Volume 10 (2009), Issue 2, Article 46, 13 pp.
ON THE COHEN p-NUCLEAR SUBLINEAR OPERATORS
ACHOUR DAHMANE, MEZRAG LAHCÈNE, AND SAADI KHALIL
L ABORATOIRE DE M ATHÉMATIQUES P URES ET A PPLIQUÉES
U NIVERSITÉ DE M’ SILA
I CHBILIA , M’ SILA , 28000, A LGÉRIE
dachourdz@yahoo.fr
lmezrag@yahoo.fr
kh_saadi@yahoo.fr
Received 29 January, 2009; accepted 22 March, 2009
Communicated by C.P. Niculescu
A BSTRACT. 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.
Key words and phrases: Banach lattice, Cohen p-nuclear operators, Pietsch’s domination theorem, Strongly p-summing operators, Sublinear operators.
2000 Mathematics Subject Classification. 46B42, 46B40, 47B46, 47B65.
1. I NTRODUCTION
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 ∗
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 p-nuclear 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 pnuclear 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
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ACHOUR DAHMANE , M EZRAG L AHCÈNE ,
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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 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 sublattice of X ∗∗ we have for
x 1 , 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
∞
X
k(xi )klp (X) =
! p1
kxi kp
<∞
1

resp. (xi )
1≤i≤n ln (X) =
p
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 46, 13 pp.
n
X
! p1 
kxi kp

1
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and by lpω (X) (resp. lpn ω (X)) the space of all sequences (xi ) in X with the norm
! p1
∞
X
k(xn )klpω (X) = sup
<∞
|hxi , ξi|p
kξkX ∗ =1
1

resp. k(xn )k n ω
lp (X) = sup
kξkX ∗ =1
n
X
! p1 
|hxi , ξi|p

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) is finite. If p = ∞, we
ω
have l∞ (X) = l∞
(X). We have also if 1 < p ≤ ∞, lpω (X) ≡ B (lp∗ , X) isometrically (where
p∗ is the conjugate of p, i.e., p1 + p1∗ = 1). In other words, let v : lp∗ −→ X be a linear operator
P
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) .
2. S UBLINEAR O PERATORS
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),
(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
∇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 2.2 below.
Consequently,
(2.2)
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,
• 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).
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 46, 13 pp.
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ACHOUR DAHMANE , M EZRAG L AHCÈNE ,
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S AADI K HALIL
We will need the following obvious properties.
Proposition 2.1. 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 2.2. 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 }).
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 2.2, 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 2.2. 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
n
X
(2.3)
|hT (xi ) , yi∗ i| ≤ C k(xi )klpn (X) sup k(yi∗ (y))kln∗ω .
y∈BY
i=1
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 2.3 ([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)} .
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 2.3. 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
(2.5)
k(T (xi ))klpn (Y ) ≤ C k(xi )klpn ω (X) .
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We put
πp+ (T ) = inf {C verifying the inequality (2.5)} .
Theorem 2.4. A sublinear operator between Banach lattices X, Y is positive p-summing (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
hx, x∗ ip dµ(x∗ )
kT (x)k ≤ C
(2.6)
+
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.
3. C OHEN p−N UCLEAR S UBLINEAR O PERATORS
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
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 ).
n
∗
Let T ∈ SB(X, Y ) and v : lp −→ Y be a bounded linear operator. By (1.3), the sublinear
operator T is Cohen p-nuclear, if and only if,
n
X
hT (xi ) , v(ei )i ≤ C sup k(x∗ (xi ))klpn kvk .
(3.2)
x∗ ∈BX ∗
i=1
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 3.1. 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 ).
(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 (xi ) , zi∗ i ≤ C sup k(x∗ (xi ))klpn kvk
x∗ ∈BX ∗
i=1
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 46, 13 pp.
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6
ACHOUR DAHMANE , M EZRAG L AHCÈNE ,
AND
S AADI K HALIL
P
where v : Z −→ lpn∗ such that v(z) = ni=1 zi∗ (z)ei . We have
n
n
X
X
huT (xi ) , zi∗ i = hT (xi ) , u∗ (zi∗ )i
i=1
i=1
≤ np (T ) sup k(x∗ (xi ))klpn kwk
x∗ ∈BX ∗
where
w(y) =
n
X
hu∗ (zi∗ ), yi ei ,
i=1
=
n
X
hzi∗ , u(y)i ei ,
i=1
= ku(y)k
n X
i=1
zi∗ ,
u(y)
ku(y)k
ei .
This implies that
kwk ≤ kuk sup (zi∗ (z))1≤i≤n y∈BY
≤ kuk kvk .
y1∗ , ..., yn∗
(ii) Let n ∈ N; e1 , ..., en ∈ E and
∈ 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
n X
S ∗ (x∗ ) ∗
∗
≤ np (T ) sup kS (x )k
ei , kS ∗ (x∗ )k x∗ ∈B ∗
X
≤ np (T ) kSk sup
e∗ ∈BE ∗
! p1
kvk
i=1
n
X
! p1
|hei , e∗ i|p
kvk .
i=1
This implies that T is Cohen p-nuclear and np (T ◦ S) ≤ kSk np (T ).
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].
Lemma 3.2. 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:
(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.
Theorem 3.3. 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.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 46, 13 pp.
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C OHEN p-N UCLEAR
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(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
1∗
Z
1 Z
p
p
∗
|y ∗ (y ∗∗ )|p dµ2 (y ∗∗ )
.
(3.3)
|hT (x) , y ∗ i| ≤ C
|x(x∗ )|p dµ1 (x∗ )
BX ∗
BY ∗∗
Moreover, in this case
np (T ) = inf {C > 0 : for all C verifying the inequality (3.3)} .
Proof. (1)⇒(2). Let T be in Np (X, Y ) and (λi ) be a scalar sequence. We have
n
X
λi hT (xi ) , yi∗ i ≤ np (T ) sup k(λi )kl∞ k(xi )klpn ω (X) sup k(yi∗ (y))kln∗ .
p
y∈BY
i=1
Taking the supremum over all sequences (λi ) with k(λi )kl∞ ≤ 1, we obtain
n
X
|hT (xi ) , yi∗ i| ≤ np (T ) k(xi )klpn ω (X) sup k(yi∗ (y))kln∗ .
y∈BY
i=1
p
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
∗
(3.4) f((xi ),(y∗ )) (µ1 , µ2 ) :=
|hT (xi ), yi i| − C
|xi (x∗ )|p dµ1 (x∗ )
i
p i=1 BX ∗
i=1
!
n Z
∗
1 X
p
+ ∗
|y ∗ (y ∗∗ )| 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
0
0∗
f((x0 ),(y0∗ )) (µ1 , µ2 ) =
|hT (xi ), yi i| − C
|hx0i , x∗ i| dµ1 (x∗ )
i
i
p i=1 BX ∗
i=1
#
k Z
∗
1 X
p
|hy 0∗ , y ∗∗ i| dµ2 (y ∗∗ ) ,
+ ∗
p i=1 BY ∗∗ i
and
g((x00 ),(y00∗ )) (µ1 , µ2 ) =
i
i
l
X
"
|hT (x00i ), yi00∗ i|
i=k+1
Z
l
1 X
p
−C
|hx00i , x∗ i| dµ1 (x∗ )
p i=k+1 BX ∗
#
l
∗
1 X
00∗ ∗∗ p
∗∗
+ ∗
|hy , y i| dµ2 (y ) .
p i=k+1 i
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 46, 13 pp.
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ACHOUR DAHMANE , M EZRAG L AHCÈNE ,
S AADI K HALIL
AND
It follows that
"
αf = α
k
X
k
|hT (x0i ), yi0∗ i|
−C
i=1
1
+ ∗
p
k Z
X
i=1
1X
p i=1
Z
p
|hx0i , x∗ i| dµ1 (x∗ )
BX ∗
!#
p∗
|hyi0∗ , y ∗∗ i| dµ2 (y ∗∗ )
BY ∗∗
k D E
X
1
1
0
∗ 0∗ p
p
=
T α xi , α yi − C
i=1
k
1X
p i=1
Z
BX ∗
D 1
E
p 0 ∗ p
α xi , x dµ1 (x∗ )
!
k Z
D 1
E ∗
1 X
p∗ 0∗ ∗∗ p
+ ∗
α yi , y dµ2 (y ∗∗ )
p i=1 BY ∗∗
= f
1
(µ , µ ),
1
1
2
α p x0i , α p∗ yi0∗
and
f +g =
k
X
k
|hT (x0i ), yi0∗ i|
1X
p i=1
−C
i=1
Z
p
|hx0i , x∗ i| dµ1 (x∗ )
BX ∗
!
k Z
l
X
∗
1 X
p
0∗ ∗∗
∗∗
|hy , y i| dµ2 (y ) +
|hT (x00i ), yi00∗ i|
+ ∗
p i=1 BY ∗∗ i
i=k+1
!
Z
l
l
X
∗
1 X
1
p
p
−C
|hx00i , x∗ i| dµ1 (x∗ ) + ∗
|hyi00∗ , y ∗∗ i| dµ2 (y ∗∗ )
p i=k+1 BX ∗
p i=k+1
k
l
n Z
X
X
1X
0
0∗
00
00∗
=
|hT (xi ), yi i| +
|hT (xi ), yi i| − C
|hxi , x∗ i|p dµ1 (x∗ )
p
i=1
i=1 BX ∗
i=k+1
!
Z
n
∗
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
with n = k + l,
(
xi =
and
x0i
if 1 ≤ i ≤ k,
x00i if k + 1 ≤ i ≤ l
(
yi∗ =
yi0∗
if 1 ≤ i ≤ k,
yi00∗ if k + 1 ≤ i ≤ l.
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
n
X
X
∗ p
sup
|hxi , x i| =
|hxi , x∗0 i|p
x∗ ∈BX ∗
i=1
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 46, 13 pp.
i=1
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C OHEN p-N UCLEAR
9
and
sup
y∈BY
∗
k(yi∗ (y))kpln∗
p
n
X
=
∗
|hyi∗ , y0 i|p .
i=1
Using the elementary identity
∀α, β ∈
(3.5)
R∗+
1 α p
1
p∗
αβ = inf
+ ∗ (β)
,
>0
p p
taking
n
X
α = sup
x∗ ∈BX ∗
! p1
p
|hxi , x∗ i|
β = sup k(yi∗ (y))kln∗
,
p
y∈BY
i=1
and = 1, then
f δx∗0 , δy0 =
n
X
i=1
≤
n
X
C
|hT (xi ), yi∗ i| −
p
n
X
sup
x∗ ∈BX ∗
|hT (xi ), yi∗ i| − C
i=1
|hxi , x∗ i|p
−
i=1
n
X
sup
x∗ ∈B
!
X∗
C
p∗
∗
sup
k(y
(y))k
i
lpn∗
p∗ y∈BY
! p1
|hxi , x∗ i|p
sup k(yi∗ (y))kln∗ .
p
y∈BY
i=1
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 ∗ ,
C
|hT (x) , y i| ≤
p
∗
Z
Z
C
|hx, x i| dµ1 (x ) + ∗
p
BX ∗
p
∗
∗
∗
|hy ∗ , y ∗∗ i|p dµ2 (y ∗∗ ).
BY ∗∗
1
Fix > 0. Replacing x by x, and y ∗ by y ∗ and taking the infimum over all > 0 (using the
elementary identity (3.5)), we find
( " Z
p1 #p
1
1
p
|hT (x) , y ∗ i| ≤ C
|hx, x∗ i| 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 )
.
∗
∗
∗∗
p∗
∗∗
BY ∗∗
BX ∗
To prove that (3)=⇒(1), let x1 , ..., xn ∈ X and y1∗ , ..., yn∗ ∈ Y ∗ . We have by (3.3)
|hT
(xi ) , yi∗ i|
Z
≤C
1 Z
p
|xi (x )| dµ1 (x )
∗
p
BX ∗
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 46, 13 pp.
∗
∗
|yi∗ (y ∗∗ )|p
1∗
p
dµ2 (y )
∗∗
BY ∗∗
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10
ACHOUR DAHMANE , M EZRAG L AHCÈNE ,
AND
S AADI K HALIL
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|
i=1
i=1
≤C
1 Z
p
|xi (x )| dµ1 (x )
n Z
X
∗
i=1
BX ∗
Z
n
X
≤C
p
!1
p
|xi (x∗ )|p dµ1 (x∗ )
! 1∗
n Z
X
i=1
n
X
x∗ ∈BX ∗
1∗
p
dµ2 (y )
∗∗
BY ∗∗
BX ∗ i=1
≤ C sup
∗
|yi∗ (y ∗∗ )|p
∗
p
p∗
|yi∗ (y ∗∗ )| dµ2 (y ∗∗ )
BY ∗∗
!1
p
|xi (x∗ )|p
sup (yi∗ (y))1≤i≤n ln .
p∗
y∈BY
i=1
This implies that T ∈ Np (X, Y ) and np (T ) ≤ C and this concludes the proof.
4. R ELATIONSHIPS B ETWEEN 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 ).
(2) Np (X, Y ) ⊆ πp (X, Y ) and πp (T ) ≤ np (T ).
Proof. (1) Let T ∈ Np (X, Y ). Let x ∈ X and y ∗ ∈ Y ∗ . We have by (3.3)
Z
1 Z
1∗
p
p
p
∗
∗
∗
∗ ∗∗ p∗
∗∗
|hT (x) , y i| ≤ np (T )
|x (x)| dµ1 (x )
|y (y )| dµ2 (y )
BX ∗
BY ∗∗
≤ np (T ) sup |x∗ (x)|
1∗
p
|y ∗ (y ∗∗ )| dµ2 (y ∗∗ )
Z
x∗ ∈BX ∗
p∗
BY ∗∗
1∗
p
|y (y )| dµ2 (y )
Z
∗
≤ np (T ) kxk
∗∗
p∗
∗∗
BY ∗∗
so
|hT (x) , y ∗ i| ≤ np (T ) kxk
1∗
p
.
|y ∗ (y ∗∗ )| dµ2 (y ∗∗ )
Z
p∗
BY ∗∗
Then, by Theorem 2.3, T is strongly p-summing and dp (T ) ≤ np (T ).
(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
∗
≤ np (T )
∗∗
p∗
∗∗
BY ∗∗
BX ∗
Z
1∗
p
|y (y )| dµ2 (y )
∗
1
p
p
∗
∗
|x (x)| dµ1 (x )
sup ky ∗ k .
BX ∗
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 46, 13 pp.
y ∗ ∈BY ∗
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C OHEN p-N UCLEAR
Then
Z
kT (x)k ≤ np (T )
11
1
p
|x (x)| dµ1 (x )
p
∗
∗
BX ∗
and by Theorem 2.4, 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).
(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 2.1(iii). Let x ∈ X and z ∗ ∈ Z ∗ . By
Theorem 2.3, we have
|hL ◦ T (x) , z ∗ i| = |hL (T (x)) , z ∗ i|
Z
≤ dp (L) kT (x)k
1∗
p
|z (z )| dλ(z )
∗
∗∗
p∗
∗∗
BZ ∗∗
and by Theorem 2.4
1 Z
p
|x(x∗ )|p dµ(x∗ )
Z
≤ dp (L)πp (T )
BX ∗
1∗
p
|z ∗ (z ∗∗ )| dλ(z ∗∗ )
,
p∗
BZ ∗∗
so
|hL ◦ T (x) , z ∗ i| ≤ dp (L)πp (T )
Z
1 Z
p
|x(x∗ )|p dµ(x∗ )
BX ∗
1∗
p
|z ∗ (z ∗∗ )| dλ(z ∗∗ )
.
p∗
BZ ∗∗
This implies that L ◦ T ∈ Np (X, Y ) and np (L ◦ T ) ≤ dp (L)πp (T ).
(2) Follows immediately by using Proposition 2.1(ii), Theorem 2.3 and Theorem 2.4.
(3) The operator v ◦ T is sublinear by Proposition 2.1(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
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)
1 Z
p
|x∗ (x)|p dµ1 (x∗ )
BX ∗
1∗
p
|z ∗∗ (z ∗ )| dµ2 (z ∗∗ )
.
p∗
BZ ∗∗
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.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 46, 13 pp.
http://jipam.vu.edu.au/
12
ACHOUR DAHMANE , M EZRAG L AHCÈNE ,
AND
S AADI K HALIL
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
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
sup {hT (x), yi∗ i , hT (−x), yi∗ i}
i=1
≤
n
X
sup {|hT (x), yi∗ i| , |hT (−x), yi∗ i|}
i=1
≤
n
X
|hT (x), yi∗ i|
i=1
+
n
X
|hT (−x), yi∗ i|
i=1
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 ∗
y∈BY+
p
+
Thus the operator S is positive Cohen p-nuclear and n+
p (S) ≤ 2np (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)]).
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.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 46, 13 pp.
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C OHEN p-N UCLEAR
13
Proof. Since ui is Cohen p-nuclear, by Theorem 3.3 there is a Radon probability measure
(µi , νi ) on K = BX ∗ × BY ∗∗ such that for all x ∈ X and y ∗ in Y ∗ , we have
Z
p1 Z
1∗
p
∗
p
p
|hui (x) , y ∗ i| ≤ np (ui )
|x (x∗ )| dµi
|y ∗ (y ∗∗ )| dνi
.
BX ∗
BY ∗∗
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
1∗
p
.
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∗
|y (y )| dν
|hT (x) , y i| ≤ C
|x (x )| dµ
.
BY ∗∗
BX ∗
This implies that np (T ) ≤ C.
R EFERENCES
[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.
[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
Berlin, 1996.
AND
L. TZAFRIRI, Classical Banach Spaces, I and II, Springer-Verlag,
[10] Y.E. LINKE, Linear operators without subdifferentials, Sibirskii Mathematicheskii Zhurnal., 32(3)
(1991), 219–221.
[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.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 46, 13 pp.
http://jipam.vu.edu.au/