Bounds of operators on the Hilbert sequence space

Abstract The author has computed the bounds of the Hilbert operator on some sequence spaces [18, 19]. Through this study the author has investigated the bounds of operators on the Hilbert sequence space and the present study is a complement of those previous research.


Introduction
Let p ≥ and ω denote the set of all real-valued sequences. The space p is the set of all real sequences x = (x k ) ∈ ω such that The operator T is called bounded, if the inequality Tx p ≤ K x p holds for all sequences x ∈ p, while the constant K is not depending on x. The constant K is called an upper bound for operator T and the smallest possible value of K is called the norm of T. We also seek the inequalities of the form Tx p > L x p , valid for every x ∈ p with x > x > · · · > . The lower bound of T is the greatest possible value of L.
Matrix domain. The matrix domain of an in nite matrix A in a sequence space X is de ned as A(X) = {x ∈ ω : Ax ∈ X}, which is also a sequence space. In especial case X = p we use the notation A(p) which has the de nition Note that, for the identity matrix A = I, A(p) = p. By using matrix domains of special triangle matrices in classical spaces, many authors have introduced and studied new Banach spaces.
Hausdor matrix. The Hausdor matrix H µ = (h j,k ) ∞ j,k= is de ned by where µ is a probability measure on [ , ]. Hardy's formula ( [9], Theorem 216) states that the Hausdor matrix is a bounded operator on p if and only if θ − p dµ(θ) < ∞ and Hausdor operator has also the following formula as its lower bound. (1. 2) The constant in (1.2) is the best possible, and there is equality only when x = or p = or dµ(θ) is the point mass at 1.
Cesàro matrix of order n. By letting dµ(θ) = n( − θ) n− dθ in the de nition of the Hausdor matrix, the Cesàro matrix of order n, C n = (c n j,k ), is de ned by Note that, C = C is the well-known Cesàro matrix for all j, k ∈ N. That is, with the p-norm C p = p * (where p * is the conjugate of p i.e. p + p * = ) and lower bound The Cesàro matrix domain ces(n, p) is the set of all sequences whose C n -transforms are in the space p, that is which is a Banach space with the norm x ces(n,p) Note that, for special case n = , we use the notation ces(p) instead of ces ( , p). For more information about the Cesàro sequence space ces(n, p) , the reader can refer to [21].
Gamma matrix. By letting dµ(θ) = nθ n− dθ in the de nition of the Hausdor matrix, the Gamma matrix of order n, Γ n = (γ n j,k ), is The sequence space associated with the matrix Γ n , is the set which is called the Gamma space of order n. In special case n = , we show the Gamma sequence space gam( , p) by the notation ces(p). Hilbert matrix. The Hilbert matrix H = (h j,k ) is de ned by for all integers j, k. That is, By theorem 323 of [7], H is a bounded operator and The sequence space associated with the Hilbert matrix, hil(p), is de ned by which has the norm According to the inequality which has proved in Theorem 11.5 of [2] x ces(p) ≤ x hil(p) ≤ π p * csc(π/p) x ces(p) , we obtain that hil(p) = ces(p). On the other hand, since ces(p) and p are isomorphic, hence the isomorphism hil(p) ∼ = p will be resulted.

Motivation.
Parallel to the several research on the nite Hilbert operator, see [1,4,8,23], there is some information about the in nite version of Hilbert matrix [24][25][26][27]. Recently the author [15,16] has introduced some factorizations for the in nite Hilbert matrix based on the generalized Cesàro matrix and Cesàro and Gamma matrices of order n. The author has also computed the norm and the lower bound of Hilbert operator on some sequence spaces [18,19]. Through this study the author has tried to compute the norm and lower bound of several operators on the Hilbert sequence space that have not been done before.

Bounds of Cesàro and Copson operators on the Hilbert sequence space
In this study, we intend to nd the bounds of some well-known operators on the Hilbert sequence space. In so doing we need the following lemma which is the combination of Lemmas 2.1 in [18] and [19].

In particular, if AT = UA, then T is a bounded operator from the matrix domain Ap into itself and T Ap = U p and L(T) Ap = L(U). Also, if T and A commute then T Ap = T p and L(T) Ap = L(T).
More recently, several mathematicians have investigated the problem of nding the norm of operators on some matrix domains [5, 6, 10-14, 17, 20, 21].
Throughout this research, we use the notations · p and L(·), for the norm and lower bound of operators on sequence space p and · X,Y and L(·) X,Y for the norm and lower bound of operators from the matrix domain X into the sequence space Y.
Hilbert matrix of order n. For a non-negative integer n, we de ne the Hilbert matrix of order n, H n = (h n j,k ), by h n j,k = j + k + n + (j, k = , , · · · ).
Note that for n = , H = H is the well-known Hilbert matrix. For more examples: Similarly, the sequence space associated with the Hilbert matrix of order n, hil(n, p), is de ned by For non-negative integers n, j and k, let us de ne the matrix B n = (b n j,k ) by b n j,k = (k + ) · · · (k + n) (j + k + ) · · · (j + k + n + ) where the β function is Consider that for n = , B = H, where H is the Hilbert matrix.
For computing the norm of Cesàro and Copson operators on the Hilbert matrix domain we need the following lemma.

In particular, the identity operator is a bounded operator from ces(p) into hil(p) and I ces(p)
,hil(p) = π p * csc( π p ).
Proof. According to Lemma 2.2, the Hilbert operator has a factorization of the form H = B n C n . Since the map ces(n, p) → p, x → C n x is an isomorphism between these two spaces, hence In particular, for n = , H = BC, where C is the Cesàro matrix and B is a bounded operator with B p = π p * csc(π/p).

Theorem 2.5. The identity operator, I, is a bounded operator from the Gamma space gam(n, p) into the Hilbert space hil(p) and
I gam(n,p),hil(p) = π − np csc(π/p).

In particular, the identity operator is a bounded operator from ces(p) into hil(p) and I ces(p),hil(p)
Proof. The map gam(n, p) → p, x → Γ n x is an isomorphism between these two spaces.
In particular, for n = , Γ = C, hence we obtain the result.
Theorem 2.6. The Cesàro operator of order n, C n , is a bounded operator from the sequence space hil(p) into p and L(C n ) hil(p), p = Γ(n + ) Γ(n + /p * )Γ( /p) .
Proof. According to Lemma 2.2, the Hilbert operator H has a factorization of the from H = B n C n , which results in the inequallity where the constant is best possible. Now, by taking in mum from both sides of the inequality C n x p Hx p ≥ Γ(n + ) Γ(n + /p * )Γ( /p) , the proof is complete.
Corollary 2.7. The Cesàro operator of order n − , C n− , is a bounded operator from the Cesàro space ces(n, p) into the Hilbert space hil(p) and C n− ces(n,p),hil(p) = π − np csc(π/p).
In particular, the identity operator is a bounded operator from ces(p) into hil(p) and I ces(p),hil(p) = π p * csc( π p ).
Proof. The sequence spaces ces(n, p) and p are isomorphic. According to the theorem 2.4 and the identity C n = Γ n C n− we have HC n− = S n C n that results in S n C n x p C n x p = sup y∈ p S n y p y p = S n p = π − np csc(π/p).
In particular, for n = , C = I and C = C, where I is the identity matrix and C is the Cesàro matrix. Therefore we have the desired result.
We say that Q = (q n,k ) is a quasi-summability matrix if it is an upper-triangular matrix, i.e. q n,k = for n < k, and k n= q n,k = for all k. The product of two quasi-summability matrices is also a quasi-summability matrix and all these matrices have the lower bound 1 on p, according to the following theorem. Theorem 2.10 ([3], Theorem 2). Let p be xed, < p < ∞, and let T be a quasi-summability matrix. If x ∈ p satis es x > x > · · · > , then Tx q ≥ x p . Now, according to the above theorem we have L(C nt ) = .
Similar to the Cesàro matrix, we have the following results for the Copson operator of order n.
In particular, the Copson operator has the lower bound L(C t ) hil( ,p), p = p π sin(π/p).

Bounds of Gamma operator on the Hilbert sequence space
In this section we intend to compute the bounds of Gamma operator on the Hilbert matrix domain, but we need the following lemma rst. Proof. (i) By applying the identity ∞ j= z j = ( − z) − for |z| < , we deduce that (ii) By applying the identity H = B n C n , commutative property of Hausdor matrices and part (i) we have HΓ n = B n C n Γ n = B n Γ n C n = B n− C n .
(iii) By applying lemma 2.2, part (i) and the identity C n = Γ n C n− we have H n Γ n = C n B n Γ n = C n B n− = Γ n H n− . So the proof is complete.

Theorem 3.3. The Gamma operator of order n, Γ n is a bounded operator from the sequence space hil(p) into
(ii) Let A be the matrix de ned in part (i). According to Lemma 2.1 part (i) ∆ n B p ,hil(p) = H∆ n B p = A p .
By a simple callculation u k = ∞ j= a j,k = (n − )! (k + ) · · · (k + n) , where u k is the k th column sum of A. Since n = u > u > · · · and A is symmetric, hence R and K are both n in Schur's theorem. Therefore A p ≤ n .