Strongly alternative Dunford–Pettis subspaces of operator ideals
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Moshtaghioun, S.M. 2020-02-13T09:19:59Z 2020-02-13T09:19:59Z 2013 Strongly alternative Dunford–Pettis subspaces of operator ideals / S.M. Moshtaghioun // Український математичний журнал. — 2013. — Т. 65, № 4. — С. 588-593. — Бібліогр.: 13 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165332 517.98 en Інститут математики НАН України Український математичний журнал Короткі повідомлення Strongly alternative Dunford–Pettis subspaces of operator ideals Сильно альтернативнi простори Данфорда – Петтiса операторних iдеалiв Article published earlier |
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Strongly alternative Dunford–Pettis subspaces of operator ideals |
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Strongly alternative Dunford–Pettis subspaces of operator ideals Moshtaghioun, S.M. Короткі повідомлення |
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Strongly alternative Dunford–Pettis subspaces of operator ideals |
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Strongly alternative Dunford–Pettis subspaces of operator ideals |
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Strongly alternative Dunford–Pettis subspaces of operator ideals |
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Strongly alternative Dunford–Pettis subspaces of operator ideals |
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strongly alternative dunford–pettis subspaces of operator ideals |
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Moshtaghioun, S.M. |
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Moshtaghioun, S.M. |
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Короткі повідомлення |
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2013 |
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Український математичний журнал |
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Сильно альтернативнi простори Данфорда – Петтiса операторних iдеалiв |
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1027-3190 |
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Strongly alternative Dunford–Pettis subspaces of operator ideals / S.M. Moshtaghioun // Український математичний журнал. — 2013. — Т. 65, № 4. — С. 588-593. — Бібліогр.: 13 назв. — англ. |
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UDC 517.98
S. M. Moshtaghioun (Yazd Univ., Iran)
STRONGLY ALTERNATIVE DUNFORD – PETTIS SUBSPACES
OF OPERATOR IDEALS
СИЛЬНО АЛЬТЕРНАТИВНI ПРОСТОРИ ДАНФОРДА – ПЕТТIСА
ОПЕРАТОРНИХ IДЕАЛIВ
Introducing the concept of strong alternative Dunford – Pettis property (strong DP1) for the subspaceM of operator ideals
U(X,Y ) between Banach spaces X and Y, we show thatM is a strong DP1 subspace if and only if all evaluation operators
φx :M→ Y and ψy∗ :M→ X∗ are DP1 operators, where φx(T ) = Tx and ψy∗(T ) = T ∗y∗ for x ∈ X, y∗ ∈ Y ∗, and
T ∈ M. Some consequences related to the concept of alternative Dunford – Pettis property in subspaces of some operator
ideals are obtained.
Введено поняття сильної альтернативної властивостi Данфорда – Петтiса (сильна DP1) для пiдпростору M опера-
торних iдеалiв U(X,Y ) мiж банаховими просторами X та Y, за допомогою якого показано, щоM є сильним DP1
пiдпростором тодi i тiльки тодi, коли всi оператори оцiнки φx :M → Y та ψy∗ :M → X∗ є DP1 операторами,
де φx(T ) = Tx та ψy∗(T ) = T ∗y∗ при x ∈ X, y∗ ∈ Y ∗ та T ∈ M. Отримано деякi наслiдки щодо поняття
альтернативної властивостi Данфорда – Петтiса в пiдпросторах деяких операторних iдеалiв.
1. Introduction. A Banach space X has the Dunford – Pettis property (DP) if for each weakly null
sequences (xn) ⊆ X and (x∗n) ⊆ X∗, one has x∗n(xn) → 0 as n → ∞. Also the Banach space X
has the alternative Dunford – Pettis property (DP1) if for each weakly convergent sequence xn → x
in X with ‖xn‖ = ‖x‖ = 1, for all integer n, and each weakly null sequence (x∗n) in X∗, we have
x∗n(xn)→ 0.
It is clear that the Banach space X has the DP1 if and only if for each weakly null sequences
(xn) in X and (x∗n) in X∗ and each x ∈ X with ‖xn + x‖ = ‖x‖ = 1, we have x∗n(xn)→ 0.
Evidently, DP implies the DP1, but the converse in general, is false. For example, every (infinite
dimensional) Hilbert space has DP1, but does not have the DP and the space of trace class operators
on an infinite dimensional Hilbert space provides another Banach space with the DP1 and without
the DP [6]. Also there are Banach spaces such as C∗-algebras and von Neumann algebras, that the
DP1 and DP on them coincide [1, 6].
A bounded linear operator T : X → Y between Banach spaces X and Y is said to be completely
continuous (or Dunford – Pettis) operator, if for each weakly convergent sequence xn → x in X, we
have ‖Txn−Tx‖ → 0, that is T carries weakly convergent sequences to norm convergent sequences.
But if under the additional condition ‖xn‖ = ‖x‖ = 1, the conclusion ‖Txn− Tx‖ → 0 is obtained,
we say that T is a DP1 operator.
The concept of DP1 for Banach spaces and for operators introduced by Freedman in [6], and
he obtained some properties of them. In particular, the Banach space X has the DP1 if and only if
every weakly compact operator T : X → Y into arbitrary Banach space Y is a DP1 operator [6]
(Theorem 1.4). This is similar to Theorem 1 of [4] which stay that a Banach space X has the DP
property if and only if every weakly compact operator T : X → Y is completely continuous. We
refer the reader for additional properties of these concepts to [1, 2, 4, 6, 8].
In [10], the author in a joint work with J. Zafarani, has introduced the concept of strongly
completely continuous for subspaces of operator ideals which is the main motivation of this article.
c© S. M. MOSHTAGHIOUN, 2013
588 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
STRONGLY ALTERNATIVE DUNFORD – PETTIS SUBSPACES OF OPERATOR IDEALS 589
In fact, if U is any operator ideal with ideal norm A(·), and by meaning of [3] or [12], for any Banach
spaces X and Y ; U(X,Y ) denotes the component of U consisting of all bounded linear operators
T : X → Y that belongs to U and K(X,Y ) denotes the Banach space of all compact operators from
X to Y ; then a linear subspaceM⊆ U(X,Y ) is called strongly completely continuous in U(X,Y )
(resp. in K(X,Y )), if for all Banach spaces W and Z and all compact operators R : Y → W and
S : Z → X, the left and right multiplication operators LR and RS as operators fromM into U(X,W )
and U(Z, Y ) (resp. into K(X,W ) and K(Z, Y )) respectively, are compact, where LR(T ) = RT and
RS(T ) = TS, for T ∈ M. Here, the linear subspace M ⊆ U(X,Y ) is called strongly DP1 in
U(X,Y ) (resp. in K(X,Y )), if under the same conditions, the operators LR and RS are DP1.
Evidently, everywhere one talks about U(X,Y ) or linear subspace M of it, the related norm is
ideal norm A(·), while the operator norm ‖ · ‖ is applied when the space is a linear subspace of
L(X,Y ) of all bounded linear operators from X into Y. Thus, ifM is a linear subspace of U(X,Y ),
we endowed always M by ideal norm A(·) and the weak topology of M is refered to this norm.
Also, DP1 ness of every operator onM, such as left and right multiplication operators, depends on
the norm of the image space.
In [9] and [10], the authors proved that for several operator ideal U , a closed subspace M ⊆
⊆ U(X,Y ) is strongly completely continuous in U(X,Y ) (or in K(X,Y )) if and only if all evalua-
tion operators φx :M→ Y and ψy∗ :M→ X∗ are compact operators, where x ∈ X and y∗ ∈ Y ∗
are arbitrary and for each T ∈ M, φx(T ) = Tx, ψy∗(T ) = T ∗y∗. Here, we will prove that similar
results hold for strongly DP1 property.
Throughout this article X, Y, Z, V and W denote arbitrary Banach spaces. The closed unit ball of
a Banach space X is denoted by X1, X
∗ is the dual of X and T ∗ refers to the adjoint of the operator
T. U is an arbitrary (Banach) operator ideal and U(X,Y ) is applied for component of U . We use
the notations ‖T‖ and A(T ) for operator norm and ideal norm of any operator T ∈ U respectively
and note that in general, ‖T‖ ≤ A(T ), for all T ∈ U . Also for arbitrary Banach spaces X and Y,
L(X,Y ) and K(X,Y ) are used for the Banach spaces of all bounded linear and compact operators
between X and Y, respectively, and Kw∗(X
∗, Y ) is the space of all compact weak∗-weak continuous
operators from X∗ to Y. The abbreviation K(X) is used for K(X,X). Our notations are standard
and we refer the reader to [3, 5, 7] for undefined notations and terminologies.
2. Main results. In this section we will show that in many operator ideals the strong DP1 ness of
its subspaces is necessary or sufficient for the DP1 ness of all evaluation operators on that subspace.
Theorem 2.1. LetM ⊆ U(X,Y ) be a closed subspace such that all of the evaluation opera-
tors φx and ψy∗ are DP1. ThenM is strongly DP1 in K(X,Y ).
Proof. Let Tn → T weakly in M and A(Tn) = A(T ) = 1. Then by assumption, for each
x ∈ X, Tnx→ Tx in norm. The boundedness of the sequence (‖Tn‖) then implies that the sequence
Tn converges uniformly to T on compact subsets of X. This shows that TnS → TS in norm of
K(Z, Y ), for every compact operator S : Z → X. So RS is DP1.
Similarly, T ∗n → T ∗ uniformly on compact subsets of Y ∗ and so T ∗nR
∗ → T ∗R∗ in norm of
K(W ∗, X∗), for every compact operator R : Y →W. Thus RTn → RT and the proof is completed.
As a consequence of the theorem, we have the following refined corollary for closed operator
ideals. Recall that an operator ideal U is closed if its components U(X,Y ) are closed in L(X,Y ).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
590 S. M. MOSHTAGHIOUN
Corollary 2.1. Let U be a closed operator ideal andM be a linear subspace of U(X,Y ) such
that all of the evaluation operators φx and ψy∗ are DP1. ThenM is strongly DP1 in U(X,Y ).
Proof. We first note that by definition of operator ideal, LR and RS are operators into U(X,W )
and U(Z, Y ), respectively. Now suppose that Tn → T weakly inM and A(Tn) = A(T ) = 1. Then
by Theorem 2.1, ‖TnS − TS‖ → 0 and ‖RTn −RT‖ → 0 as n→∞. Since U is a closed operator
ideal, by open mapping theorem, there exists a δ > 0 such that A(K) ≤ δ‖K‖ for all operator K in
U . So A(TnS − TS) and A(RTn −RT ) tend to 0. This shows thatM is strongly DP1 in U(X,Y )
and the proof is completed.
Although, by Theorem 2.1, the strong DP1 ness ofM⊆ U(X,Y ) in K(X,Y ) follows from the
DP1 ness of all point evaluations onM, but we do not know that in general, the same question about
strong DP1 ness of M ⊆ U(X,Y ) in U(X,Y ) is true or false. In the following two theorems, we
partially give an affirmative answer to this question.
Theorem 2.2. Let X and Y ∗ have the approximation property andM⊆ U(X,Y ) be a linear
subspace. If all of the evaluation operators φx and ψy∗ are DP1, thenM is strongly DP1 in U(X,Y ).
Proof. Let R : Y → W be a compact operator. Since Y ∗ has the approximation property, there
exists a sequence Rn : Y →W of finite rank operators such that ‖Rn −R‖ → 0 as n→∞.
We claim that each multiplication operator LRn :M → U(X,W ) is a DP1 operator. Since any
finite sum of DP1 operators is also DP1, it is enough to consider the particular case Rn = y∗ ⊗ w,
where y∗ ∈ Y ∗ and w ∈W are arbitrary.
Now suppose that (Tk) is a weakly null sequence inM, T ∈ M and A(T + Tk) = A(T ) = 1.
Then by assumption ‖T ∗k y∗‖ → 0 and so
A(LRnTk) = A(RnTk) = A(T ∗k y
∗ ⊗ w) = ‖T ∗k y∗‖ · ‖w‖ → 0,
as k →∞. Thus LRn is DP1, for each n. Also,
A(LRTk) ≤ A((LR − LRn)Tk) +A(LRnTk) ≤ ‖Rn −R‖A(Tk) +A(LRnTk)→ 0,
as k →∞ for suitable n. Hence LR is a DP1 operator.
Similarly, if S : Z → X is a compact operator, the approximation property of X yields the
existence of a sequence Sn : Z → X of finite rank operators such that ‖Sn − S‖ → 0 as n→∞.
Since each Sn is a finite sum of operators z∗ ⊗ x with z∗ ∈ Z∗ and x ∈ X, a similar method
finishes the proof of the argument.
Now we will show that for each two arbitrary Banach spaces X and Y (without any approximate
assumption), if U is an injective operator ideal, then a similar result holds for closed subspaces of
U(X,Y ). Recall that an operator ideal U is injective if for each Banach spaces X, V and W and
each isometric embedding J : V →W, the operator LJ : U(X,V )→ U(X,W ) is also an (isometric)
embedding, and furthermore, an operator T ∈ L(X,V ) belongs to U if JT ∈ U . Many usual op-
erator ideals are injective. For instance, the (weakly) compact operators, the (weakly) Banach – Saks
operators, the unconditionally converging operators and the p-summing operators between Banach
spaces, with 1 ≤ p < ∞, are standard examples. For additional examples see [3, 12]. So we have
the following theorem:
Theorem 2.3. If U is an injective operator ideal andM⊆ U(X,Y ) is a linear subspace such
that all of the evaluation operators are DP1, thenM is strongly DP1 in U(X,Y ).
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STRONGLY ALTERNATIVE DUNFORD – PETTIS SUBSPACES OF OPERATOR IDEALS 591
Proof. Let R : Y → V be a compact operator and W = l∞(V ∗1 ). Then W has the approximation
property and there exists a natural isometric embedding J : V →W. Since JR : Y →W is compact,
the proof of Theorem 2.2 shows that the left multiplication operator LJR :M→ U(X,W ) is a DP1
operator. But LJR = LJ ◦ LR and LJ : U(X,V )→ U(X,W ) is an isometric embedding. So LR is
a DP1 operator.
Similarly, for each compact operator S : Z → X, if J : Z∗ → l∞((Z∗∗)1) is an isometric
embedding and (Sn) is an approximated sequence of finite rank operators for JS∗, then the DP1
ness of all φx with the fact that each Sn is a finite sum of one rank operators x⊗ z with x ∈ X and
z ∈ l∞((Z∗∗)1), shows that each LSn as operator from M̃ = {T ∗ : T ∈ M} is DP1. Hence LJS∗
and so LS∗ as operator from M̃ is DP1.
Theorem 2.3 is proved.
The following theorem shows that the converse of the above theorems is also valid in every
operator ideal U .
Theorem 2.4. LetM ⊆ U(X,Y ) be a linear subspace such that for some Banach spaces W
and Z and each finite rank operators R : Y → W and S : Z → X, the left and right multiplication
operators LR and RS , as operators fromM into U(X,W ) and U(Z, Y ) respectively, be DP1. Then
all of the evaluation operators φx and ψy∗ are DP1.
Proof. Suppose that x ∈ X is a fixed element. We will show that the evaluation operator φx is
DP1. Consider an element z∗ ∈ Z∗ such that ‖z∗‖ = 1 and J : Y → U(Z, Y ) via Jy = z∗⊗ y as an
isometric embedding. If one define the operator S on Z by S = z∗ ⊗ x, then RS = Jφx and then by
assumption is DP1 operator. Since J is an isometric embedding, φx is also DP1. The same argument
proves that all evaluation operators ψy∗ is DP1.
Corollary 2.2. Let X and Y be two Banach spaces and U be an operator ideal that satisfy one
of the following assertions:
(1) X and Y ∗ have the approximation property,
(2) U is a closed operator ideal or,
(3) U is an injective operator ideal.
IfM⊆ U(X,Y ) is a normed linear subspace, then the following are equivalent:
(a) all evaluation operators φx and ψy∗ are DP1,
(b) M is strongly DP1 in U(X,Y ),
(c) M is strongly DP1 in K(X,Y ),
(d) for some Banach spaces W and Z and all finite rank operators R : Y →W and S : Z → X,
left and right multiplication operators LR and RS , as operators fromM into U(X,W ) and U(Z, Y )
(or into K(X,W ) and K(Z, Y )) respectively, are DP1 operators.
Corollary 2.3. Let X and Y be two reflexive Banach spaces and M ⊆ U(X,Y ) be a closed
subspace. IfM has the DP1, thenM is strongly DP1 in K(X,Y ).
Proof. Since X and Y are reflexive Banach spaces, Theorem 2.2 of [8] shows that all evaluation
operators are DP1. So by Theorem 2.1,M is strongly DP1 in K(X,Y ).
Evidently, if one of the additional conditions of Corollary 2.2 satisfies, then M is strongly DP1
in U(X,Y ).
The following corollary is similar to Theorem 6 of [13] which stay that if A is a closed subalgebra
of the space K(X) of all compact operators on a reflexive Banach space X, which has the DP, then
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
592 S. M. MOSHTAGHIOUN
A is completely continuous; that is, for each S ∈ A, the left and right multiplication operators
LS : A → A and RS : A → A are compact.
Corollary 2.4. LetX be a reflexive Banach space andA ⊆ K(X) be a closed subalgebra which
has the DP1 property. Then for each S ∈ A, the left and right multiplication operators LS : A → A
and RS : A → A are DP1.
Proof. By Corollary 2.3, A is strongly DP1 in K(X). Since the concept of strongly DP1 in
K(X) is in general stronger than the DP1 of the left and right multiplication operators on closed
subalgebras of K(X), the prove of this corollary is completed.
The next Corollary 2.5 proves that for some Banach spaces X and Y, having the Schauder de-
compositions [7], and some operator ideal between them, the strong DP1 is also a sufficient condition
for the DP1 property. The concept of Schauder decomposition, as a generalization of Schauder basis
of Banach spaces, provides a good location for introducing the concept of P-property for subspaces
of operator spaces, which has an essential role in the results of [8, 11].
If
∑∞
n=1
⊕Xn and
∑∞
n=1
⊕Yn are Schauder decompositions of X and Y respectively, by
meaning of [11], we say that a closed subspaceM⊆ U(X,Y ) has the P-property if for all integers
m0 and n0 and every operators T, S ∈M,
‖PWTPV + PW ′SPV ′‖ ≤ max{‖PWTPV ‖, ‖PW ′SPV ′‖},
where V = X1 ⊕ ... ⊕ Xm0 and W = Y1 ⊕ ... ⊕ Yn0 , V
′ and W ′ are complementary subspaces
of V and W in X and Y respectively. Also for each complemented subspace V of X, the symbols
PV and PV ′ refer to the the natural projections of X onto V and complementary subspace V ′ of V
respectively.
Finally, we need to remember the following theorem of [8]. For undefined terminologies about
Schauder decomposition, we refer the reader to [7].
Theorem 2.5 (Theorems 2.4 and 2.6 of [8]). Let X and Y have monotone finite dimensional
Schauder decompositions such that the decomposition of X is shrinking. LetM be a closed subspace
of Kw∗(X
∗, Y ) or K(X,Y ) which has the P-property. If all of the related evaluation operators are
DP1 operators, thenM has the DP1 property.
There is a similar result for a closed subspace M ⊆ K(H1, H2), where H1 and H2 are two
arbitrary Hilbert spaces and one can find it in Theorem 2.10 of [8].
Corollary 2.5. Let X and Y satisfy the hypothesis of Theorem 2.5. If M is a closed subspace
of Kw∗(X
∗, Y ) or K(X,Y ) which has the P-property and is strongly DP1, then M has the DP1
property.
Proof. By Theorem 2.4, all related evaluation operators on M are DP1. Now an appeal to
Theorem 2.5 finishes the proof.
There are Banach spaces X and Y having the Schauder decompositions such that some classes
of operators between them have the P-property and one can fined them in [8] or [11]. These leads to
the following two corollaries:
Corollary 2.6. Let X be an lp-direct sum and Y be an lq-direct sum of finite dimensional
Banach spaces with 1 < p ≤ q <∞. IfM is a closed subspace of K(X,Y ), then the following are
equivalent:
(a) M has the DP1 property,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
STRONGLY ALTERNATIVE DUNFORD – PETTIS SUBSPACES OF OPERATOR IDEALS 593
(b) M is a strongly DP1 subspace of K(X,Y ),
(c) all evaluation operators φx and ψy∗ are DP1 operators.
Proof. The statement (a) ⇒ (b) follows from Corollary 2.3, and (b) ⇒ (c) follows from
Theorem 2.4. Also an appeal to Corollary 2.7 of [8] proves (c)⇒ (a).
Corollary 2.7. LetH1 andH2 be two Hilbert spaces andM be a closed subspace ofK(H1, H2).
Then the following are equivalent:
(a) M has the DP1 property,
(b) M is a strongly DP1 subspace of K(H1, H2),
(c) all evaluation operators φx and ψy are DP1 operators.
Proof. Apply Theorems 2.1, 2.4 and also Theorem 2.10 of [8].
Finally, we give an example of a closed linear subspace of an operator ideal such that all eval-
uation operators are DP1 but all are not completely continuous. So the results of this article can be
informative.
Example 2.1. Let U be an arbitrary operator ideal and consider the closed linear subspace
M := l2 of U(l2, l2) via the isometrically embedding
x 7→ Tx, Tx(y) = 〈y, e1〉x,
where x, y ∈ l2 and e1 is the first element of the standard orthonormal basis (en) of l2.
Since the Hilbert space l2 has the Kadec – Klee property ( that is, weak and norm convergence of
sequences in the unit sphere of l2 coincide); for each weakly convergent sequence xn → x in l2 with
‖xn‖ = ‖x‖ = 1, we have ‖xn − x‖ → 0 as n → ∞. Thus all of the bounded evaluation operators
onM = l2 are DP1. On the other hand, for each n, en = Ten(e1) ∈ φe1(M1), so that the evaluation
operator φe1 is not compact and so is not completely continuous.
1. Bunce L., Peralta A. M. The alternative Dunford – Pettis property in C∗-algebras and von Neumann preduals // Proc.
Amer. Math. Soc. – 2002. – 131. – P. 1251 – 1255.
2. Chu C. H., Iochum B. The Dunford – Pettis property in C∗-algebras // Stud. Math. – 1990. – 97. – P. 59 – 64.
3. Defant A., Floret K. Tensor norms and operator ideals // Math. Stud. – 1993. – 179.
4. Diestel J. A survey of results related to the Dunford – Pettis property // Contemp. Math. – 1980. – 2. – P. 15 – 60.
5. Diestel J. Sequences and series in Banach spaces // Grad. Texts Math. – 1984. – 92.
6. Freedman W. An alternative Dunford – Pettis property // Stud. Math. – 1997. – 125. – P. 143 – 159.
7. Lindenstrauss J., Tzafriri L. Classical Banach spaces I, II. – Berlin: Springer-Verlag, 1996.
8. Moshtaghioun S. M. The alternative Dunford – Pettis property in subspaces of operator ideals // Bull. Korean Math.
Soc. – 2010. – 47, № 4. – P. 743 – 750.
9. Moshtaghioun S. M. Weakly completely continuous subspaces of operator ideals // Taiwan. J. Math. – 2007. – 11,
№ 2. – P. 523 – 530.
10. Moshtaghioun S. M., Zafarani J. Completely continuous subspaces of operator ideals // Taiwan. J. Math. – 2006. –
10, № 3. – P. 691 – 698.
11. Moshtaghioun S. M., Zafarani J. Weak sequential convergence in the dual of operator ideals // J. Operator Theory. –
2003. – 49. – P. 143 – 151.
12. Pietsch A. Operator ideals // North-Holland Math. Libr. – 1980. – 20.
13. Ülger A. Subspaces and subalgebras of K(H) whose duals have the Schur property // J. Operator Theory. – 1997. –
37. – P. 371 – 378.
Received 28.01.11
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 4
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