recently Roopaei et al. [25] have discussed the Köthe dual of Cesàro sequence spaces of order n, C n p = p(C n ) with ≤ p < ∞ as well as the problem of nding the norm of operators on this sequence space.
This manuscript investigates and characterizes spaces associated with a new matrix, who has a very close de nition to the Cesàro matrix, their duals and their linear operators. The topic has been studied for more than half a century by now, and the author's aim is to ll in some empty spots in the theory. There is also a challenging problem about the norm of the new de ned fractional Cesàro matrix that the authors invite the readers to involve and help to solve that.
Fractional Cesàro Banach spaces C n/m p and C n/m ∞ Suppose that n, m are two non-negative integers that n ≥ m. Let us de ne the fractional Cesàro matrix C n/m = (c n/m j,k ) by Note that for n = m, C n/m = C n , where C n is the Cesàro matrix of order n de ned by relation (1.2). We say that A = (a j,k ) is a summability matrix if it is a lower-triangular matrix, i.e. a j,k = for j < k, and j k= a j,k = for all j. One can easily investigate that the Hausdor matrices and consequently Cesàro matrix of order n are summability matrix, while for n ≠ m, the fractional Cesàro matrix is not a summability matrix.
Proof. By applying the identity now the result is obvious.

Lemma 2.2. The fractional Cesàro matrix C n/m , is invertible and it's inverse C
Proof. By applying Lemma 2.1 we have where I is the identity matrix. So the proof is complete. Now the sequence spaces C n/m p ( ≤ p < ∞) and C n/m ∞ are introduced by using the fractional Cesàro matrix as the set of all sequences whose C n/m -transforms are in the spaces p and ∞, respectively; that is Throughout the study, v = (v j ) will be the C n/m -transform of a sequence u = (u j ); that is, for all j ∈ N . Also, the relation respectively.
Proof. We omit the proof which is a routine veri cation.

Remark 2.4.
If we choose n = m, then we obtain the Cesàro sequence spaces C n p and C n ∞ de ned in [25] which are Banach spaces endowed with the norms respectively.
Theorem 2.5. The spaces C n/m p and C n/m ∞ are linearly isomorphic to p and ∞, respectively.
Proof. Let de ne a mapping T : C n/m p → p with T(u) = ((C n/m u) j ) for any u ∈ C n/m p . It is clear that T is linear and one-to-one. Now consider the sequence u = (u k ) given as onto. Also, since u C n/m p = C n/m u p holds we obtain that T preserves norms. This completes the proof.
Remark 2.6. If we choose n = m, then we obtain that the Cesàro sequence spaces C n p and C n ∞ are linearly isomorphic to p and ∞, respectively. Remark 2.7. The space C n/m is an inner product space with the inner product de ned as u,ũ C n/m = C n/m u, C n/mũ , where ., . is the inner product on .
Theorem 2.8. The space C n/m p is not an inner product space except for p = . Thus, the space C n/m p is not a Hilbert space except for p = .
Proof. Let us de ne the sequences u = (u j ) andũ = (ũ j ) as Proof. Let u ∈ C n/m p . Then, we have C n/m u ∈ p. Since the inclusion p ⊂ q holds for ≤ p < q < ∞, we have C n/m u ∈ q. This implies that u ∈ C n/m q . Hence, we conclude that the inclusion C n/m p ⊂ C n/m q holds. Now, we show that the inclusion is strict. Since the inclusion p ⊂ q is strict, we can choose v = (v j ) ∈ q\ p. De ne a sequence u = (u j ) as Then, we have (C n/m u) j = v j for every j ∈ N which means C n/m u = v and so C n/m u ∈ q\ p. Hence, we conclude that u ∈ C n/m q \C n/m p and so the inclusion C n/m p ⊂ C n/m q is strict.
Proof. Let u ∈ C n/m p . Then, we have C n/m u ∈ p. Since the inclusion p ⊂ ∞ holds for ≤ p < ∞, we have C n/m u ∈ ∞. This implies that u ∈ C n/m ∞ . Hence, we conclude that the inclusion C n/m p ⊂ C n/m ∞ holds. Now, we show that the inclusion is strict. Consider the sequence u = (u j ) de ned as We deduce that Since we have C n/m u = ((− ) j ) ∈ ∞\ p, we obtain that u ∈ C n/m ∞ \C n/m p . Consequently, the inclusion C n/m p ⊂ C n/m ∞ is strict.
Theorem 2.11. De ne the sequence (e (k) ) = (e (k) j ) for each k ∈ N by Then, the sequence (e (k) ) is a basis for the space C n/m p , and each u ∈ C n/m p has a unique representation of the form u = k (C n/m u) k e (k) .
Proof. Let A be a triangle. By Theorem 2.3 of Jarrah and Malkowsky [17], the matrix domain U A has a basis if and only if the normed sequence space U has a basis. Hence the proof follows immediately.
We use the following lemma to compute the dual spaces. By N, we denote the family of all nite subsets of N. The α-dual of a sequence space U consists of all sequences a = (a k ) ∈ ω such that au = (a k u k ) ∈ for all u = (u k ) ∈ U.
Theorem 2.13. The α-duals of the spaces C n/m , C n/m p ( < p < ∞) and C n/m ∞ are as follows: Proof. Let a = (a j ) ∈ ω and de ne the matrix D = (d j,k ) as For any u = (u j ) ∈ C n/m p ( < p < ∞), we have a j u j = (Dv) j for all j ∈ N. Thus au ∈ with u ∈ C n/m p if and only if Dv ∈ with v ∈ p. Hence, we conclude that a ∈ (C n/m p ) α if and only if D ∈ ( p , ). This completes the proof by part (ii) of Lemma 2.12. The other cases can be proved similarly.
The β-dual of a sequence space U consists of all sequences a = (a k ) ∈ ω such that n k= a k u k ∈ c for all u = (u k ) ∈ U. Theorem 2.14. Let de ne the following sets.
Proof. a = (a k ) ∈ (C n/m ) β if and only if the series ∞ k= a k u k is convergent for all u = (u k ) ∈ C n/m . From the equality Hence, we conclude by part (iv) of Lemma 2.12 that The γ-dual of a sequence space U consists of all sequences a = (a k ) ∈ ω such that n k= a k u k ∈ ∞ for all u = (u k ) ∈ U.
Theorem 2.15. The γ-duals of the spaces C n/m , C n/m p ( < p < ∞) and C n/m ∞ are as follows: Proof. By using the same technique in the proof of Theorem 2.14, we compute the gamma duals.

Factorization of the Hilbert matrix based on Cesàro matrices
In this section, we introduce several factorizations for the Hilbert matrix based on fractional Cesàro matrix. Throughout this section we suppose that m and n are two non-negative integers that n ≥ m. Let us de ne the matrix B n/m = (b n/m j,k ) by where the β function is Note that, in case n = m we have the matrix B m that has the entries b m j,k = k + m k β(j + k + , m + ), and according to [24] has the p-norm B n p = Γ(n+ /p * )Γ( /p) Proof. By di erentiating n − times the identity ( − z) − = ∞ j= z j , we get the result. .
Proof. By applying Lemma 3.1 we have Proof. By multiplying both sides of the identity in x m− and also integrating from to , we have the desired result.

Remark 3.4. By applying Lemma 3.3 one can straightly obtain
which results the factor B n/m in Theorem 3.2.
We are ready to generalize the Hilbert's inequality. Proof. According to Theorem 3.2 it is su cient to prove U n/m = (C −n/m ) t B n/m . By applying the identity n j= (− ) j n j z j = ( − z) n we have . Remark 3.7. By letting n = m in Corollary 3.6 and using the notation U n,n = U n we obtain the Corollary 2.6 of [24] which states that the Hilbert matrix H, has a symmetric factorization of the form H = (C n ) t U n C n , where the matrix U n is a bounded operator on p with u n j,k = n + j j n + k k β(j + k + , n + ) = n n j + k k β(j + n + , k + n + ) = n n (j + ) · · · (j + n)(k + ) · · · (k + n) (j + k + ) · · · (j + k + n + ) , and U n p = Γ(n + /p)Γ(n + /p * ) (Γ(n + )) .

. Factorization of the Hilbert operator based on Gamma matrices
In this part of our study, we state some factorizations for the Hilbert matrix based on fractional Gamma matrix. Suppose that n, m are two non-negative integers that n ≥ m. We de ne the fractional Gamma matrix Γ n/m = (γ n/m j,k ) by Obviously, for n = m, Γ n/m = Γ n . The fractional Gamma matrix, Γ n/m , is invertible and it's inverse Γ −n/m = (γ −n/m j,k ) is de ned by otherwise.

Lemma 3.8. For the fractional Cesàro and Gamma matrices the following identity holds:
C n/m = Γ n/m C n− .
In particular, for n = m, we have the identity C n = Γ n C n− = C n− Γ n . is a bounded operator on p and S n/m p ≤ π( − np ) csc(π/p).
But Roopaei in [24] Theorem 2.8 has proved that S n is a bounded operator and S n p = π( − np ) csc(π/p), which completes the proof.