ON THE LIMITED p-SCHUR PROPERTY OF SOME OPERATOR SPACES

We introduce and study the notion of limited p-Schur property (1 ≤ p ≤ ∞) of Banach spaces. Also, we establish some necessary and sufficient conditions under which some operator spaces have the limited p-Schur property. In particular, we prove that if X and Y are two Banach spaces such that X contains no copy of `1 and Y has the limited p-Schur property, then K(X,Y ) (the space of all compact operators from X into Y ) has the limited p-Schur property.


Introduction
A non-empty subset K of a Banach space X is said to be limited (resp Dunford-Pettis (DP)), if for every weak * -null (resp.weakly null) sequence (x * n ) in the dual space X * of X converges uniformly on K, that is, where x, x * denotes the duality between x ∈ X and x * ∈ X * .In particular, a sequence (x n ) ⊂ X is limited if and only if x n , x * n → 0, for all weak * -null sequences (x * n ) in X * .
A subset K of a Banach space X is a limited set if and only if for any Banach space Y , every pointwise convergent sequence (T n ) ⊂ L(X, Y ) converges uniformly on K, where L(X, Y ) denoted the space of all bounded operators from X into Y [17, Corollary 1.1.2].
It is easily seen that every relatively compact subset of a Banach space is limited.But the converse is not true, in general.If every limited subset of Banach space X is relatively compact, then X has the Gelfand-Phillips (GP ) property.For example, the classical Banach space c 0 and 1 have the GP property and every reflexive space and dual space containing no copy of 1 have the same property.
A sequence (x n ) in Banach space X is called weakly p-summable with 1 ≤ p < ∞, if for each x * ∈ X * , the sequence ( x n , x * ) ∈ p and a sequence (x n ) in X is said to be weakly p-convergent to x ∈ X if the sequence (x n − x) ∈ weak p (X), where weak p (X) denoted the space of all weakly p-summable sequence in X.Also a bounded set K in a Banach space is said to be relatively weakly p-compact, 1 ≤ p ≤ ∞ if every sequence in K has a weakly p-convergent subsequence.If the limit point of each weakly p-convergent subsequence is in K, then we call K weakly p-compact set.Also, a Banach space X is weakly p-compact if the closed unit ball B X of X is a weakly p-compact set.An operator T ∈ L(X, Y ) is said to be p-converging if it transfers weakly p-summable sequence into norm null sequences.The class of all p-converging operators from X into An operator T ∈ L(X, Y ) is limited p-converging if it transfers limited and weakly p-summable sequences into norm null sequences.we denote the space of all limited p-converging operators from X into Y by C lp (X, Y ) [7].
A Banach space X has the Schur property if every weakly null sequence in X converges in norm.The simplest Banach space with the Schur property is 1 .Also a banach space X has the p-Schur property (X) containing all weakly null sequences in X.So ∞-Schur property coincides with the Schur property.Also one note that every Schur space has the p-Schur property [6].
In this note, we study the limited p-Schur property of some operator spaces, specially, the space of compact operators.We prove that if X and Y are two Banach spaces such that X contains no copy of 1 and Y has the limited p-Schur property, then K(X, Y ) has the limited p-Schur property.Finally, we conclude that if (X α ) α∈I are Banach spaces and X = (⊕ α∈I X α ) 1 their 1 -sum, then the space X has the p-Schur property if and only if each factor X α has the same property.

Main results
Recall that the Banach space X has the limited p-Schur property if every limited weakly p-compact subset of X is relatively compact.More precisely, the Banach space X has the limited p- ) if all weakly p-compact sets in X are limited [10].
Recall that if M is a closed subspace of L(X, Y ), then for arbitrary elements x ∈ X and y * ∈ Y * , the evaluation operators φ x : M → Y and ψ y * : M → X * on M are defined by Theorem 2.1.Let X and Y be two Banach spaces such that X is weakly p-compact and Y has the p-Schur property.Then L(X, Y ) has the limited p-Schur property.
Proof.Suppose that (T n ) is a limited weakly p-summable sequence in L(X, Y ).We have to prove that (T n ) is norm null.We first observe that for every x ∈ X the evaluation operator φ x from L(X, Y ) to Y maps the sequence (T n ) to the sequence (T n x).So the latter is also a limited weakly p-summable sequence in Y .
Therefore T n x → 0, since Y has the limited p-Schur property.Now, suppose that (T n ) is not norm null.Then there is a sequence (x n ) in X and ε > 0 such that for all n ∈ N. Since X is weakly p-compact we may assume that there exists x ∈ X such that (x n − x) ∈ weak p (X).As T n x → 0, we may finally suppose that In the following theorem we give a necessary and sufficient condition for which a Banach space has the limited p-Schur property.
Theorem 2.2.The Banach space X has the limited p-Schur property if and only if for every Banach space Y .
Proof.Suppose that X has the limited p-Schur Conversely, If Y = X, then the identity operator on X is belongs to C lp .Therefore X has the limited p-Scuhr property.
Similarly, we can prove that the Banach space X has the limited p-Schur property if and only if L(Y, X) = C lp (Y, X) for every Banach space Y .
Hence (T n x n ) is weakly null and so is norm null.This contradiction shows that L(X, Y ) has the limited p-Schur property.
We can choose a sequence and so (T n x * n ) is limited.Finally, the GP property of Y yields that T n x * n → 0 which implies T n → 0.
Note that the map T → T * * is an isometric isomorphism from K(X, Y ) into K w * (X * , Y ).Therefore we have the following result.Since X ⊗ ε Y may be identified with a closed subspace of K w * (X * , Y ) via the isometric embedding (a) X * * has the GP property and for every weak * -null sequence (b) X contains no copy of 1 and for every weakly null sequence (x n ) ⊆ X, (T x n ) is norm null uniformly with respect T ∈ M .
Recall that the operator T ∈ L(X, Y ) is said to be limited operator if T (B X ) is a limited set in Y .The class of all limited operator from X into Y is denoted by L(X, Y ).On the other hand, T ∈ L(X, Y ) if and Theorem 2.6.Let X be a Banach space such that X * has the GP property.If F is a closed subspace of K(X, Y ) and for every x * * ∈ X * * , the evaluation operator φ x * * on F is limited p-converging, then F has the limited p-Schur property.
Proof.First, observe that the evaluation operator φ x * * , as a generalization of φ x is denoted by φ x * * (T ) = T * * x * * , for all T ∈ M and x * * ∈ X * * .
Let M ⊂ F be a limited weakly p-compact set.Since for every x ∈ X, the evaluation map φ x is limited pconverging, we conclude that M (x) = {T x : T ∈ M } is relatively compact.Since the adjoint of every limited operator is weak * -norm sequentially continuous, it follows that for every compact operator T ∈ K(X, Y ), the operator T * is also compact and so is limited.This shows that T * * is weak * -norm sequentially continuous and therefore for each weak * -null sequence (x * * n ) in X * * , the sequence ( Theorem 2.7.Let X be a Banach space containing no copy of 1 .If F is a closed subspace of K(X, Y ) such that for each x ∈ X, the evaluation operator φ x is limited p-converging, then F has the limited p-Schur property.
Recall that a subset H of L(X, Y ) is uniformly completely continuous, if for every weakly null sequence We remember the following theorem, which has a main role in the proof of the Theorem 2.9.
Theorem 2.8.[13] If X contains no copy of 1 , then a subset H ⊆ K(X, Y ) is relatively compact if and only if H is uniformly completely continuous and for each x ∈ X, the set φ x (H) is relatively compact in Y .
Theorem 2.9.If X contains no copy of 1 and Y has the limited p-Schur property, then K(X, Y ) has the limited p-Schur property.
Proof.If Y has the limited p-Schur property, then Theorem 2.2 shows that each φ This shows that H is uniformly completely continuous.Hence Theorem 2.5 (a) shows that H is relatively compact in K(X, Y ) and so K(X, Y ) has the limited p-Schur property.
Recall that if 1 ≤ p ≤ ∞, the Banach space X has the Dunford-Pettis property of order p (DP p ) if for each Banach space Y, every weakly compact operator T : X → Y is p-converging.For more information about DP p property of Banach spaces the reader is referred to [3].
We also notice that by Theorem 2.2, if the closed subspace M of L(X, Y ) has the limited p-Schur property, then all operators on M , such as evaluation operators, are limited p-converging.Therefore the converse of Theorem 2.6 is also true.Moreover, in the following two theorems 2.11 and 2.12, we will give another sufficient conditions for the limited p-Schur property of closed subspace M of some operator spaces with respect to the limited p-converging of evaluation operators.
To obtain our next result we need the following well known theorem.
Theorem 2.10.[9] Let X and Y be two Banach spaces and H be a subset of L(X, Y ) such that Then H is relatively compact.Proof.Suppose that H is a limited weakly p-compact subset of M .By Theorem 2.10, it is enough to show that H(B X ) and all ψ y * (H) are relatively compact in Y and X * , respectively.For every y * ∈ Y * , the evaluation operator ψ y * is limited p-converging.Therefore ψ y * (H) is relatively compact.On the other hand, if (y * n ) is a weak * -null sequence in Y * , then the weak * -norm sequential continuity of the adjoint of eah T ∈ H implies that ψ y * n (T ) = T * y * n → 0 as n → ∞.Therefore (ψ y * n ) converges pointwise on H an so it is converges uniformly on the subset Thus H(B X ) is limited and so is relatively compact.Now, we give a sufficient condition for the limited p-Schur property of subspaces of L w * (X * , Y ) of all bounded weak * -weak continuous operator from X * to Y .Clearly, if T ∈ L w * (X * , Y ), then T * transfers Y * into X.The proof of this theorem is similar to the proof of Theorem 3.6 of [6].So we omit its proof.
Theorem 2.12.Let X and Y be Banach spaces such that X has the Schur property.If M is a closed subspace of L w * (X * , Y ) such that every evaluation operator φ x * is limited p-converging on M , then M has the limited p-Schur property.
Recall that according to [6], a bounded subset K of a Banach space X is p-Limited if Also, a sequence (x * n ) in X * is an L p -set if and only if lim n→∞ x n , x * n = 0 for all (x n ) ∈ weak p (X) [7].It is clear that for every limited subset and every p-limited subset of a dual space is an L p -set.Moreover, the following result has been proved in [7].
Theorem 2.13.A Banach space X is weakly p-compact if and only if every L p -set in X * is relatively compact.
Theorem 2.14.Let X and Y be Banach spaces.If X contains no copy of 1 , Y * is weakly p-compact and for every h ∈ L(X, Y * * ), for every weakly null sequence (x n ) ⊂ X, the sequence (hx n ) is an L p -set, then K(X, Y ) has the GP property and so has the limited p-Schur property.
Proof.Let M ⊂ K(X, Y ) be a limited set.We have to prove that M is relatively compact.Since M because (T n ) is a limited set and so is a DP set.So we have actually proved that (T n x n ) is a p-limited set and so L p -set.It follows from Theorem 2.13 that it must be a relatively compact set.Since it is a weakly null sequence, there is a norm null subsequence and it is a contradiction.
In [18] the authors have been proved that for Banach spaces (X α ) α∈I , if X = (⊕ α∈I X α ) 1 is their 1 -direct sum, then X has the Schur property if and only if each factor X α has the same property.Here, by a similar idea, we prove that the same condition holds for (limited) p-Schur property.
Theorem 2.15.Let (X α ) α∈I be Banach spaces and X = (⊕ α∈I X α ) 1 .Then the space X has the p-Schur property if and only if each X α has the p-Schur property.
Proof.If X = (⊕ α∈I X α ) 1 has the p-Schur property, then clearly, every closed subspace of X has the p-Schur property.Hence each X α has the p-Schur property.On the other hand, a straightforward computations shows that a Banach space has the p-Schur property if and only if all of its closed separable subspaces have the p-Schur property.Therefore we can assume that each X α is separable and take I = N. Hence X = (⊕X k ) 1 is separable and so has the GP property.where k ≥ 1 and M k−1 + 1 ≤ j ≤ M k .Let h = (h j ) j≥1 = (λ 1 a n1,1 , λ 2 a n1,2 , ..., λ M1 a n1,M1 , λ M1+1 a n2,M1+1 , ...).

1 . 2 . 2 .Corollary 2 . 3 .
Contradicting the assumption that (T n ) is limited.Corollary 2.1.Let X and Y be two Banach spaces.If X is reflexive and Y has the Schur property, then L(X, Y ) has the GP property.Proof.Let p = ∞ in Theorem 2.Corollary Let X and Y be two Banach spaces.If X is a weakly p-compact and Y * has the p-Schur property, then (X ⊗ π Y ) * has the limited p-Schur property.Proof.It follows easily from the fact that L(X, Y * ) = (X ⊗ π Y ) * .Let X and Y be two Banach spaces.If X * has the p-Schur property and Y * is weakly p-compact, then L(X, Y ) has the limited p-Schur property.Proof.The mapping T → T * maps L(X, Y ) onto a closed subspace of L(Y * , X * ), which has the limited p-Schur property by virtue of Theorem 2.1.

Theorem 2 . 3 .
If X * has the limited p-Schur property and Y has the Schur property, then L(X, Y ) has the limited p-Schur property.Proof.Since X * has the limited p-Schur property, Theorem 2.2 implies that each ψ y * : L(X, Y ) → X * is limited p-converging.It follows that L(X, Y ) has the limited p-Schur property.In fact, if L(X, Y ) does not have the limited p-Schur property, then there exists a limited weakly p-summable sequence

1 (
X * ) has the limited p-Schur property.If we take p = ∞ in Theorem 2.3 we obtain the following result.Corollary 2.4.If X * has the GP property and Y has the Schur property, then L(X, Y ) has the GP property.Theorem 2.4.Let X and Y be Banach spaces.If X has the limited p-Schur property and Y has the GP property, then the space K w * (X * , Y ) of all compact weak * -weak continuous operators from X * into Y has the limited p-Schur property.

Corollary 2 . 5 .
Let X and Y be two Banach spaces.If X * has the limited p-Schur property and Y has the GP property, then K(X, Y ) has the limited p-Schur property.

Corollary 2 . 6 .
which is defined by x ⊗ y → θ x⊗y , where θ x⊗y (x * ) = x, x * y, we have the following corollary.If X has the limited p-Schur property and Y has the GP property, then injective tensor product X ⊗ ε Y has the limited p-Schur property.Theorem 2.5.[9,13]Let X and Y be two Banach spaces and M ⊆ K(X, Y ) such that for all x ∈ X, M (x) := {T x : T ∈ M } is relatively compact in Y .Then under each of the following conditions, M is a relatively compact subset of K(X, Y ).
is a pointwise norm null sequence of bounded linear operators.Hence (φ x * * n ) converges uniformly on the limited set M [17, Corollary 1.1.2].It follows that lim n→∞ sup T ∈M φ x * * n (T ) = 0. Then by Theorem 2.5 (a) M is relatively compact and so F has the p-Schur property.If one use Theorem 2.5 (b) instead of Theorem 2.5 (a), we can prove the following theorem.
Y ) is a limited weakly p-compact set.Therefore φ x (H) is relatively compact for all x ∈ X.On the other hand, if (x n ) is weakly null in X, then complete continuity of each operator T ∈ H implies that φ xn (T ) = T x n → 0. Therefore (φ xn ) is a norm null sequence at each element T ∈ H and then it is uniformly convergent on the limited set H [17, Corolarry 1.1.2

Theorem 2 . 11 .
Let M be a closed linear subspace of L(X, Y ) such that the closed linear span of the set M (X) = {T x : T ∈ M, x ∈ X} has the GP property.If all evaluation operator ψ y * are limited p-converging, then M has the limited p-Schur property.

k=1 δ k < ε 4 . 1 ( 2 )M k− 1 j=1| 3 )j>M k− 1 |
If (x n ) ∈ weak p (X), where x n = (b n,k ) k∈N , then (b n,k ) ∈ weak p (X k ) for all k ∈ N. Since X k has the p-Schur property, therefore ||b n,k || → 0 as n → ∞, for all k ∈ N. We have to prove that ||x n || → 0 or the weakly null sequence (x n ) is relatively compact.Let {f n } n∈N be a w * -null sequence in B X * .If we show that lim n→∞x n , f n = 0, then the proof is completed, thanks to the GP property of X.Each f n is of the form f n = (a n,k ) k∈N and for all k ∈ N, a n,k w * −→ 0 in X * k as n → ∞.To prove that lim n→∞ x n , f n = 0, it is enough to show that sup n k>M | a n,k , b n,k | → 0 as M → ∞.Therefore we have to show that for each ε > 0 there existsM ∈ N such that k>M | a n,k , b n,k | < ε,(2.1)forall sufficiently large enough n ∈ N. Let (2.1) is false.Then there is an ε > 0 such thatk>M | a n,k , b n,k | ≥ ε,(2.2)forall M ∈ N and some sufficiently large enough n ∈ N. Consider a sequence of positive number, (δ k ) such that ∞ By the technique given in the proof of main theorem of[18] one can construct two strictly increasing sequences, (n k ) k≥1 and (M k ) k≥0 such that (1)j>M k ||b n k ,j || ≤ δ k for each k ≥ a n,j , b n k−1 , j | ≤ δ k for each n ≥ n k (a n k ,j , b n k , j | ≥ ε.Now, let us choose a sequence (λ j ) such that |λ j | = 1, for all j and λ j a n k ,j , b n k ,j = | a n k ,j , b n k ,j |, Banach space X have the limited p-Schur and DP * p properties, then it has the p-Schur property.Indeed, a Banach space X is said to have the DP * -property of order p (DP * p