ON THE COHEN p-NUCLEAR SUBLINEAR
OPERATORS
ACHOUR DAHMANE, MEZRAG LAHCÈNE AND SAADI KHALIL
Laboratoire de Mathématiques Pures et Appliquées
Université de M’sila
Ichbilia, M’sila, 28000, Algérie
EMail: dachourdz@yahoo.fr lmezrag@yahoo.fr kh_saadi@yahoo.fr
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
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Received:
29 January, 2009
Accepted:
22 March, 2009
Communicated by:
JJ
II
C.P. Niculescu
J
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2000 AMS Sub. Class.:
46B42, 46B40, 47B46, 47B65
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Key words:
Banach lattice, Cohen p-nuclear operators, Pietsch’s domination theorem,
Strongly p-summing operators, Sublinear operators.
Abstract:
Let SB(X, Y ) be the set of all bounded sublinear operators from a Banach space
X into a complete Banach lattice Y . In the present paper, we will introduce to
this category the concept of Cohen p-nuclear operators. We give an analogue to
“Pietsch’s domination theorem” and we study some properties concerning this
notion.
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Contents
1
Introduction and terminology
3
2
Sublinear Operators
7
3
Cohen p−Nuclear Sublinear Operators
4
Relationships Between πp (X, Y ), Dp (X, Y ) and Np (X, Y )
11
21
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
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1.
Introduction and terminology
The notion of Cohen p-nuclear operators (1 ≤ p ≤ ∞) was initiated by Cohen in
[7] and generalized to Cohen (p, q)-nuclear (1 ≤ q ≤ ∞) by Apiola in [4]. A linear
operator u between two Banach spaces X, Y is Cohen p-nuclear for (1 < p < ∞) if
there is a positive constant C such that for all n ∈ N; x1 , ..., xn ∈ X and y1∗ , ..., yn∗ ∈
Y ∗ we have
n
X
hu (xi ) , yi∗ i ≤ C sup k(x∗ (xi ))klpn sup k(yi∗ (y))kln∗ .
p
y∈BY
x∗ ∈BX ∗
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
i=1
The smallest constant C which is noted by np (u), such that the above inequality
holds, is called the Cohen p-nuclear norm on the space Np (X, Y ) of all Cohen pnuclear operators from X into Y which is a Banach space. For p = 1 and p = ∞
we have N1 (X, Y ) = π1 (X, Y ) (the Banach space of all 1-summing operators) and
N∞ (X, Y ) = D∞ (X, Y ) (the Banach space of all strongly ∞-summing operators).
In [7, Theorem 2.3.2], Cohen proves that, if u verifies a domination theorem then
u is p-nuclear and he asked if the statement of this theorem characterizes p-nuclear
operators. The reciprocal of this statement is given in [8, Theorem 9.7, p.189], but
these operators are called p-dominated operators. In this work, we generalize this
notion to the sublinear maps and we give an analogue to “Pietsch’s domination theorem” for this category of operators which is one of the main results of this paper. We
study some properties concerning this class and treat some related results concerning
the relations between linear and sublinear operators.
This paper is organized as follows. In the first section, we give some basic definitions and terminology concerning Banach lattices. We also recall some standard
notations. In the second section, we present some definitions and properties concerning sublinear operators. We give the definition of positive p-summing operators
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introduced by Blasco [5, 6] and we present the notion of strongly p-summing sublinear operators initiated in [3].
In Section 3, we generalize the class of Cohen p-nuclear operators to the sublinear
operators. This category verifies a domination theorem, which is the principal result.
We use Ky Fan’s lemma to prove it.
We end in Section 4, by studying some relations between the different classes of
sublinear operators (p-nuclear, strongly p-summing and p-summing). We study also
the relation between T and ∇T concerning the notion of Cohen p-nuclear sublinear
operators, where ∇T = {u ∈ L(X, Y ) : u ≤ T } (L(X, Y ) is the space of all linear
operators from X into Y ). We prove that, if T is a Cohen positive p-nuclear sublinear
operator, then u is Cohen positive p-nuclear and consequently u∗ is positive p∗ summing. For the converse, we add one condition concerning T .
We start by recalling the abstract definition of Banach lattices. Let X be a Banach space. If X is a vector lattice and kxk ≤ kyk whenever |x| ≤ |y| (|x| =
sup {x, −x}) we say that X is a Banach lattice. If the lattice is complete, we say that
X is a complete Banach lattice. Note that this implies obviously that for any x ∈ X
the elements x and |x| have the same norm. We denote by X+ = {x ∈ X : x ≥ 0}.
An element x of X is positive if x ∈ X+ .
The dual X ∗ of a Banach lattice X is a complete Banach lattice endowed with
the natural order
(1.1)
x∗1 ≤ x∗2 ⇐⇒ hx∗1 , xi ≤ hx∗2 , xi ,
∀x ∈ X+
where h·, ·i denotes the bracket of duality.
By a sublattice of a Banach lattice X we mean a linear subspace E of X so that
sup {x, y} belongs to E whenever x, y ∈ E. The canonical embedding i : X −→
X ∗∗ such that hi(x), x∗ i = hx∗ , xi of X into its second dual X ∗∗ is an order isometry
from X onto a sublattice of X ∗∗ , see [9, Proposition 1.a.2]. If we consider X as a
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
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sublattice of X ∗∗ we have for x1 , x2 ∈ X
(1.2)
x1 ≤ x2 ⇐⇒ hx1 , x∗ i ≤ hx2 , x∗ i ,
∀x∗ ∈ X+∗ .
For more details on this, the interested reader can consult the references [9, 11].
We continue by giving some standard notations. Let X be a Banach space and
1 ≤ p ≤ ∞. We denote by lp (X) (resp. lpn (X)) the space of all sequences (xi ) in
X with the norm
! p1
∞
X
k(xi )klp (X) =
<∞
kxi kp
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
1

resp. (xi )
1≤i≤n ln (X) =
p
n
X
! p1 
kxi kp
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1
and by lpω (X) (resp. lpn ω (X)) the space of all sequences (xi ) in X with the norm
k(xn )klpω (X) = sup
∞
X
kξkX ∗ =1
! p1
|hxi , ξi|p
resp. k(xn )k n ω
lp (X) = sup
kξkX ∗ =1
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<∞
1

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n
X
! p1 
|hxi , ξi|p
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
1
where X ∗ denotes the dual (topological) of X and BX denotes the closed unit ball
of X. We know (see [8]) that lp (X) = lpω (X) for some 1 ≤ p < ∞ iff dim (X)
ω
is finite. If p = ∞, we have l∞ (X) = l∞
(X). We have also if 1 < p ≤ ∞,
ω
∗
∗
lp (X) ≡ B (lp , X) isometrically (where p is the conjugate of p, i.e., p1 + p1∗ = 1).
Close
In other
P words, let v : lp∗ −→ X be a linear operator such that v (ei ) = xi (namely,
v= ∞
1 ej ⊗ xj , ej denotes the unit vector basis of lp ) then
(1.3)
kvk = k(xn )klpω (X) .
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
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2.
Sublinear Operators
For our convenience, we give in this section some elementary definitions and fundamental properties relative to sublinear operators. For more information see [1, 2, 3].
We also recall some notions concerning the summability of operators.
Definition 2.1. A mapping T from a Banach space X into a Banach lattice Y is said
to be sublinear if for all x, y in X and λ in R+ , we have
(i) T (λx) = λT (x) (i.e., positively homogeneous),
vol. 10, iss. 2, art. 46, 2009
(ii) T (x + y) ≤ T (x) + T (y) (i.e., subadditive).
Note that the sum of two sublinear operators is a sublinear operator and the multiplication by a positive number is also a sublinear operator.
Let us denote by
SL(X, Y ) = {sublinear mappings T : X −→ Y }
and we equip it with the natural order induced by Y
(2.1)
T1 ≤ T2 ⇐⇒ T1 (x) ≤ T2 (x),
∀x ∈ X
and
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∇T = {u ∈ L(X, Y ) : u ≤ T
(i.e., ∀x ∈ X, u(x) ≤ T (x))} .
A very general case when the set ∇T is not empty is provided by Proposition 2.3
below.
Consequently,
(2.2)
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
u ≤ T ⇐⇒ −T (−x) ≤ u(x) ≤ T (x),
∀x ∈ X.
Let T be sublinear from a Banach space X into a Banach lattice Y . Then we
have,
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• T is continuous if and only if there is C > 0 such that for all x ∈ X, kT (x)k ≤
C kxk .
In this case we say that T is bounded and we put
kT k = sup kT (x)k : kxkBX = 1 .
We will denote by SB(X, Y ) the set of all bounded sublinear operators from X into
Y.
We say that a sublinear operator T is positive if for all x in X, T (x) ≥ 0; is
increasing if for all x, y in X, T (x) ≤ T (y) when x ≤ y.
Also, there is no relation between positive and increasing like the linear case (a
linear operator u ∈ L(X, Y ) is positive if u(x) ≥ 0 for x ≥ 0).
We will need the following obvious properties.
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
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Proposition 2.2. Let X be an arbitrary Banach space. Let Y, Z be Banach lattices.
(i) Consider T in SL(X, Y ) and u in L(Y, Z). Assume that u is positive. Then,
u ◦ T ∈ SL(X, Z).
(ii) Consider u in L(X, Y ) and T in SL(Y, Z). Then, T ◦ u ∈ SL(X, Z).
(iii) Consider S in SL(X, Y ) and T in SL(Y, Z). Assume that S is increasing.
Then, S ◦ T ∈ SL(X, Z).
The following proposition will be used implicitly in the sequel. For its proof, see
[1, Proposition 2.3].
Proposition 2.3. Let X be a Banach space and let Y be a complete Banach lattice.
Let T ∈ SL(X, Y ). Then, for all x in X there is ux ∈ ∇T such that T (x) = ux (x)
(i.e., the supremum is attained, T (x) = sup{u(x) : u ∈ ∇T }).
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We have thus that ∇T is not empty if Y is a complete Banach lattice. If Y is
simply a Banach lattice then ∇T is empty in general (see [10]).
As an immediate consequence of Proposition 2.3, we have:
• the operator T is bounded if and only if for all u ∈ ∇T , u ∈ B (X, Y ) (the
space of all bounded linear operators).
We briefly continue by defining the notion of strongly p-summing introduced by
Cohen [7] and generalized to sublinear operators in [3].
Definition 2.4. Let X be a Banach space and Y be a Banach lattice. A sublinear
operator T : X −→ Y is strongly p-summing (1 < p < ∞), if there is a positive
constant C such that for any n ∈ N; x1 , ..., xn ∈ X and y1∗ , ..., yn∗ ∈ Y ∗ we have
(2.3)
n
X
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
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|hT
(xi ) , yi∗ i|
≤ C k(xi )klpn (X) sup
y∈BY
i=1
k(yi∗
(y))kln∗ω .
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p
We denote by Dp (X, Y ) the class of all strongly p-summing sublinear operators
from X into Y and by dp (T ) the smallest constant C such that the inequality (2.3)
holds. For p = 1, we have D1 (X, Y ) = SB(X, Y ).
Theorem 2.5 ([3]). Let X be a Banach space and Y be a Banach lattice. An operator T ∈ SB (X, Y ) is strongly p-summing (1 < p < ∞), if and only if, there exists
a positive constant C > 0 and a Radon probability measure µ on BY ∗∗ such that for
all x ∈ X, we have
Z
1∗
p
∗
∗ ∗∗ p∗
∗∗
(2.4)
|hT (x) , y i| ≤ C kxk
|y (y )| dµ(y )
.
BY ∗∗
Moreover, in this case
dp (T ) = inf {C > 0 : for all C verifying the inequality (2.4)} .
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For the definition of positive strongly p-summing, we replace Y ∗ by Y+∗ and dp (T )
by d+
p (T ).
To conclude this section, we recall the definition of positive p-summing sublinear
operators, which was first stated in the linear case by Blasco in [5]. For the definition
of p-summing and related properties, the reader can see [1].
Definition 2.6. Let X, Y be Banach lattices. Let T : X −→ Y be a sublinear
operator. We will say that T is “positive p-summing” (0 ≤ p ≤ ∞) (we write
T ∈ πp+ (X, Y )), if there exists a positive constant C such that for all n ∈ N and all
{x1 , ..., xn } ⊂ X+ , we have
vol. 10, iss. 2, art. 46, 2009
k(T (xi ))klpn (Y ) ≤ C k(xi )klpn ω (X) .
(2.5)
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We put
πp+
(T ) = inf {C verifying the inequality (2.5)} .
Theorem 2.7. A sublinear operator between Banach lattices X, Y is positive psumming (1 ≤ p < ∞), if and only if, there exists a positive constant C > 0 and a
+
Borel probability µ on BX
∗ such that
!1
Z
p
(2.6)
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
∗ p
kT (x)k ≤ C
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∗
hx, x i dµ(x )
+
BX
∗
for every x ∈ X+ . Moreover, in this case
πp+ (T ) = inf {C > 0 : for all C verifying the inequality (2.6)} .
Proof. It is similar to the linear case (see [5, 12]).
If T is positive p-summing then u is positive p-summing for all u ∈ ∇T and by
[1, Corollary 2.4], we have πp+ (u) ≤ 2πp+ (T ). We do not know if the converse is
true.
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3.
Cohen p−Nuclear Sublinear Operators
We introduce the following generalization of the class of Cohen p-nuclear operators.
We give the domination theorem for such a category by using Ky Fan’s Lemma.
Definition 3.1. Let X be a Banach space and Y be a Banach lattice. A sublinear
operator T : X −→ Y is Cohen p-nuclear (1 < p < ∞), if there is a positive
constant C such that for any n ∈ N, x1 , ..., xn ∈ X and y1∗ , ..., yn∗ ∈ Y ∗ , we have
n
X
(3.1)
hT (xi ) , yi∗ i ≤ C sup k(x∗ (xi ))klpn sup k(yi∗ (y))kln∗ .
p
x∗ ∈BX ∗
y∈BY
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
i=1
We denote by Np (X, Y ) the class of all Cohen p-nuclear sublinear operators from
X into Y and by np (T ) the smallest constant C such that the inequality (3.1) holds.
For the definition of positive Cohen p-nuclear, we replace Y ∗ by Y+∗ and np (T ) by
n+
p (T ).
Let T ∈ SB(X, Y ) and v : lpn −→ Y ∗ be a bounded linear operator. By (1.3), the
sublinear operator T is Cohen p-nuclear, if and only if,
n
X
(3.2)
hT (xi ) , v(ei )i ≤ C sup k(x∗ (xi ))klpn kvk .
x∗ ∈BX ∗
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Similar to the linear case, for p = 1 and p = ∞, we have N1 (X, Y ) = π1 (X, Y )
and N∞ (X, Y ) = D∞ (X, Y ).
Proposition 3.2. Let X be a Banach space and Y, Z be two Banach lattices. Consider T in SB(X, Y ), u a positive operator in B (Y, Z) and S in B (E, X).
(i) If T is a Cohen p-nuclear sublinear operator, then u ◦ T is a Cohen p-nuclear
sublinear operator and np (u ◦ T ) ≤ kuk np (T ).
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(ii) If T is a Cohen p-nuclear sublinear operator, then T ◦ S is a Cohen p-nuclear
sublinear operator and np (T ◦ S) ≤ kSk np (T ).
Proof. (i) Let n ∈ N; x1 , ..., xn ∈ X and z1∗ , ..., zn∗ ∈ Z ∗ . It suffices by (3.2) to prove
that
n
X
∗ ∗
huT
(x
)
,
z
i
i
i ≤ C sup k(x (xi ))klpn kvk
∗
x ∈BX ∗
i=1
P
n
where v : Z −→ lpn∗ such that v(z) = i=1 zi∗ (z)ei . We have
n
n
X
X
huT (xi ) , zi∗ i = hT (xi ) , u∗ (zi∗ )i
i=1
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
i=1
≤ np (T ) sup k(x∗ (xi ))klpn kwk
x∗ ∈B
X∗
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where
w(y) =
=
n
X
i=1
n
X
hu
∗
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(zi∗ ), yi ei ,
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hzi∗ , u(y)i ei ,
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i=1
= ku(y)k
n X
i=1
zi∗ ,
u(y)
ku(y)k
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ei .
This implies that
kwk ≤ kuk sup (zi∗ (z))1≤i≤n y∈BY
≤ kuk kvk .
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(ii) Let n ∈ N; e1 , ..., en ∈ E and y1∗ , ..., yn∗ ∈ Y ∗ . We have
! p1
n
n
X
X
hT ◦ S (ei ) , yi∗ i ≤ np (T ) sup
|hS (ei ) , x∗ i|p
kvk
∗
x ∈BX ∗
i=1
i=1
p ! p1
n ∗
∗
X
S
(x
)
ei ,
≤ np (T ) sup kS ∗ (x∗ )k
kvk
∗
∗
kS (x )k x∗ ∈B ∗
X
≤ np (T ) kSk sup
e∗ ∈BE ∗
i=1
n
X
! p1
|hei , e∗ i|p
kvk .
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
i=1
This implies that T is Cohen p-nuclear and np (T ◦ S) ≤ kSk np (T ).
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The main result of this section is the next extension of “Pietsch’s domination
theorem” for the class of sublinear operators. For the proof we will use the following
lemma due to Ky Fan, see [8].
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Lemma 3.3. Let E be a Hausdorff topological vector space, and let C be a compact
convex subset of E. Let M be a set of functions on C with values in (−∞, ∞] having
the following properties:
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(a) each f ∈ M is convex and lower semicontinuous;
(b) if g ∈ conv(M ), there is an f ∈ M with g(x) ≤ f (x), for every x ∈ C;
(c) there is an r ∈ R such that each f ∈ M has a value not greater than r.
Then there is an x0 ∈ C such that f (x0 ) ≤ r for all f ∈ M .
We now give the domination theorem by using the above lemma.
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Theorem 3.4. Let X be a Banach space and Y be a Banach lattice. Consider
T ∈ SB (X, Y ) and C a positive constant.
1. The operator T is Cohen p-nuclear and np (T ) ≤ C.
2. For any n in N, x1 , ..., xn in X and y1∗ , ..., yn∗ in Y ∗ we have
n
X
|hT (xi ) , yi∗ i| ≤ C sup k(x∗ (xi ))klpn sup k(yi∗ (y))kln∗ .
x∗ ∈BX ∗
i=1
p
y∈BY
3. There exist Radon probability measures µ1 on BX ∗ and µ2 on BY ∗∗ , such that
for all x ∈ X and y ∗ ∈ Y ∗ , we have
(3.3)
|hT (x) , y ∗ i| ≤ C
Z
1
p
|x(x∗ )|p dµ1 (x∗ )
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Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
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BX ∗
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1∗
p
∗ ∗∗ p∗
∗∗
|y (y )| dµ2 (y )
.
Z
×
BY ∗∗
Moreover, in this case
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np (T ) = inf {C > 0 : for all C verifying the inequality (3.3)} .
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Proof. (1)⇒(2). Let T be in Np (X, Y ) and (λi ) be a scalar sequence. We have
n
X
∗ ∗
λ
hT
(x
)
,
y
i
i
i
i ≤ np (T ) sup k(λi )kl∞ k(xi )klpn ω (X) sup k(yi (y))klpn∗ .
y∈BY
i=1
Taking the supremum over all sequences (λi ) with k(λi )kl∞ ≤ 1, we obtain
n
X
i=1
|hT (xi ) , yi∗ i| ≤ np (T ) k(xi )klpn ω (X) sup k(yi∗ (y))kln∗ .
y∈BY
p
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To prove that (2) implies (3). We consider the sets P (BX ∗ ) and P (BY ∗∗ ) of probability measures in C(BX ∗ )∗ and C(BY ∗∗ )∗ , respectively. These are convex sets which
are compact when we endow C(BX ∗ )∗ and C(BY ∗∗ )∗ with their weak∗ topologies.
We are going to apply Ky Fan’s Lemma with E = C(BX ∗ )∗ × C(BY ∗∗ )∗ and
C = P (BX ∗ ) × P (BY ∗∗ ).
Consider the set M of all functions f : C → R of the form
n
n Z
X
1X
∗
|hT (xi ), yi i| − C
(3.4) f((xi ),(y∗ )) (µ1 , µ2 ) :=
|xi (x∗ )|p dµ1 (x∗ )
i
p
i=1
i=1 BX ∗
!
Z
n
∗
1 X
+ ∗
|y ∗ (y ∗∗ )|p dµ2 (y ∗∗ ) ,
p i=1 BY ∗∗ i
where x1 , ..., xn ∈ X and y1∗ , ..., yn∗ ∈ Y ∗ .
These functions are convex and continuous. We now apply Ky Fan’s Lemma
(the conditions (a) and (b) of Ky Fan’s Lemma are satisfied). Let f, g be in M and
α ∈ [0, 1] such that
" k Z
k
X
1X
p
f((x0 ),(y0∗ )) (µ1 , µ2 ) =
|hT (x0i ), yi0∗ i| − C
|hx0i , x∗ i| dµ1 (x∗ )
i
i
p
i=1
i=1 BX ∗
#
k Z
X
∗
1
p
+ ∗
|hyi0∗ , y ∗∗ i| dµ2 (y ∗∗ ) ,
p i=1 BY ∗∗
and
g((x00 ),(y00∗ )) (µ1 , µ2 )
i
i
"
Z
l
l
X
1 X
p
00
00∗
=
|hT (xi ), yi i| − C
|hx00i , x∗ i| dµ1 (x∗ )
p i=k+1 BX ∗
i=k+1
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
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#
l
X
∗
1
p
+ ∗
|hyi00∗ , y ∗∗ i| dµ2 (y ∗∗ ) .
p i=k+1
It follows that
" k
X
αf = α
|hT (x0i ), yi0∗ i| − C
k
1X
p i=1
i=1
+
1
p∗
k Z
X
i=1
Z
p
|hx0i , x∗ i| dµ1 (x∗ )
BX ∗
!#
p∗
|hyi0∗ , y ∗∗ i| dµ2 (y ∗∗ )
vol. 10, iss. 2, art. 46, 2009
BY ∗∗
k D E
X
1
1
p x0
p∗ y 0∗
=
T
α
,
α
i
i −C
i=1
k
1X
p i=1
Z
BX ∗
D 1
E
p 0 ∗ p
α
x
,
x
dµ1 (x∗ )
i
!
k Z
∗
D
E
X
p
1
1
p∗ 0∗ ∗∗ + ∗
α yi , y dµ2 (y ∗∗ )
p i=1 BY ∗∗
= f
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
1
(µ , µ ),
1
1
2
α p x0i , α p∗ yi0∗
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and
f +g =
k
X
i=1
k
|hT (x0i ), yi0∗ i| − C
1X
p i=1
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Z
p
|hx0i , x∗ i| dµ1 (x∗ )
Full Screen
BX ∗
!
k Z
l
X
∗
1 X
0∗ ∗∗ p
∗∗
+ ∗
|hy , y i| dµ2 (y ) +
|hT (x00i ), yi00∗ i|
p i=1 BY ∗∗ i
i=k+1
!
Z
l
l
X
X
∗
1
1
p
p
−C
|hx00i , x∗ i| dµ1 (x∗ ) + ∗
|hyi00∗ , y ∗∗ i| dµ2 (y ∗∗ )
p i=k+1 BX ∗
p i=k+1
Close
=
k
X
|hT (x0i ), yi0∗ i|
l
X
+
i=1
n
|hT (x00i ), yi00∗ i|
−C
i=k+1
1X
p i=1
Z
|hxi , x∗ i|p dµ1 (x∗ )
BX ∗
!
n Z
∗
1 X
+ ∗
|hyi∗ , y ∗∗ i|p dµ2 (y ∗∗ )
p i=1 BY ∗∗
n
n Z
X
1X
∗
=
|hT (xi ), yi i| − C
|hxi , x∗ i|p dµ1 (x∗ )
p i=1 BX ∗
i=1
!
n Z
∗
1 X
p
+ ∗
|hy ∗ , y ∗∗ i| dµ2 (y ∗∗ )
p i=1 BY ∗∗ i
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
Title Page
with n = k + l,
x0i
if
1 ≤ i ≤ k,
x00i
if
k+1≤i≤l
yi0∗
if 1 ≤ i ≤ k,
yi00∗
if k + 1 ≤ i ≤ l.
(
xi =
and
(
yi∗
=
Contents
sup
x∗ ∈BX ∗
∗
p
|hxi , x i| =
i=1
sup
y∈BY
|hxi , x∗0 i|p
i=1
and
∗
k(yi∗ (y))kpln∗
p
n
X
=
n
X
i=1
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For the condition (c), since BX ∗ and BY ∗∗ are weak∗ compact and norming sets,
there exist for f ∈ M two elements, x∗0 ∈ BX ∗ and y0 ∈ BY ∗∗ such that
n
X
JJ
∗
|hyi∗ , y0 i|p .
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Using the elementary identity
∀α, β ∈
(3.5)
1 α p
1
p∗
αβ = inf
+ ∗ (β)
,
>0
p p
R∗+
taking
α = sup
x∗ ∈B
X∗
n
X
! p1
|hxi , x∗ i|p
β = sup k(yi∗ (y))kln∗
,
i=1
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
p
y∈BY
and = 1, then
f δx∗0 , δy0 =
n
X
i=1
≤
n
X
i=1
vol. 10, iss. 2, art. 46, 2009
|hT (xi ), yi∗ i|
C
−
p
|hT (xi ), yi∗ i| − C
sup
x∗ ∈BX ∗
sup
x∗ ∈BX ∗
n
X
!
|hxi , x∗ i|p
−
i=1
n
X
i=1
∗
C
sup k(yi∗ (y))kpln∗
∗
p
p y∈BY
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Contents
! p1
|hxi , x∗ i|p
sup k(yi∗ (y))kln∗ .
y∈BY
p
The last quantity is less than or equal to zero (by hypothesis (2)) and hence condition
(c) is verified by taking r = 0. By Ky Fan’s Lemma, there is (µ1 , µ2 ) ∈ C with
f (µ1 , µ2 ) ≤ 0 for all f ∈ M . Then, if f is generated by the single elements
x ∈ X and y ∗ ∈ Y ∗ ,
Z
Z
∗
C
C
∗
∗ p
∗
|hT (x) , y i| ≤
|hx, x i| dµ1 (x ) + ∗
|hy ∗ , y ∗∗ i|p dµ2 (y ∗∗ ).
p BX ∗
p BY ∗∗
1
Fix > 0. Replacing x by x, and y ∗ by y ∗ and taking the infimum over all > 0
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(using the elementary identity (3.5)), we find
( " Z
p1 #p
1
1
|hT (x) , y ∗ i| ≤ C
|hx, x∗ i|p dµ1 (x∗ )
p BX ∗

" Z
p1∗ #p∗ 
∗
1
+ ∗ |hy ∗ , y ∗∗ i|p dµ2 (y ∗∗ )

p
BY ∗∗
p1 Z
|hx, x i| dµ1 (x )
Z
p
∗
≤C
p1∗
|hy , y i| dµ2 (y )
.
∗
BX ∗
∗
∗∗
p∗
∗∗
|hT
1 Z
p
|xi (x )| dµ1 (x )
Z
∗
≤C
p
∗
|yi∗ (y ∗∗ )|p
∗
BX ∗
1∗
p
dµ2 (y )
Contents
BY ∗∗
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i=1
i=1
≤C
Title Page
∗∗
for all 1 ≤ i ≤ n. Thus we obtain by using Hölder’s inequality
n
n
X
X
∗ hT (xi ) , yi i ≤
|hT (xi ) , yi∗ i|
≤C
vol. 10, iss. 2, art. 46, 2009
BY ∗∗
To prove that (3)=⇒(1), let x1 , ..., xn ∈ X and y1∗ , ..., yn∗ ∈ Y ∗ . We have by (3.3)
(xi ) , yi∗ i|
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
1 Z
p
|xi (x∗ )| dµ1 (x∗ )
n Z
X
1∗
p
|yi∗ (y ∗∗ )| dµ2 (y ∗∗ )
i=1
BX ∗
Z
n
X
BX ∗ i=1
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p∗
p
Close
BY ∗∗
!1
p
∗
p
∗
|xi (x )| dµ1 (x )
n Z
X
i=1
BY ∗∗
! 1∗
p∗
|yi∗ (y ∗∗ )|
p
∗∗
dµ2 (y )
≤ C sup
x∗ ∈BX ∗
n
X
i=1
!1
p
|xi (x∗ )|p
sup (yi∗ (y))1≤i≤n ln .
y∈BY
p∗
This implies that T ∈ Np (X, Y ) and np (T ) ≤ C and this concludes the proof.
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
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4.
Relationships Between πp (X, Y ), Dp (X, Y ) and Np (X, Y )
In this section we investigate the relationships between the various classes of sublinear operators discussed in Section 2 and 4. We also give a relation between T and
∇T concerning the notion of Cohen p-nuclear.
Theorem 4.1. Let X be a Banach space and Y be a Banach lattice. We have:
1. Np (X, Y ) ⊆ Dp (X, Y ) and dp (T ) ≤ np (T ).
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
2. Np (X, Y ) ⊆ πp (X, Y ) and πp (T ) ≤ np (T ).
vol. 10, iss. 2, art. 46, 2009
Proof. (1) Let T ∈ Np (X, Y ). Let x ∈ X and y ∗ ∈ Y ∗ . We have by (3.3)
Title Page
|hT (x) , y ∗ i| ≤ np (T )
1 Z
p
p
|x∗ (x)| dµ1 (x∗ )
Z
BX ∗
1∗
p
∗
p
|y ∗ (y ∗∗ )| dµ2 (y ∗∗ )
BY ∗∗
≤ np (T ) sup |x∗ (x)|
1∗
p
∗
|y ∗ (y ∗∗ )|p dµ2 (y ∗∗ )
Z
x∗ ∈BX ∗
BY ∗∗
Z
≤ np (T ) kxk
1∗
p
∗
|y ∗ (y ∗∗ )|p dµ2 (y ∗∗ )
BY ∗∗
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so
∗
Contents
Z
|hT (x) , y i| ≤ np (T ) kxk
1∗
p
|y ∗ (y ∗∗ )| dµ2 (y ∗∗ )
.
p∗
BY ∗∗
Then, by Theorem 2.5, T is strongly p-summing and dp (T ) ≤ np (T ).
Close
(2) Let T be an operator in Np (X, Y )
kT (x)k = sup |hT (x) , y ∗ i|
y ∗ ∈BY ∗
Z
≤ sup np (T )
y ∗ ∈BY ∗
1 Z
p
|x∗ (x)| dµ1 (x∗ )
p
BY ∗∗
BX ∗
Z
≤ np (T )
∗
1∗
p
|y ∗ (y ∗∗ )| dµ2 (y ∗∗ )
p∗
p
∗
1
p
|x (x)| dµ1 (x )
BX ∗
Then
Z
kT (x)k ≤ np (T )
∗
sup ky k .
y ∗ ∈BY ∗
1
p
|x∗ (x)|p dµ1 (x∗ )
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
Title Page
BX ∗
and by Theorem 2.7, T is p-summing and πp (T ) ≤ np (T ). The proof is complete.
Theorem 4.2. Let X be Banach space and Y, Z be two Banach lattices. Let 1 <
p < ∞.
1. Let T ∈ SB(X, Y ) and L ∈ SB(Y, Z). Assume that L is increasing. If L
is a strongly p-summing sublinear operator, and T is a p-summing sublinear
operator, then L ◦ T is a Cohen p-nuclear sublinear operator and np (L ◦ T ) ≤
dp (L)πp (T ).
2. Consider u in B(Z, X) a p-summing operator and T in SB(X, Y ) a strongly
p-summing one. Then, T ◦ u is a Cohen p-nuclear sublinear operator and
np (T ◦ u) ≤ dp (T )πp (u).
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3. Consider T in SB(X, Y ) a p-summing operator and v in B(Y, Z) a strongly
p-summing one. Assume that v is positive. Then, v ◦ T is a Cohen p-nuclear
sublinear operator and np (v ◦ T ) ≤ dp (v)πp (T ).
Proof. (1) The operator L ◦ T is sublinear by Proposition 2.2(iii). Let x ∈ X and
z ∗ ∈ Z ∗ . By Theorem 2.5, we have
|hL ◦ T (x) , z ∗ i| = |hL (T (x)) , z ∗ i|
Z
≤ dp (L) kT (x)k
1∗
p
∗
p
|z ∗ (z ∗∗ )| dλ(z ∗∗ )
BZ ∗∗
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
and by Theorem 2.7
1∗
p
∗
p
|z ∗ (z ∗∗ )| dλ(z ∗∗ )
,
1 Z
p
p
|x(x∗ )| dµ(x∗ )
Z
≤ dp (L)πp (T )
BX ∗
BZ ∗∗
so
|hL ◦ T (x) , z ∗ i|
Z
≤ dp (L)πp (T )
1 Z
p
p
|x(x∗ )| dµ(x∗ )
BX ∗
1∗
p
∗
p
|z ∗ (z ∗∗ )| dλ(z ∗∗ )
.
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BZ ∗∗
This implies that L ◦ T ∈ Np (X, Y ) and np (L ◦ T ) ≤ dp (L)πp (T ).
(2) Follows immediately by using Proposition 2.2(ii), Theorem 2.5 and Theorem 2.7.
(3) The operator v ◦T is sublinear by Proposition 2.2(i). Letting x ∈ X and z ∗ ∈ Z ∗ ,
we have
|hv(T (x)), z ∗ i| = |hT (x) , v ∗ (z ∗ )i|
≤ kT (x)k kv ∗ (z ∗ )k
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because, v is strongly p−summing iff v ∗ is p∗ −summing and dp (v) = πp∗ (v ∗ ) (see
[7, Theorem 2.2.1 part(ii)]), so
kT (x)k kv ∗ (z ∗ )k
Z
≤ dp (v) kT (x)k
1∗
p
|z ∗∗ (z ∗ )| dµ2 (z ∗∗ )
p∗
BZ ∗∗
Z
≤ πp (T )dp (v)
BX ∗
1 Z
p
p
∗
∗
|x (x)| dµ1 (x )
1∗
p
∗∗ ∗
∗∗
|z (z )| dµ2 (z )
.
p∗
BZ ∗∗
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
This implies that v ◦ T ∈ Np (X, Z) and np (v ◦ T ) ≤ dp (v)πp (T ).
We now present an example of Cohen p-nuclear sublinear operators.
Example 4.1. Let 1 ≤ p < ∞ and n, N ∈ N. Let u be a linear operator from l2n into
lpN such that S(x) = |u (x)| . Let v be a linear operator from Lq (µ) (1 ≤ q < ∞)
into l2n . Then T = S ◦ v is a Cohen 2-nuclear sublinear operator.
Proof. Indeed, S (x) = |u (x)| is a strongly 2-summing sublinear operator by [3],
and by [7, Lemma 3.2.2], v is 2-summing. Then by Theorem 4.2 part (2), T = S ◦ v
is a Cohen 2-nuclear sublinear operator.
Proposition 4.3. Let X be a Banach lattice and Y be a complete Banach lattice.
Let T be a bounded sublinear operator from X into Y . Suppose that T is positive
Cohen p-nuclear (1 < p < ∞). Then for all S ∈ SB (X, Y ) such that S ≤ T , S is
positive Cohen p-nuclear.
Proof. Letting xi ∈ X1 and yi∗ ∈ Y+∗ , by (1.2), we have
hS(xi ), yi∗ i ≤ hT (xi ), yi∗ i
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and consequently, by (2.2),
− hS(xi ), yi∗ i ≤ hT (−xi ), yi∗ i
for all 1 ≤ i ≤ n. This implies that
n
X
|hS(xi ), yi∗ i|
≤
i=1
≤
≤
n
X
i=1
n
X
i=1
n
X
sup {hT (x), yi∗ i , hT (−x), yi∗ i}
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
sup {|hT (x), yi∗ i| , |hT (−x), yi∗ i|}
|hT (x), yi∗ i|
+
n
X
i=1
vol. 10, iss. 2, art. 46, 2009
|hT (−x), yi∗ i|
Title Page
i=1
Contents
and hence
n
X
i=1
∗
∗
|hS(xi ), yi∗ i| ≤ 2n+
p (T ) sup k(x (xi ))klpn sup k(yi (y))kln∗ .
x∗ ∈BX ∗
Thus the operator S is positive Cohen p-nuclear and
p
y∈BY+
n+
p
(S) ≤
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2n+
p (T ).
Remark 1. If S, T are any sublinear operators, we have no answer.
Corollary 4.4. If T is positive Cohen p-nuclear (1 < p < ∞), then for all u ∈ ∇T ,
u is positive Cohen p-nuclear and consequently u∗ is positive p∗ -summing.
Proof. Let T be a positive Cohen p-nuclear sublinear operator. Then for all u ∈ ∇T ,
u is positive Cohen p-nuclear (replacing S by u in Proposition 4.3). If u is positive
Cohen p-nuclear (by Theorem 4.1, u is positive strongly p-summing), then u∗ is
p∗ -summing (see [7, Theorem 2.2.1 part(ii)]).
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We now study the converse of the preceding corollary with some conditions.
Theorem 4.5. Let X be Banach space and Y be a complete Banach lattice. Let
T : X → Y be a sublinear operator. Suppose that there is a constant C > 0, a set
I, an ultrafilter U on I and {ui }i∈I ⊂ ∇T such that for all x in X and y ∗ in Y ∗ ,
|hui (x) , y ∗ i| −→ |hT (x) , y ∗ i|
U
and np (ui ) ≤ C uniformly. Then, T ∈ Np (X, Y ) and np (T ) ≤ C.
Proof. Since ui is Cohen p-nuclear, by Theorem 3.4 there is a Radon probability
measure (µi , νi ) on K = BX ∗ × BY ∗∗ such that for all x ∈ X and y ∗ in Y ∗ , we have
p1 Z
Z
1∗
p
∗
|hui (x) , y ∗ i| ≤ np (ui )
|x (x∗ )|p dµi
|y ∗ (y ∗∗ )|p dνi
.
BX ∗
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
Title Page
BY ∗∗
Contents
As we have for all x in X and y ∗ ∈ Y ∗ ,
|hui (x) , y ∗ i| −→ |hT (x) , y ∗ i|
U
thus we obtain that for all x in X and y ∗ ∈ Y ∗ ,
Z
p1 Z
∗
∗ p
|hT (x) , y i| ≤ limnp (ui )
|x (x )| dµi
U
BX ∗
p∗
|y ∗ (y ∗∗ )| dνi
This implies that np (T ) ≤ C.
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1∗
p
.
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BY ∗∗
The set K = BX ∗ × BY ∗∗ is weak∗ compact, hence (µi , νi ) converge weak∗ to a
probability (µ, ν) on K = BX ∗ ×BY ∗∗ and consequently, for all x in X and y ∗ ∈ Y ∗
Z
p1 Z
1∗
p
∗
∗ p
∗ ∗∗ p∗
|hT (x) , y i| ≤ C
|x (x )| dµ
|y (y )| dν
.
BX ∗
JJ
BY ∗∗
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References
[1] D. ACHOUR AND L. MEZRAG, Little Grothendieck’s theorem for sublinear
operators, J. Math. Anal. Appl., 296 (2004), 541–552.
[2] D. ACHOUR AND L. MEZRAG, Factorisation des opérateurs sous-linéaires
par Lp∞ (Ω, ν) et Lq1 (Ω, ν), Ann. Sci. Math., Quebec, 26(2) (2002), 109–121.
[3] D. ACHOUR, L. MEZRAG AND A. TIAIBA, On the strongly p-summing sublinear operators, Taiwanese J. Math., 11(4) (2007), 959–973.
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
and Saadi Khalil
vol. 10, iss. 2, art. 46, 2009
[4] H. APIOLA, Duality between spaces of p-summable sequences, (p, q)summing operators and characterizations of nuclearity, Math. Ann., 219 (1976),
53–64.
[5] O. BLASCO, A class of operators from a Banach lattice into a Banach space,
Collect. Math., 37 (1986), 13–22.
[6] O. BLASCO, Positive p-summing operators from Lp -spaces, Proc. Amer. Math.
Soc., 31 (1988), 275–280.
[7] J.S. COHEN, Absolutely p-summing, p-nuclear operators and their conjugates,
Math. Ann., 201 (1973), 177–200.
[8] J. DIESTEL, H. JARCHOW AND A. TONGE, Absolutely Summing Operators,
Cambridge University Press, 1995.
[9] J. LINDENSTRAUSS AND L. TZAFRIRI, Classical Banach Spaces, I and II,
Springer-Verlag, Berlin, 1996.
[10] Y.E. LINKE, Linear operators without subdifferentials, Sibirskii Mathematicheskii Zhurnal., 32(3) (1991), 219–221.
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[11] P. MEYER-NIEBERG, Banach Lattices, Springer-Verlag, Berlin, Heidelberg,
New-York, 1991.
[12] A. PIETSCH, Absolut p-summierende Abbildungen in normierten Räumen.
Studia Math., 28 (1967), 1–103.
Cohen p-Nuclear
Achour Dahmane, Mezrag Lahcène
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