The p -Gelfand Phillips Property in Spaces of Operators and Dunford-Pettis like sets
aa r X i v : . [ m a t h . F A ] M a r THE p -GELFAND PHILLIPS PROPERTY IN SPACES OFOPERATORS AND DUNFORD-PETTIS LIKE SETS IOANA GHENCIU
Abstract.
The p -Gelfand Phillips property (1 ≤ p < ∞ ) is studied in spacesof operators. Dunford - Pettis type like sets are studied in Banach spaces. Wediscuss Banach spaces X with the property that every p -convergent operator T : X → Y is weakly compact, for every Banach space Y . Introduction
Numerous papers have investigated whether spaces of operators inherit the Gelfand-Phillips property when the co-domain and the dual of the domain possess the re-spective property; e.g., see [10], [12], [16], and [29]. In [14] the authors introducedthe p -Gelfand-Phillips property (1 ≤ p < ∞ ), a property which is in general weakerthan the Gelfand-Phillips property. In this paper limited p -convergent evaluationoperators are used to study the p -Gelfand-Phillips property in spaces of operators.We show that if Y has the Gelfand-Phillips property and M is a closed subspace of K w ∗ ( X ∗ , Y ) such that the evaluation operator ψ y ∗ : M → X is limited p -convergentfor each y ∗ ∈ Y ∗ , then M has the p -Gelfand-Phillip property. We prove that if X ∗ has the p -Gelfand-Phillip property and Y has the Schur property, and M is a closedsubspace of L ( X, Y ), then M has the p -Gelfand-Phillip property (1 ≤ p < ∞ ). Wealso prove that if L w ∗ ( X ∗ , Y ) has the p -Gelfand-Phillip property, then at least oneof the spaces X and Y does not contain ℓ .We study weakly-p-Dunford-Pettis sets and weakly-p-L-sets (1 ≤ p < ∞ ). Weshow that every operator with p -convergent adjoint is weakly precompact and thatweakly- p -Dunford-Pettis sets are weakly precompact (for 2 < p < ∞ ). We provethat a bounded subset A of X ∗ is an weakly- p - L -subset of X ∗ if and only if T ∗ ( A )is relatively compact whenever Y is a Banach space and T : Y → X is a weakly- p -precompact operator (1 ≤ p < ∞ ).We study two Banach space properties, called the reciprocal Dunford-Pettis prop-erty of order p (or property RDP p ) and property RDP ∗ p (1 ≤ p < ∞ ).We prove that a Banach space X has property RDP p (1 ≤ p < ∞ ) if andonly if for every Banach space Y , every p -convergent operator T : X → Y is weaklycompact. We also prove that a Banach space X has property RDP ∗ p (1 ≤ p < ∞ ) ifand only if for every Banach space Y , every operator T : Y → X with p -convergentadjoint is weakly compact. These results are motivated by results in [6]. Mathematics Subject Classification.
Primary:46B20; Secondary: 46B25, 46B28.
Key words and phrases.
The p -Gelfand Phillips property; p -convergent operators; the recipro-cal Dunford-Pettis property of order p . Definitions and Notation
Throughout this paper, X , Y , E and F will denote Banach spaces. The unitball of X will be denoted by B X and X ∗ will denote the continuous linear dual of X . The space X embeds in Y (in symbols X ֒ → Y ) if X is isomorphic to a closedsubspace of Y . An operator T : X → Y will be a continuous and linear function.The set of all operators, weakly compact operators, and compact operators from X to Y will be denoted by L ( X, Y ), W ( X, Y ), and K ( X, Y ). The w ∗ − w continuous(resp. compact) operators from X ∗ to Y will be denoted by L w ∗ ( X ∗ , Y ) (resp. K w ∗ ( X ∗ , Y )). The injective tensor product of two Banach spaces X and Y will bedenoted by X ⊗ ǫ Y . The space X ⊗ ǫ Y can be embedded into the space K w ∗ ( X ∗ , Y ),by identifying x ⊗ y with the rank one operator x ∗ → h x ∗ , x i y (see [9] for the theoryof tensor products).A subset S of X is said to be weakly precompact provided that every sequencefrom S has a weakly Cauchy subsequence. An operator T : X → Y is called weaklyprecompact (or almost weakly compact) if T ( B X ) is weakly precompact.An operator T : X → Y is called completely continuous (or Dunford-Pettis) if T maps weakly convergent sequences to norm convergent sequences.A Banach space X has the Dunford-Pettis property (DPP) if every weakly com-pact operator T : X → Y is completely continuous, for any Banach space Y . If X is a C ( K )-space or an L -space, then X has the DP P . The reader can check [8]and [9] for results related to the
DP P .For 1 ≤ p < ∞ , p ∗ denotes the conjugate of p . If p = 1, ℓ p ∗ plays the role of c .The unit vector basis of ℓ p will be denoted by ( e n ).Let 1 ≤ p ≤ ∞ . A sequence ( x n ) in X is called weakly p-summable if ( h x ∗ , x n i ) ∈ ℓ p for each x ∗ ∈ X ∗ [7, p. 32]. Let ℓ wp ( X ) denote the set of all weakly p -summablesequences in X . The space ℓ wp ( X ) is a Banach space with the norm k ( x n ) k wp = sup { ( ∞ X n =1 |h x ∗ , x n i| p ) /p : x ∗ ∈ B X ∗ } We recall the following isometries: L ( ℓ p ∗ , X ) ≃ ℓ wp ( X ) for 1 < p < ∞ ; L ( c , X ) ≃ ℓ wp ( X ) for p = 1; T → ( T ( e n )) [21], [7, Proposition 2.2, p. 36].A series P x n in X is said to be weakly unconditionally convergent (wuc) if forevery x ∗ ∈ X ∗ , the series P | x ∗ ( x n ) | is convergent. An operator T : X → Y is unconditionally converging if it maps weakly unconditionally convergent series tounconditionally convergent ones.Let 1 ≤ p ≤ ∞ . An operator T : X → Y is called p-convergent if T mapsweakly p -summable sequences into norm null sequences. The set of all p -convergentoperators is denoted by C p ( X, Y ) [6].The 1-convergent operators are precisely the unconditionally converging opera-tors and the ∞ -convergent operators are precisely the completely continuous oper-ators. If p < q , then C q ( X, Y ) ⊆ C p ( X, Y ).A sequence ( x n ) in X is called weakly-p-convergent to x ∈ X if the sequence( x n − x ) is weakly p -summable [6]. The weakly- ∞ -convergent sequences are preciselythe weakly convergent sequences.Let 1 ≤ p ≤ ∞ . A bounded subset K of X is relatively weakly- p -compact (resp. weakly-p-compact ) if every sequence in K has a weakly- p -convergent subsequencewith limit in X (resp. in K ). An operator T : X → Y is weakly- p -compact if HE p -GELFAND PHILLIPS PROPERTY IN SPACES OF OPERATORS AND DUNFORD-PETTIS LIKE SETS3 T ( B X ) is relatively weakly- p -compact [6]. The set of weakly- p -compact operators T : X → Y will be denoted by W p ( X, Y ).If p < q , then W p ( X, Y ) ⊆ W q ( X, Y ). A Banach space X ∈ C p (resp. X ∈ W p )if id ( X ) ∈ C p ( X, X ) (resp. id ( X ) ∈ W p ( X, X )) [6], where id ( X ) is the identitymap on X .Let 1 ≤ p ≤ ∞ . A sequence ( x n ) in X is called weakly-p-Cauchy if ( x n k − x m k ) isweakly p -summable for any increasing sequences ( n k ) and ( m k ) of positive integers.Every weakly- p -convergent sequence is weakly- p -Cauchy, and the weakly ∞ -Cauchy sequences are precisely the weakly Cauchy sequences.Let 1 ≤ p ≤ ∞ . We say that a subset S of X is called weakly- p -precompact if every sequence from S has a weakly- p -Cauchy subsequence. The weakly- ∞ -precompact sets are precisely the weakly precompact sets.Let 1 ≤ p ≤ ∞ . An operator T : X → Y is called weakly- p -precompact if T ( B X )is weakly- p -precompact. The set of all weakly- p -precompact operators T : X → Y is denoted by W P C p ( X, Y ). We say that X ∈ W P C p if id ( X ) ∈ W P C p ( X, X ).The weakly- ∞ -precompact operators are precisely the weakly precompact oper-ators. If p < q , then ℓ wp ( X ) ⊆ ℓ wq ( X ), thus W P C p ( X, Y ) ⊆ W P C q ( X, Y ).The bounded subset A of X is called a Dunford-Pettis (resp. limited ) subset of X if each weakly null (resp. w ∗ -null) sequence ( x ∗ n ) in X ∗ tends to 0 uniformly on A ; i.e. sup x ∈ A | x ∗ n ( x ) | → . Every DP (resp. limited) subset of X is weakly precompact [1], [26, p. 377] (resp.[4], [30]).The sequence ( x n ) in X is called limited if the corresponding set of its terms is alimited set. If the sequence ( x n ) is also weakly null (resp. weakly p -summable), then( x n ) is called a limited weakly null (resp. limited weakly p -summable) sequence in X .The space X has the Gelfand-Phillips (GP) property (or is a
Gelfand-Phillipsspace ) if every limited subset of X is relatively compact.Banach spaces having the Gelfand-Phillips property include, among others, Schurspaces, separably complemented spaces, spaces with w ∗ -sequential compact dualunit balls, separable spaces, reflexive spaces, spaces whose duals do not contain ℓ ,and dual spaces X ∗ whith X not containing ℓ ([4], [11], [30, p. 31]).A Banach space X has the DP ∗ -property ( DP ∗ P ) if all weakly compact sets in X are limited [5]. The space X has the DP ∗ -property if and only if L ( X, c ) = CC ( X, c ) [5], [17]. If X is a Schur space or if X has the DP P and the Grothendieckproperty, then X has the DP ∗ P .Let 1 ≤ p ≤ ∞ . A Banach space X has the Dunford-Pettis property of order p ( DP P p ) (1 ≤ p ≤ ∞ ) if every weakly compact operator T : X → Y is p -convergent,for any Banach space Y [6].Let 1 ≤ p ≤ ∞ . A Banach space X has the DP ∗ - property of order p ( DP ∗ P p )if all weakly- p -compact sets in X are limited [15].If X has the DP ∗ P , then X has the DP ∗ P p , for all 1 ≤ p ≤ ∞ . If X has the DP ∗ P p , then X has the DP P p .Let 1 ≤ p < ∞ . A Banach space X has the p - Gelfand-Phillips ( p - GP ) property (or is a p - Gelfand-Phillips space ) if every limited weakly p -summable sequence in X is norm null [14]. IOANA GHENCIU
If 1 ≤ p < q and X has the q - GP property, then X has the p - GP property. If X has the GP property, then X has the p - GP property for any 1 ≤ p < ∞ . Separablespaces with the DP ∗ P p have the p - GP property [14].An operator T : X → Y is called limited p -convergent if it carries limited weakly p -summable sequences in X to norm null ones in Y [14].The bounded subset A of X ∗ is called an L -subset of X ∗ if each weakly nullsequence ( x n ) in X tends to 0 uniformly on A .A bounded subset A of X ∗ (resp. of X ) is called a V -subset of X ∗ (resp. a V ∗ -subset of X ) provided thatsup { | x ∗ ( x n ) | : x ∗ ∈ A } → { | x ∗ n ( x ) | : x ∈ A } → P x n in X (resp. P x ∗ n in X ∗ ).The Banach space X has property ( V ) (resp. ( V ∗ )) if every V -subset of X ∗ (resp. V ∗ -subset of X ) is relatively weakly compact. The following results wereestablished in [25]: C ( K ) spaces have property ( V ); reflexive Banach spaces haveboth properties ( V ) and ( V ∗ ); L - spaces have property ( V ∗ ); X has property ( V )if and only if every unconditionally converging operator T from X to any Banachspace Y is weakly compact.Let 1 ≤ p < ∞ . We say that a bounded subset A of X is called a weakly-p-Dunford-Pettis set if for all weakly p -summable sequences ( x ∗ n ) in X ∗ ,sup x ∈ A | x ∗ n ( x ) | → . Let 1 ≤ p < ∞ . We say that a bounded subset A of X ∗ is called a weakly-p-L -setif for all weakly p -summable sequences ( x n ) in X ,sup x ∗ ∈ A | x ∗ ( x n ) | → . The weakly-1- L -subsets of X ∗ are precisely the V -subsets and the weakly-1-Dunford-Pettis subsets of X are precisely the V ∗ -subsets. If p < q , then a weakly- q - L -subset is a weakly- p - L -subset, since ℓ wp ( X ) ⊆ ℓ wq ( X ). Similarly, a weakly- q -DPset is a weakly- p -DP set, if p < q .The Banach space X has the reciprocal Dunford-Pettis ( RDP ) property if everycompletely continuous operator T from X to any Banach space Y is weakly compact[22, p. 153]. The space X has the RDP property if and only if every L -subset of X ∗ is relatively weakly compact [23], [19]. Banach spaces with property ( V ) ofPe lczy´nski, in particular reflexive spaces and C ( K ) spaces, have the RDP property[25]. A Banach space X does not contain ℓ if and only if every L -subset of X ∗ isrelatively compact if and only if every DP subset of X ∗ is relatively compact [11].The Banach space X has property RDP ∗ if every DP subset of X is relativelyweakly compact [2]. The space X has RDP ∗ whenever X has property ( V ∗ ) or X is weakly sequentially complete [2]. Also, X ∗ has RDP ∗ whenever X has property( V ).Let 1 ≤ p < ∞ . We say that the space X has the reciprocal Dunford-Pettisproperty of order p or RDP p (resp. the weak reciprocal Dunford-Pettis property oforder p or wRDP p ) if every weakly- p - L -subset of X ∗ is relatively weakly compact(resp. weakly precompact). HE p -GELFAND PHILLIPS PROPERTY IN SPACES OF OPERATORS AND DUNFORD-PETTIS LIKE SETS5 If X has the RDP p property, then X has the RDP property (since any L -subsetof X ∗ is a weakly- p - L -set). If p < q and X has the RDP p property, then X has the RDP q property.We say that X has property RDP ∗ p if every weakly- p -Dunford Pettis subset of X is relatively weakly compact.If p < q and X has property RDP ∗ p , then X has property RDP ∗ q . If X hasproperty RDP ∗ p , then X has property RDP ∗ (since every DP subset of X is aweakly- p -Dunford Pettis set). Note that c does not have property RDP ∗ p , since itdoes not have property RDP ∗ . Consequently, if X has property RDP ∗ p , then X does not contain a copy of c .3. The p -Gelfand Phillips property in spaces of operators In the following we give sufficient conditions for the p - GP property of some spacesof operators in terms of the limited p -convergence of the evaluation operators.We refer to [10] for the following two facts:(A) A sequence ( x n ) is limited if and only if x ∗ n ( x n ) → w ∗ -null sequence( x ∗ n ) in X ∗ .(B) A Banach space X has the GP property if and only if every limited weaklynull sequence in X is norm null.We recall the following well-known isometries ([27, p.60]):1) L w ∗ ( X ∗ , Y ) ≃ L w ∗ ( Y ∗ , X ), K w ∗ ( X ∗ , Y ) ≃ K w ∗ ( Y ∗ , X ) ( T → T ∗ )2) W ( X, Y ) ≃ L w ∗ ( X ∗∗ , Y ) and K ( X, Y ) ≃ K w ∗ ( X ∗∗ , Y ) ( T → T ∗∗ ).Suppose that X and Y are Banach spaces and M is a closed subspace of L ( X, Y ).If x ∈ X and y ∗ ∈ Y ∗ , the evaluation operators φ x : M → Y and ψ y ∗ : M → X ∗ are defined by φ x ( T ) = T ( x ) , ψ y ∗ ( T ) = T ∗ ( y ∗ ) , T ∈ M. Theorem 1.
Let ≤ p < ∞ .(i) Suppose that Y has the GP property. If M is a closed subspace of K w ∗ ( X ∗ , Y ) such that the evaluation operator ψ y ∗ : M → X is limited p -convergent for each y ∗ ∈ Y ∗ , then M has the p - GP property.(ii) Suppose that Y has the GP property. If M is a closed subspace of K ( X, Y ) such that the evaluation operator ψ y ∗ : M → X ∗ is limited p -convergent for each y ∗ ∈ Y ∗ , then M has the p - GP property.Proof. (i) Suppose not and let ( T n ) be a limited weakly p -summable sequence in M such that k T n k = 1 for each n . Let ( x ∗ n ) be a sequence in B X ∗ so that k T n ( x ∗ n ) k > / n .Let y ∗ ∈ Y ∗ . Since ψ y ∗ : M → X is limited p -convergent, ( T ∗ n ( y ∗ )) = ( ψ y ∗ ( T n ))is norm null. Hence h T n ( x ∗ n ) , y ∗ i = h T ∗ n ( y ∗ ) , x ∗ n i ≤ k T ∗ n ( y ∗ ) k →
0. Thus ( T n ( x ∗ n ))is weakly null.Let ( y ∗ n ) be a w ∗ -null sequence in Y ∗ . Let T ∈ K w ∗ ( X ∗ , Y ). Since T ∗ is w ∗ -normsequentially continuous, h x ∗ n ⊗ y ∗ n , T i ≤ k T ∗ ( y ∗ n ) k → . Thus ( x ∗ n ⊗ y ∗ n ) is w ∗ - null in ( K w ∗ ( X ∗ , Y )) ∗ . Since ( T n ) is limited in K w ∗ ( X ∗ , Y ), h x ∗ n ⊗ y ∗ n , T n i = h T n ( x ∗ n ) , y ∗ n i → . Thus ( T n ( x ∗ n )) is limited. Then ( T n ( x ∗ n )) is norm null, since Y has the GP property.This contradiction concludes the proof. IOANA GHENCIU (ii) Apply (i) and the isometry K ( X, Y ) ≃ K w ∗ ( X ∗∗ , Y ). (cid:3) We note that if X has the p - GP property, then any operator T : E → X is limited p -convergent. Indeed, if ( x n ) is limited weakly p -summable, then ( T ( x n )) is limitedweakly p -summable, and thus norm null. Thus, if X has the p - GP property and M is a closed subspace of K w ∗ ( X ∗ , Y ), then the evaluation operator ψ y ∗ : M → X is limited p -convergent for each y ∗ ∈ Y ∗ . Corollary 2.
Let ≤ p < ∞ .(i) Suppose X has the p - GP property and Y has the GP property. If M is aclosed subspace of K w ∗ ( X ∗ , Y ) , then M has the p - GP property.(ii) Suppose Y has the p - GP property and X has the GP property. If M is aclosed subspace of K w ∗ ( X ∗ , Y ) , then M has the p - GP property.Proof. (i) Since X has the p - GP property, ψ y ∗ : M → X is limited p -convergentfor each y ∗ ∈ Y ∗ . Apply Theorem 1.(ii) Apply (i) and the isometry K w ∗ ( X ∗ , Y ) ≃ K w ∗ ( Y ∗ , X ). (cid:3) Corollary 3.
Let ≤ p < ∞ . Suppose X has the p - GP property and Y has the GP property (or X has the GP property and Y has the p - GP property). Then X ⊗ ǫ Y has the p - GP property.Proof. The space X ⊗ ǫ Y can be embedded into the space K w ∗ ( X ∗ , Y ), by identi-fying x ⊗ y with the rank one operator x ∗ → h x ∗ , x i y . Apply Corollary 2. (cid:3) Corollary 4.
Let ≤ p < ∞ . Suppose X ∗ has the p - GP property and Y has the GP property (or X ∗ has the GP property and Y has the p - GP property). If M isa closed subspace of K ( X, Y ) , then M has the p - GP property.Proof. Apply Corollary 2 and the isometry K ( X, Y ) ≃ K w ∗ ( X ∗∗ , Y ). (cid:3) Corollary 5.
Let ≤ p < ∞ . If X has the p - GP property, then so has ℓ [ X ] , thespace of all unconditionally convergent series P x n in X equipped with the norm k ( x n ) k = sup { P | x ∗ ( x n ) | : x ∗ ∈ B X ∗ } .Proof. It is known that ℓ [ X ] is isometrically isomorphic to K ( c , X ) [13]. Since c ∗ ≃ ℓ has the GP property, Corollary 4 gives the conclusion. (cid:3) Corollary 6.
Let ≤ p < ∞ . If µ is a finite measure and X has the p - GP property, then so has L ( µ ) ⊗ ǫ X .Proof. The space L ( µ ) where µ is a finite measure, has the GP property [10]. Itis known that L ( µ ) ⊗ ǫ X ≃ K w ∗ ( X ∗ , L ( µ )) [9, Theorem 5, p. 224]. By Corollary2, this space has the p - GP property. (cid:3) Corollary 7.
Let ≤ p < ∞ . If X has the p - GP property, then so has c ( X ) ,the Banach space of sequences in X that converge to zero equipped with the norm k ( x n ) k = sup n k x n k .Proof. The space c has the GP property [4]. It is known that c ⊗ ǫ X ≃ c ( X )[28, p. 47]. Then c ⊗ ǫ X has the p - GP property, by Corollary 3. (cid:3) Theorem 8.
Let ≤ p < ∞ . Let X and Y be Banach spaces.(i) Let M be a closed subspace of L ( X, Y ) such that the evaluation operator ψ y ∗ : M → X ∗ is limited p -convergent for each y ∗ ∈ Y ∗ . If M does not have HE p -GELFAND PHILLIPS PROPERTY IN SPACES OF OPERATORS AND DUNFORD-PETTIS LIKE SETS7 the p - GP property, then there is a separable subspace Y of Y and an operator A : Y → c which is not completely continuous.(ii) Let M be a closed subspace of L w ∗ ( X ∗ , Y ) such that the evaluation operator ψ y ∗ : M → X is limited p -convergent for each y ∗ ∈ Y ∗ . If M does not have the p - GP property, then there is a separable subspace Y of Y and an operator A : Y → c which is not completely continuous.Proof. (i) Suppose M is a closed subspace of L ( X, Y ) which does not have the p - GP property. Let ( T n ) be a limited weakly p -summable sequence in M such that k T n k = 1 for each n . Let ( x n ) be a sequence in B X so that k T n ( x n ) k > / n .Let y ∗ ∈ Y ∗ . Since ψ y ∗ : M → X ∗ is limited p -convergent, ( T ∗ n ( y ∗ )) = ( ψ y ∗ ( T n ))is norm null. Then h y ∗ , T n ( x n ) i ≤ k T ∗ n ( y ∗ ) k →
0. Therefore ( y n ) := ( T n ( x n )) isweakly null in Y .By the Bessaga-Pelczynski selection principle [8], we may (and do) assume that( y n ) is a seminormalized weakly null basic sequence in Y . Let Y = [ y n ] be theclosed linear span of ( y n ) and let ( y ∗ n ) be the sequence of coefficient functionalsassociated with ( y n ). Define A : Y → c by A ( y ) = ( y ∗ k ( y )), y ∈ Y . Note that k A ( y n ) k ≥ n . Then A is a bounded linear operator defined on a separablespace, and A is not completely continuous.(ii) Suppose M a closed subspace of L w ∗ ( X ∗ , Y ) which does not have the p - GP property. Let ( T n ) be a limited weakly p -summable sequence in M such that k T n k = 1 for each n . Let ( x ∗ n ) be a sequence in B X ∗ so that k T n ( x ∗ n ) k > / n .Let y ∗ ∈ Y ∗ . Since ψ y ∗ : M → X is limited p -convergent, ( T ∗ n ( y ∗ )) = ( ψ y ∗ ( T n ))is norm null. Therefore ( y n ) := ( T n ( x ∗ n )) is weakly null in Y . Continue as in (i). (cid:3) Corollary 9.
Let ≤ p < ∞ .(i) Suppose X ∗ has the p - GP property and M is a closed subspace of L ( X, Y ) .If M does not have the p - GP property, then there is a separable subspace Y of Y and an operator A : Y → c which is not completely continuous.(ii) Suppose X has the p - GP property and M is a closed subspace of L w ∗ ( X ∗ , Y ) .If M does not have the p - GP property, then there is a separable subspace Y of Y and an operator A : Y → c which is not completely continuous.Proof. (i) Since X ∗ has the p - GP property, ψ y ∗ : M → X ∗ is limited p -convergentfor each y ∗ ∈ Y ∗ . Apply Theorem 8.(ii) Since X has the p - GP property, ψ y ∗ : M → X is limited p -convergent foreach y ∗ ∈ Y ∗ . Apply Theorem 8. (cid:3) Corollary 10.
Let ≤ p < ∞ .(i) Suppose X ∗ has the p - GP property and Y has the Schur property. If M is aclosed subspace of L ( X, Y ) , then M has the p - GP property.(ii) Suppose X has the p - GP property and Y has the Schur property. If M is aclosed subspace of L w ∗ ( X ∗ , Y ) = K w ∗ ( X ∗ , Y ) , then M has the p - GP property.(iii) Suppose Y has the p - GP property and X has the Schur property. If M is aclosed subspace of L w ∗ ( X ∗ , Y ) = K w ∗ ( X ∗ , Y ) , then M has the p - GP property.Proof. (i) Suppose that M does not have the p - GP property. By Corollary 9, thereis a non-completely continuous operator defined on a closed linear subspace Y of Y . This is a contradiction since Y has the Schur property. IOANA GHENCIU (ii) Let T ∈ L w ∗ ( X ∗ , Y ). Since T is weakly compact and Y has the Schurproperty, T is compact. Continue as above.(iii) It follows from (ii) and the isometries 1) on page 5. (cid:3) It is known that ℓ ∞ does not have the GP property. Further, ℓ ∞ does not havethe p - GP property for any 1 ≤ p < ∞ [14]. Theorem 11.
Let ≤ p < ∞ . Suppose that L w ∗ ( X ∗ , Y ) has property GP (resp. p - GP ). Then X and Y have property GP (resp. p - GP ) and either ℓ ֒ → X or ℓ ֒ → Y . If moreover Y is a dual space Z ∗ , the condition ℓ ֒ → Y implies ℓ ֒ → Z .Proof. We only prove the result for the p - GP property. The other proof is similar.Suppose that L w ∗ ( X ∗ , Y ) has property p - GP . Then X and Y have property p - GP , since property p - GP is inherited by closed subspaces. Suppose ℓ ֒ → X and ℓ ֒ → Y . Then c ֒ → K w ∗ ( X ∗ , Y ) by [18, Theorem 20]. Since c ֒ → L w ∗ ( X ∗ , Y )and X and Y do not have the Schur property, ℓ ∞ ֒ → L w ∗ ( X ∗ , Y ) by [18, Corollary2]. This contradiction proves the first assertion.Now suppose Y = Z ∗ and ℓ ֒ → Z . Then L ֒ → Z ∗ [8, p. 212]. Also, theRademacher functions span ℓ inside of L , hence ℓ ֒ → Z ∗ . (cid:3) Corollary 12.
Let ≤ p < ∞ . Suppose that W ( X, Y ) has property GP (resp. p - GP ). Then X ∗ and Y have property GP (resp. p - GP ) and either ℓ ֒ → X or ℓ ֒ → Y . If moreover Y is a dual space Z ∗ , the condition ℓ ֒ → Y implies ℓ ֒ → Z .Proof. Apply Theorem 11 and the isometries 2) on page 5. (cid:3) Weakly- p - L -sets and weakly- p -Dunford-Pettis sets The following result gives a characterization of p -convergent operators. The case1 < p < ∞ of the following result [6, p. 45] appeared with no proof. We include aproof for the convenience of the reader. Proposition 13.
Let ≤ p < ∞ . An operator T : X → Y is p-convergent if andonly if for any operator S : ℓ p ∗ → X if < p < ∞ (resp. S : c → X if p = 1 ), theoperator T S is compact.Proof.
Suppose T : X → Y is p -convergent. Let 1 < p < ∞ and let S : ℓ p ∗ → X bean operator. Then S is weakly- p -compact, since ℓ p ∗ ∈ W p [6]. Hence T S : ℓ p ∗ → Y is compact. Let p = 1 and let S : c → X be an operator. Then T S : c → Y is unconditionally converging, and P T S ( e n ) is unconditionally convergent. Hence T S is compact ([7, Theorem 1.9, p. 9], [8, p. 113]).Conversely, let ( x n ) be weakly p -summable in X . Then ℓ wp ( X ) ≃ L ( ℓ p ∗ , X ) if1 < p < ∞ (resp. ℓ wp ( X ) ≃ L ( c , X ) if p = 1) ([21], [7, Proposition 2.2, p. 36]).Let S : ℓ p ∗ → X if 1 < p < ∞ (resp. S : c → X if p = 1) be an operator suchthat S ( e n ) = x n . Since T S is compact, k T ( x n ) k = k T S ( e n ) k →
0, and thus T is p -convergent. (cid:3) Theorem 14.
Let ≤ p < ∞ . Let T : Y → X be an operator. The following areequivalent:1. (i) T ( B Y ) is a weakly- p -Dunford-Pettis set.(ii) T ∗ : X ∗ → Y ∗ is p -convergent.(iii) If < p < ∞ and S : ℓ p ∗ → X ∗ (resp. p = 1 and S : c → X ∗ ) is anoperator, then T ∗ S is compact. HE p -GELFAND PHILLIPS PROPERTY IN SPACES OF OPERATORS AND DUNFORD-PETTIS LIKE SETS9
2. (i) T ∗ ( B X ∗ ) is a weakly- p - L -set.(ii) T is p -convergent.(iii) If < p < ∞ and S : ℓ p ∗ → Y (resp. p = 1 and S : c → Y ) is an operator,then T S is compact.Proof.
We only prove case 1. The other proof is similar.( i ) ⇔ ( ii ) This follows directly from the equality k T ∗ ( x ∗ n ) k = sup {|h T ( y ) , x ∗ n i| : y ∈ B Y } , for all weakly p -summable sequences ( x ∗ n ) in X ∗ .( ii ) ⇔ ( iii ) by Proposition 13. (cid:3) In the next theorem we give elementary operator theoretic characterizations ofweak precompactness, relative weak compactness, and relative norm compactnessfor weakly- p -Dunford-Pettis sets. Theorem 15.
Let ≤ p < ∞ . Let X be a Banach space. The following statementsare equivalent:(i) For every Banach space Y , if T : Y → X is an operator such that T ∗ : X ∗ → Y ∗ is p -convergent, then T is weakly precompact (weakly compact, resp. compact).(ii) same as (i) with Y = ℓ .(iii) Every weakly- p -Dunford-Pettis subset of X is weakly precompact (relativelyweakly compact, resp. relatively compact).Proof. We will show that ( i ) ⇒ ( ii ) ⇒ ( iii ) ⇒ ( i ) in the relatively weakly compactcase. The arguments for the remaining implications of the theorem follow the samepattern.( i ) ⇒ ( ii ) is obvious.( ii ) ⇒ ( iii ) Let K be a weakly- p -Dunford-Pettis subset of X and let ( x n ) be asequence in K . Define T : ℓ → X by T ( b ) = P b i x i . Note that T ∗ : X ∗ → ℓ ∞ , T ∗ ( x ∗ ) = ( x ∗ ( x i )). Suppose ( x ∗ n ) is a weakly p -summable sequence in X ∗ . Since K is a weakly- p -Dunford-Pettis set, k T ∗ ( x ∗ n ) k = sup i | x ∗ n ( x i ) | → . Therefore T ∗ is p -convergent and thus T is weakly compact. Let ( e ∗ n ) be the unitbasis of ℓ . Then ( T ( e ∗ n )) = ( x n ) has a weakly convergent subsequence.( iii ) ⇒ ( i ) Let T : Y → X be an operator such that T ∗ : X ∗ → Y ∗ is p -convergent. Then T ( B Y ) is a weakly- p -Dunford-Pettis set, thus relatively weaklycompact. Hence T is weakly compact. (cid:3) Odell, Rosenthal, and Stegall [26, p. 377], showed that an operator T : Y → X is weakly precompact if LT : Y → L is compact whenever L : X → L is acompletely continuous map. Maps with range in ℓ p (1 < p < ∞ ) can also beemployed to identify such operators. Theorem 16.
Let < p < ∞ .(i) If T : Y → X is an operator such that JT : Y → ℓ p is compact for alloperators J : X → ℓ p , then T is weakly precompact.(ii) If T : Y → X is an operator such that T ∗ : X ∗ → Y ∗ is p -convergent, then T is weakly precompact. Proof. (i) Let T be an operator as in the statement of the theorem. Suppose (by wayof contradiction) that ( y n ) is a sequence in B Y and ( T ( y n )) has no weakly Cauchysubsequence. By Rosenthal’s ℓ - theorem, we can assume that ( T ( y n )) ∼ ( e ∗ n ),where ( e ∗ n ) is the unit vector basis of ℓ . Let S = [ T ( y n )], an isomorph of ℓ .Let j : S → ℓ p be the natural inclusion. The canonical injection j : ℓ → ℓ is(absolutely) 1-summing [28, Example 6.17, p. 145], and thus it is 2-summing. Since j naturally factors through ℓ , j is 2-summing. Now use the fact that all closedlinear subspaces of an L - space are complemented and the constructions on p. 60 -61 of [8] to obtain an operator J : X → ℓ p which extends j (see also [28, Proposition6.24, p. 150]). Then JT : Y → ℓ p is compact. But JT ( y n ) = j ( e ∗ n ) = e n , and ( e n )is not relatively compact in ℓ p .(ii) Let T : Y → X is an operator such that T ∗ : X ∗ → Y ∗ is p -convergent andlet J : X → ℓ p be an operator. By Theorem 14, T ∗ J ∗ : ℓ p ∗ → Y ∗ is compact. Then JT : Y → ℓ p is compact. Apply (i). (cid:3) The weak precompactness of a DP set is well known; e.g., see [1], [26, p. 377].We obtain an analogous result for weakly- p -Dunford-Pettis sets (2 < p < ∞ ). Corollary 17.
Let < p < ∞ . Every weakly- p -Dunford-Pettis subset of X isweakly precompact.Proof. Let T : Y → X be an operator such that T ∗ : X ∗ → Y ∗ is p -convergent. ByTheorem 16, T is weakly precompact. Apply Theorem 15. (cid:3) A Banach space X is called weakly sequentially complete if every weakly Cauchysequence in X is weakly convergent. Corollary 18.
Let < p < ∞ .(i) If X is weakly sequentially complete, then X has property RDP ∗ p .(ii) If X has the Schur property, then every weakly-p-Dunford-Pettis subset of X is relatively compact. The following theorem gives a characterization of weakly- p - L -sets. For any p , 1 ≤ p < ∞ , a bounded subset K of ℓ p is relatively compact if and only iflim n P ∞ i = n | k i | p = 0, uniformly for k ∈ K [8, p. 6]. Theorem 19.
Let < p < ∞ . Suppose that A is a bounded subset of X ∗ . Thefollowing are equivalent:(i) A is a weakly- p - L -subset of X ∗ .(ii) T ∗ ( A ) is relatively compact whenever Y is a Banach space and T : Y → X is a weakly p -precompact operator.(iii) T ∗ ( A ) is relatively compact whenever Y ∈ W P C p and T : Y → X is anoperator.(iv) T ∗ ( A ) is relatively compact whenever T : ℓ p ∗ → X is an operator.(v) If ( x n ) is a weakly p -summable sequence in X and ( x ∗ n ) is a sequence in A ,then lim x ∗ n ( x n ) = 0 .Proof. ( i ) ⇒ ( ii ) Suppose that A is a weakly- p - L -subset of X ∗ and let ( x ∗ n ) be asequence in A . Let T : Y → X be a weakly- p -precompact operator. Define S : X → ℓ ∞ by S ( x ) = ( x ∗ n ( x )). Let ( x n ) be a weakly p -summable sequence in X . Since A is a weakly- p - L -subset of X ∗ , lim n k S ( x n ) k = lim n sup i | x ∗ i ( x n ) | = 0, and thus S is p -convergent. Then ST : Y → ℓ ∞ is compact, since T is weakly- p -precompact. HE p -GELFAND PHILLIPS PROPERTY IN SPACES OF OPERATORS AND DUNFORD-PETTIS LIKE SETS11 Let ( e ∗ n ) be the unit basis of ℓ . Since T ∗ S ∗ is compact, ( T ∗ ( x ∗ n )) = ( T ∗ S ∗ ( e ∗ n )) isrelatively compact.( ii ) ⇒ ( iii ) If Y ∈ W P C p , then any operator T : Y → X is weakly- p -precompact.( iii ) ⇒ ( iv ) Suppose T : ℓ p ∗ → X is an operator. Since 1 < p ∗ < ∞ , ℓ p ∗ ∈ W p [6, Proposition 1.4]. Then T ∗ ( A ) is relatively compact.( iv ) ⇒ ( i ) Let ( x n ) be a weakly p -summable sequence in X . Let T : ℓ p ∗ → X such that T ( e n ) = x n ([21], [7, Proposition 2.2]). Since T ∗ ( A ) is relatively compactin ℓ p , sup x ∗ ∈ A |h T ∗ ( x ∗ ) , e n i| = sup x ∗ ∈ A |h x ∗ , x n i| → i ) ⇒ ( v ) is obvious.( v ) ⇒ ( i ) Let ( x n ) be a weakly p -summable sequence in X . Since A is bounded,for every n we can choose x ∗ n in A such that sup x ∗ ∈ A | x ∗ ( x n ) | ≤ | x ∗ n ( x n ) | . Thensup x ∗ ∈ A | x ∗ ( x n ) | ≤ | x ∗ n ( x n ) | →
0, and A is an weakly- p - L -set. (cid:3) The following result gives a characterization of weakly- p -DP sets. Corollary 20.
Let < p < ∞ . Suppose that A is a bounded subset of a Banachspace X . Then the following assertions are equivalent:(i) A is a weakly- p -DP set.(ii) T ( A ) is relatively compact whenever Y is a Banach space and T : X → Y isan operator with weakly- p -precompact adjoint.(iii) T ( A ) is relatively compact whenever Y ∗ ∈ W P C p and T : X → Y is anoperator.(iv) T ( A ) is relatively compact whenever T : X → ℓ p is an operator.(v) If ( x ∗ n ) is a weakly p -summable sequence in X ∗ and ( x n ) is a sequence in A ,then lim x ∗ n ( x n ) = 0 .Proof. ( i ) ⇒ ( ii ) Let T : X → Y be an operator such that T ∗ : Y ∗ → X ∗ is weakly- p -precompact. Since A is a weakly- p -DP subset of X , A is an weakly- p - L -subsetof X ∗∗ . By Theorem 19, T ∗∗ ( A ) is relatively compact. Hence T ( A ) is relativelycompact.( ii ) ⇒ ( iii ) If Y ∗ ∈ W P C p , then any operator T : X → Y has a weakly- p -precompact adjoint.( iii ) ⇒ ( iv ) Let T : X → ℓ p be an operator. Since 1 < p ∗ < ∞ , ℓ p ∗ ∈ W p [6].Thus T ( A ) is relatively compact.( iv ) ⇒ ( i ) Let ( x ∗ n ) be a weakly p -summable sequence in X ∗ . Let T : ℓ p ∗ → X ∗ such that T ( e n ) = x ∗ n ([21], [7, Proposition 2.2]). Let T = T ∗ | X : X → ℓ p . Since T ( A ) is relatively compact in ℓ p , sup x ∈ A |h T ∗ ( x ) , e n i| = sup x ∈ A |h x, x ∗ n i| → v ) ⇒ ( i ) Let ( x ∗ n ) be a weakly p -summable sequence in X ∗ . Since A is bounded,for every n we can choose x n in A such that sup x ∈ A | x ∗ n ( x ) | ≤ | x ∗ n ( x n ) | . Thensup x ∗ ∈ A | x ∗ ( x n ) | ≤ | x ∗ n ( x n ) | →
0, and A is a weakly- p -DP set. (cid:3) If X is any infinite dimensional Schur space, then all bounded subsets of X ∗ areweakly- p - L -subsets, and thus there are weakly- p - L -subsets of X ∗ which fail to beweakly precompact. Theorem 21.
Let ≤ p < ∞ . Suppose that X is a Banach space. The followingare equivalent:(i) For every Banach space Y , if T : X → Y is a p-convergent operator, then T ∗ : Y ∗ → X ∗ is weakly precompact (weakly compact, resp. compact).(ii) Same as (i) with Y = ℓ ∞ . (iii) Every weakly-p- L -subset of X ∗ is weakly precompact (relatively weakly com-pact, resp. relatively compact).Proof. We will show that ( i ) ⇒ ( ii ) ⇒ ( iii ) ⇒ ( i ) in the weakly precompact case.The arguments for all the remaining implications in the theorem follow the samepattern.( i ) ⇒ ( ii ) is obvious. ( ii ) ⇒ ( iii ) Let A be a weakly- p - L -subset of X ∗ and let( x ∗ n ) be a sequence in A . Define T : X → ℓ ∞ by T ( x ) = ( x ∗ i ( x )), x ∈ X . Suppose( x n ) is weakly p -summable in X . Since A is a weakly- p - L -subset,lim n k T ( x n ) k = lim n sup i | x ∗ i ( x n ) | = 0 . Therefore T is p -convergent, and thus T ∗ : ℓ ∗∞ → X ∗ is weakly precompact. Let( e ∗ n ) be the unit vector basis of ℓ . Hence ( T ∗ ( e ∗ n )) = ( x ∗ n ) has a weakly Cauchysubsequence.( iii ) ⇒ ( i ) Suppose that every weakly- p - L -subset of X ∗ is weakly precompactand let T : X → Y be a p -convergent operator. Hence T ∗ ( B Y ∗ ) is a weakly- p - L -subset of X ∗ , thus weakly precompact. Therefore T ∗ is weakly precompact. (cid:3) Corollary 22.
Let ≤ p < ∞ .(i) If X has property ( V ∗ ) , then X has property RDP ∗ p .(ii) If X has property ( V ) , then X has property RDP p .Proof. (i) Let A be a weakly- p -DP subset of X . Then A is a weakly-1-DP subset,thus a V ∗ -subset of X . Since X has property ( V ∗ ), A is relatively weakly compact.(ii) Suppose T : X → Y is a p -convergent operator. Then T is unconditionallyconverging. Since X has property ( V ), T is weakly compact [25]. Hence T ∗ isweakly compact. Apply Theorem 21. (cid:3) Corollary 23.
Let ≤ p < ∞ . If X has property RDP p , then every quotient spaceof X has property RDP p .Proof. Suppose that X has property RDP p , Z is a quotient of X and Q : X → Z is a quotient map. Let T : Z → E be a p -convergent operator. Then T Q : X → E is p -convergent, and thus ( T Q ) ∗ is weakly compact by Theorem 21. Since Q ∗ is anisomorphism and Q ∗ T ∗ ( B E ∗ ) is relatively weakly compact, T ∗ ( B E ∗ ) is relativelyweakly compact. Apply Theorem 21. (cid:3) Corollary 24.
Let < p < ∞ . Let T : Y → X be an operator. The following areequivalent:(i) T ( B Y ) is a weakly-p-Dunford-Pettis set.(ii) T ∗ : X ∗ → Y ∗ is p-convergent.(iii) If A : X → ℓ p is an operator, then AT is compact.Proof. ( i ) ⇔ ( ii ) by Theorem 14.( ii ) ⇒ ( iii ) Let A : X → ℓ p be an operator. By Theorem 14, T ∗ A ∗ : ℓ p ∗ → Y ∗ is compact. Then AT : Y → ℓ p is compact.( iii ) ⇒ ( ii ) Let S : ℓ p ∗ → X ∗ be an operator. Let A = S ∗ | X , A : X → ℓ p . Then A ∗ = S . Since AT is compact, T ∗ A ∗ = T ∗ S is compact. Apply Theorem 14. (cid:3) Corollary 25.
Let < p < ∞ . The following are equivalent:1. (i) B X is a weakly-p-Dunford-Pettis set.(ii) L ( ℓ p ∗ , X ∗ ) = K ( ℓ p ∗ , X ∗ ) . HE p -GELFAND PHILLIPS PROPERTY IN SPACES OF OPERATORS AND DUNFORD-PETTIS LIKE SETS13 (iii) L ( X, ℓ p ) = K ( X, ℓ p ) .(iv) X ∗ ∈ C p .2. (i) B X ∗ is a weakly-p-L-set.(ii) L ( ℓ p ∗ , X ) = K ( ℓ p ∗ , X ) .(iii) X ∈ C p .Proof. We only prove case 1. The other proof is similar. Apply Theorem 14 andCorollary 24 to the identity map i on X . (cid:3) Corollary 26.
Let < p < ∞ .1. Suppose L ( ℓ p ∗ , X ∗ ) = K ( ℓ p ∗ , X ∗ ) . Then X has the RDP ∗ p property if andonly if X is reflexive.2. Suppose L ( ℓ p ∗ , X ) = K ( ℓ p ∗ , X ) . Then X has the RDP p property if and onlyif X is reflexive.Proof. We only prove case 1. The other proof is similar. Suppose L ( ℓ p ∗ , X ∗ ) = K ( ℓ p ∗ , X ∗ ) and X has the RDP ∗ p property. By Corollary 25, B X is a weakly- p -Dunford-Pettis set, and thus relatively weakly compact. (cid:3) Corollary 27.
Let < p < ∞ .(i) If X ∈ W P C p , then every weakly-p-L-subset of X ∗ is relatively compact.(ii) If X ∗ ∈ W P C p , then every weakly-p-Dunford Pettis subset of X is relativelycompact.(iii) If X is infinite dimensional and X ∈ C p , then X ∗ contains weakly-p- L -setswhich are not relatively compact.(iv) If X is infinite dimensional and X ∗ ∈ C p , then X contains weakly-p-DunfordPettis sets which are not relatively compact.Proof. Let i : X → X be the identity map on X .(i) Since X ∈ W P C p , i is weakly- p -precompact. Let A be a weakly- p - L subsetof X ∗ . By Theorem 19, i ∗ ( A ) = A is relatively compact.(ii) Let A be a weakly- p -Dunford Pettis subset of X . Since X ∗ ∈ W P C p , i ∗ : X ∗ → X ∗ is weakly- p -precompact. By Corollary 20, i ( A ) = A is relativelycompact.(iii) Since X ∈ C p and X is infinite dimensional, i is p -convergent and notcompact. Apply Theorem 21.(iv) Since X ∗ ∈ C p , i ∗ : X ∗ → X ∗ is p -convergent. Further, i is not compact.Apply Theorem 15. (cid:3) If 1 < p < ∞ and 1 < r < p ∗ , then the identity map on ℓ r is p -convergent ([6,Corollary 1.7]). Hence B ℓ r ∗ is a weakly- p - L -set which is not relatively compact.Let 1 < p < ∞ . Then ℓ p ∗ ∈ W p [6]. By Corollary 27, every weakly- p - L -subsetof ℓ p is relatively compact. Further, every weakly- p -Dunford Pettis subset of ℓ p isrelatively compact.Let T be the Tsirelson’s space [6]. Then T ∗ ∈ W p and T ∈ C p . Hence ev-ery weakly- p -Dunford Pettis subset of T is relatively compact (by Corollary 27).Further, T ∗ contains weakly- p - L -sets which are not relatively compact. Corollary 28.
Let ≤ p < ∞ .(i) Suppose Y is a closed subspace of X ∗ and X has the RDP p . Then Y hasproperty RDP ∗ p . (ii) If Y ∗ has the RDP p , then Y has property RDP ∗ p .Proof. (i) Let K be a weakly- p -DP subset of Y . Then K is a weakly- p -DP subsetof X ∗ , and thus a weakly- p - L -subset of X ∗ . Hence K is relatively weakly compact.Then Y has property RDP ∗ p .(ii) Consider Y a closed subspace of Y ∗∗ and apply (i). (cid:3) The converse of Corollary 28 (i) is not true. Let X be the first Bourgain-Delbaenspace [3]. Then X is an infinite dimensional L ∞ -space with the Schur property and X ∗ is weakly sequentially complete. Since X has the Schur property, the identitymap i on X is completely continuous, hence p -convergent (1 < p < ∞ ), and notweakly compact. Thus X does not have property RDP p . Since X ∗ is weaklysequentially complete, X ∗ has property RDP ∗ p by Corollary 18. Corollary 29.
Let ≤ p < ∞ .(i) Suppose F is a closed subspace of Z ∗ and Z has the RDP p . If T : E → F isan operator such that T ∗ : F ∗ → E ∗ is p -convergent, then T is weakly compact.(ii) Suppose F ∗ has the RDP p . If T : E → F is an operator such that T ∗ : F ∗ → E ∗ is p -convergent, then T is weakly compact.Proof. (i) By Corollary 28, F has property RDP ∗ . Let T : E → F be an operatorsuch that T ∗ : F ∗ → E ∗ is p -convergent. By Theorem 15, T is weakly compact.(ii) Consider F a subspace of F ∗∗ and Z = F ∗ . Apply (i). (cid:3) In the following two results we need the following result.
Lemma 30. ( [20, Theorem 2.7] ) Let X be a Banach space, Y a reflexive subspaceof X (resp. a subspace not containing copies of ℓ ), and Q : X → X/Y the quotientmap. Let ( x n ) be a bounded sequence in X such that ( Q ( x n )) is weakly convergent(resp. weakly Cauchy). Then ( x n ) has a weakly convergent subsequence (resp.weakly Cauchy). Theorem 31.
Let < p < ∞ . Let X be a Banach space and Y be a reflexivesubspace of X . If X/Y has the
RDP ∗ p property, then X has the RDP ∗ p property.Proof. Let Q : X → X/Y be the quotient map. Let A be a weakly- p -DP subset of X and ( x n ) be a sequence in A . Then ( Q ( x n )) is a weakly- p -DP subset of X/Y , andthus relatively weakly compact. By passing to a subsequence, suppose ( Q ( x n )) isweakly convergent. By Lemma 30, ( x n ) has a weakly convergent subsequence. (cid:3) Let E be a Banach space and F be a subspace of E ∗ . Let ⊥ F = { x ∈ E : y ∗ ( x ) = 0 for all y ∗ ∈ F } . Theorem 32.
Let < p < ∞ . Let E be a Banach space and F be a reflexivesubspace of E ∗ (resp. a subspace not containing ℓ ). If ⊥ F has property RDP p (resp. wRDP p ), then E has property RDP p (resp. wRDP p ).Proof. We will only prove the result for property
RDP p . The other proof is similar.Suppose that ⊥ F has property RDP p . Let Q : E ∗ → E ∗ /F be the quotientmap and i : E ∗ /F → ( ⊥ F ) ∗ be the natural surjective isomorphism [24, Theorem1.10.16]. It is known that iQ : E ∗ → ( ⊥ F ) ∗ is w ∗ − w ∗ continuous, since iQ ( x ∗ )is the restriction of x ∗ to ⊥ F [24, Theorem 1.10.16]. Then there is an operator S : ⊥ F → E such that iQ = S ∗ . HE p -GELFAND PHILLIPS PROPERTY IN SPACES OF OPERATORS AND DUNFORD-PETTIS LIKE SETS15 Let T : E → X be a p -convergent operator. Since ⊥ F has property RDP p andthe operator T S : ⊥ F → X is p -convergent, it is weakly compact. Since S ∗ T ∗ = iQT ∗ is weakly compact and i is a surjective isomorphism, QT ∗ is weakly compact.Let ( x ∗ n ) be a sequence in B X ∗ . By passing to a subsequence, we can assumethat ( QT ∗ ( x ∗ n )) is weakly convergent. Hence ( T ∗ ( x ∗ n )) has a weakly convergentsubsequence by Lemma 30. Then E has property RDP p by Theorem 21. (cid:3) References [1] Kevin Andrews, Dunford-Pettis sets in the space of Bochner integrable functions,
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