ON THE COHEN p-NUCLEAR SUBLINEAR OPERATORS

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Cohenp-Nuclear Achour Dahmane, Mezrag Lahcène

and Saadi Khalil vol. 10, iss. 2, art. 46, 2009

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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

Received: 29 January, 2009

Accepted: 22 March, 2009

Communicated by: C.P. Niculescu

2000 AMS Sub. Class.: 46B42, 46B40, 47B46, 47B65

Key words: Banach lattice, Cohen p-nuclear operators, Pietsch’s domination theorem, Stronglyp-summing operators, Sublinear operators.

Abstract: LetSB(X, Y)be the set of all bounded sublinear operators from a Banach space Xinto a complete Banach latticeY. In the present paper, we will introduce to this category the concept of Cohenp-nuclear operators. We give an analogue to

“Pietsch’s domination theorem” and we study some properties concerning this notion.

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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 operatorubetween two Banach spacesX, Y is Cohenp-nuclear for (1< p <∞) if there is a positive constantCsuch that for alln ∈N;x1, ..., xn ∈Xandy^∗₁, ..., y_n^∗ ∈ Y^∗we have

n

X

i=1

hu(xi), y_i^∗i

≤C sup

x^∗∈B_X∗

k(x^∗(xi))k_ln p sup

y∈BY

k(y^∗_i(y))k_ln p∗.

The smallest constantC which is noted byn_p(u), such that the above inequality holds, is called the Cohenp-nuclear norm on the spaceN_p(X, Y) of all Cohen p- nuclear operators fromX intoY which is a Banach space. For p = 1 andp = ∞ we haveN₁(X, Y) = π₁(X, Y)(the Banach space of all1-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, ifuverifies a domination theorem then uisp-nuclear and he asked if the statement of this theorem characterizesp-nuclear operators. The reciprocal of this statement is given in [8, Theorem 9.7, p.189], but these operators are calledp-dominated operators. In this work, we generalize this notion to the sublinear maps and we give an analogue to “Pietsch’s domination theo- rem” 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 positivep-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 Section3, we generalize the class of Cohenp-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 Section4, by studying some relations between the different classes of sublinear operators (p-nuclear, stronglyp-summing andp-summing). We study also the relation betweenT and∇T concerning the notion of Cohenp-nuclear sublinear operators, where∇T ={u∈ L(X, Y) :u≤T}(L(X, Y)is the space of all linear operators fromXintoY). We prove that, ifT is a Cohen positivep-nuclear sublinear operator, then u is Cohen positive p-nuclear and consequently u^∗ is positive p^∗- summing. For the converse, we add one condition concerningT.

We start by recalling the abstract definition of Banach lattices. Let X be a Ba- nach space. If X is a vector lattice and kxk ≤ kyk whenever |x| ≤ |y| (|x| = sup{x,−x}) we say thatXis a Banach lattice. If the lattice is complete, we say that Xis a complete Banach lattice. Note that this implies obviously that for anyx∈X the elementsxand|x|have the same norm. We denote byX₊ ={x∈X :x≥0}.

An elementxofX is positive ifx∈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₊ whereh·,·idenotes the bracket of duality.

By a sublattice of a Banach latticeX we mean a linear subspaceE ofX so that sup{x, y} belongs toE whenever x, y ∈ E. The canonical embeddingi : X −→

X^∗∗such thathi(x), x^∗i=hx^∗, xiofXinto its second dualX^∗∗is an order isometry fromX onto a sublattice of X^∗∗,see [9, Proposition 1.a.2]. If we consider X as a

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sublattice ofX^∗∗we have forx₁, 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 byl_p(X)(resp. l_pⁿ(X)) the space of all sequences(x_i)in Xwith the norm

k(x_i)k_l

p(X)=

∞

X

1

kx_ik^p

!¹_p

<∞



resp.

(xi)_1≤i≤n

lⁿ_p(X) =

n

X

1

kxik^p

!¹_p



and byl^ω_p (X)(resp. l^{n ω}_p (X)) the space of all sequences(x_i)inXwith the norm k(x_n)k_lω

p(X)= sup

kξk_X∗=1

∞

X

1

|hx_i, ξi|^p

!_p¹

<∞



resp.k(x_n)k_ln ω

p (X)= sup

kξk_X∗=1 n

X

1

|hx_i, ξi|^p

!¹_p



whereX^∗ denotes the dual (topological) of X andB_X denotes the closed unit ball ofX. We know (see [8]) that l_p(X) = l^ω_p (X)for some 1 ≤ p < ∞ iff dim (X) is finite. If p = ∞, we have l∞(X) = l^ω_∞(X). We have also if 1 < p ≤ ∞, l^ω_p (X)≡ B(l_p^∗, X)isometrically (wherep^∗ is the conjugate ofp, i.e., ¹_p + _p¹∗ = 1).

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In other words, letv : l_p^∗ −→ Xbe a linear operator such thatv(e_i) = x_i (namely, v =P∞

1 e_j ⊗x_j,e_jdenotes the unit vector basis ofl_p) then

(1.3) kvk=k(x_n)k_lω

p(X).

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2. Sublinear Operators

For our convenience, we give in this section some elementary definitions and funda- mental 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 mappingT from a Banach spaceXinto a Banach latticeY is said to be sublinear if for allx, y inX andλinR+, 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 mul- tiplication by a positive number is also a sublinear operator.

Let us denote by

SL(X, Y) ={sublinear mappingsT :X −→Y} and we equip it with the natural order induced byY

(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 2.3 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,

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• T is continuous if and only if there isC >0such that for allx∈X,kT(x)k ≤ Ckxk.

In this case we say thatT is bounded and we put kTk= sup

kT(x)k:kxk_B

X = 1 .

We will denote bySB(X, Y)the set of all bounded sublinear operators fromX into Y.

We say that a sublinear operator T is positive if for all x in X, T(x) ≥ 0; is increasing if for allx, y inX,T(x)≤T(y)whenx≤y.

Also, there is no relation between positive and increasing like the linear case (a linear operatoru∈ L(X, Y)is positive ifu(x)≥0forx≥0).

We will need the following obvious properties.

Proposition 2.2. LetX be an arbitrary Banach space. LetY, Z be Banach lattices.

(i) ConsiderT inSL(X, Y) andu inL(Y, Z). Assume that uis positive. Then, u◦T ∈ SL(X, Z).

(ii) ConsideruinL(X, Y)andT inSL(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. LetXbe a Banach space and letY be a complete Banach lattice.

LetT ∈ SL(X, Y).Then, for allxinX there isu_x ∈ ∇T such thatT(x) =u_x(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 ifY 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 Proposition2.3, 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 stronglyp-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 constantCsuch that for anyn∈N;x₁, ..., x_n∈X andy^∗₁, ..., y_n^∗ ∈Y^∗we have (2.3)

n

X

i=1

|hT (x_i), y_i^∗i| ≤Ck(x_i)k_ln

p(X) sup

y∈B_Y

k(y^∗_i (y))k_ln ω p∗ .

We denote byD_p(X, Y)the class of all stronglyp-summing sublinear operators fromX intoY and byd_p(T)the smallest constant C such that the inequality (2.3) holds. Forp= 1, we haveD₁(X, Y) =SB(X, Y).

Theorem 2.5 ([3]). LetX be a Banach space andY be a Banach lattice. An oper- atorT ∈ SB(X, Y)is stronglyp-summing (1< p <∞), if and only if, there exists a positive constantC > 0and a Radon probability measureµonB_Y^∗∗ such that for allx∈X, we have

(2.4) |hT (x), y^∗i| ≤Ckxk Z

B_Y∗∗

|y^∗(y^∗∗)|^p^∗dµ(y^∗∗) _p¹∗

.

Moreover, in this case

dp(T) = inf{C >0 : for allCverifying the inequality (2.4)}.

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For the definition of positive stronglyp-summing, we replaceY^∗byY₊^∗andd_p(T) byd⁺_p(T).

To conclude this section, we recall the definition of positivep-summing sublinear operators, which was first stated in the linear case by Blasco in [5]. For the definition ofp-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 ∈π⁺_p (X, Y)), if there exists a positive constantC such that for alln ∈Nand all {x1, ..., xn} ⊂X+, we have

(2.5) k(T (x_i))k_ln

p(Y)≤Ck(x_i)k_ln ω p (X). We put

π_p⁺(T) = inf{Cverifying the inequality (2.5)}.

Theorem 2.7. A sublinear operator between Banach lattices X, Y is positive p- summing (1≤ p < ∞), if and only if, there exists a positive constantC > 0and a Borel probabilityµonB_X⁺∗ such that

(2.6) kT (x)k ≤C

Z

B⁺_X∗

hx, x^∗i^pdµ(x^∗)

!¹_p

for everyx∈X+.Moreover, in this case

π⁺_p(T) = inf{C > 0 : for allC verifying the inequality (2.6)}. Proof. It is similar to the linear case (see [5,12]).

IfT is positivep-summing thenuis positivep-summing for allu ∈ ∇T and by [1, Corollary 2.4], we have π⁺_p(u) ≤ 2π_p⁺(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 Cohenp-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 constantCsuch that for anyn∈N,x₁, ..., x_n ∈Xandy₁^∗, ..., y^∗_n∈Y^∗,we have (3.1)

n

X

i=1

hT(xi), y^∗_ii

≤C sup

x^∗∈B_X∗

k(x^∗(xi))k_ln p sup

y∈BY

k(y_i^∗(y))k_ln p∗.

We denote byN_p(X, Y)the class of all Cohenp-nuclear sublinear operators from XintoY and byn_p(T)the smallest constantCsuch that the inequality (3.1) holds.

For the definition of positive Cohenp-nuclear, we replace Y^∗ by Y₊^∗ andn_p(T)by n⁺_p(T).

LetT ∈ SB(X, Y)andv :lⁿ_p −→Y^∗be a bounded linear operator. By (1.3), the sublinear operatorT is Cohenp-nuclear, if and only if,

(3.2)

n

X

i=1

hT (x_i), v(e_i)i

≤C sup

x^∗∈B_X∗

k(x^∗(x_i))k_ln pkvk.

Similar to the linear case, forp= 1andp=∞, we haveN₁(X, Y) =π₁(X, Y) andN∞(X, Y) = D∞(X, Y).

Proposition 3.2. LetX be a Banach space andY, Z be two Banach lattices. Con- siderT inSB(X, Y),ua positive operator inB(Y, Z)andSinB(E, X).

(i) IfT is a Cohenp-nuclear sublinear operator, thenu◦T is a Cohenp-nuclear sublinear operator andnp(u◦T)≤ kuknp(T).

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(ii) IfT is a Cohenp-nuclear sublinear operator, thenT ◦S is a Cohenp-nuclear sublinear operator andn_p(T ◦S)≤ kSkn_p(T).

Proof. (i) Letn∈N;x₁, ..., x_n∈Xandz₁^∗, ..., z^∗_n∈Z^∗. It suffices by (3.2) to prove

that

n

X

i=1

huT (x_i), z_i^∗i

≤C sup

x^∗∈B_X∗

k(x^∗(x_i))k_ln pkvk wherev :Z −→l_pⁿ∗ such thatv(z) = Pn

i=1z_i^∗(z)e_i. We have

n

X

i=1

huT (x_i), z_i^∗i

=

n

X

i=1

hT(x_i), u^∗(z_i^∗)i

≤n_p(T) sup

x^∗∈B_X∗

k(x^∗(x_i))k_ln pkwk where

w(y) =

n

X

i=1

hu^∗(z_i^∗), yie_i,

=

n

X

i=1

hz^∗_i, u(y)ie_i,

=ku(y)k

n

X

i=1

z_i^∗, u(y) ku(y)k

e_i.

This implies that

kwk ≤ kuk sup

y∈B_Y

(z_i^∗(z))_1≤i≤n

≤ kuk kvk.

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(ii) Letn∈N;e₁, ..., e_n ∈Eandy₁^∗, ..., y^∗_n∈Y^∗. We have

n

X

i=1

hT ◦S(ei), y^∗_ii

≤np(T) sup

x^∗∈B_X∗

n

X

i=1

|hS(ei), x^∗i|^p

!¹_p kvk

≤n_p(T) sup

x^∗∈B_X∗

kS^∗(x^∗)k

n

X

i=1

e_i, S^∗(x^∗) kS^∗(x^∗)k

p!¹_p kvk

≤n_p(T)kSk sup

e^∗∈B_E∗

n

X

i=1

|he_i, e^∗i|^p

!¹_p kvk. This implies thatT is Cohenp-nuclear andn_p(T ◦S)≤ kSkn_p(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 3.3. LetEbe a Hausdorff topological vector space, and letC be a compact convex subset ofE. LetM be a set of functions onC with values in(−∞,∞]having the following properties:

(a) eachf ∈M is convex and lower semicontinuous;

(b) ifg ∈conv(M), there is anf ∈M withg(x)≤f(x), for everyx∈ C;

(c) there is anr∈Rsuch that eachf ∈M has a value not greater thanr.

Then there is anx0 ∈ C such thatf(x0)≤rfor allf ∈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)andC a positive constant.

1. The operatorT is Cohenp-nuclear andn_p(T)≤C.

2. For anyninN,x₁, ..., x_ninX andy^∗₁, ..., y_n^∗ inY^∗we have

n

X

i=1

|hT (x_i), y_i^∗i| ≤C sup

x^∗∈B_X∗

k(x^∗(x_i))k_ln p sup

y∈B_Y

k(y^∗_i(y))k_ln p∗.

3. There exist Radon probability measures µ₁ onB_X^∗ andµ₂ onB_Y^∗∗,such that for allx∈Xandy^∗ ∈Y^∗, we have

(3.3) |hT (x), y^∗i| ≤C Z

B_X∗

|x(x^∗)|^pdµ1(x^∗) ¹_p

× Z

BY∗∗

|y^∗(y^∗∗)|^p^∗dµ₂(y^∗∗) _p¹∗

.

Moreover, in this case

n_p(T) = inf{C >0 : for allC verifying the inequality (3.3)}. Proof. (1)⇒(2). LetT be inN_p(X, Y)and(λ_i)be a scalar sequence. We have

n

X

i=1

λ_ihT (x_i), y^∗_ii

≤n_p(T) supk(λ_i)k_l

∞k(x_i)k_ln ω

p (X) sup

y∈BY

k(y_i^∗(y))k_ln p∗. Taking the supremum over all sequences(λ_i)withk(λ_i)k_l

∞ ≤1, we obtain

n

X

i=1

|hT (xi), y_i^∗i| ≤np(T)k(xi)k_ln ω

p (X) sup

y∈BY

k(y_i^∗(y))k_ln p∗.

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To prove that (2) implies (3). We consider the setsP(B_X^∗)andP(B_Y^∗∗)of probability measures inC(B_X^∗)^∗ andC(B_Y^∗∗)^∗,respectively. These are convex sets which are compact when we endowC(B_X^∗)^∗ andC(B_Y^∗∗)^∗ with their weak∗topologies.

We are going to apply Ky Fan’s Lemma with E = C(B_X^∗)^∗ × C(B_Y^∗∗)^∗ and C =P(B_X^∗)×P(B_Y^∗∗).

Consider the setM of all functionsf :C →Rof the form (3.4) f(^(xi),(^y^∗i)) (µ₁, µ₂) :=

n

X

i=1

|hT(x_i), y_i^∗i| −C 1 p

n

X

i=1

Z

B_X∗

|x_i(x^∗)|^pdµ₁(x^∗)

+ 1 p^∗

n

X

i=1

Z

B_Y∗∗

|y_i^∗(y^∗∗)|^p^∗dµ₂(y^∗∗)

! ,

wherex₁, ..., x_n∈Xandy₁^∗, ..., y_n^∗ ∈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). Letf, g be inM and α∈[0,1]such that

f((^x⁰i)^,(^yi^0∗))(µ₁, µ₂) =

k

X

i=1

|hT(x⁰_i), y_i^0∗i| −C

"

1 p

k

X

i=1

Z

B_X∗

|hx⁰_i, x^∗i|^pdµ₁(x^∗)

+ 1 p^∗

k

X

i=1

Z

B_Y∗∗

|hy_i^0∗, y^∗∗i|^p^∗dµ₂(y^∗∗)

# ,

and

g((^x⁰⁰i)^,(^y^00∗i ))(µ₁, µ₂)

=

l

X

i=k+1

|hT(x⁰⁰_i), y^00∗_i i| −C

"

1 p

l

X

i=k+1

Z

B_X∗

|hx⁰⁰_i, x^∗i|^pdµ₁(x^∗)

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+ 1 p^∗

l

X

i=k+1

|hy^00∗_i , y^∗∗i|^p^∗dµ₂(y^∗∗)

# .

It follows that αf =α

" _k X

i=1

|hT(x⁰_i), y_i^0∗i| −C 1 p

k

X

i=1

Z

B_X∗

|hx⁰_i, x^∗i|^pdµ₁(x^∗)

+ 1 p^∗

k

X

i=1

Z

B_Y∗∗

|hy^0∗_i , y^∗∗i|^p^∗dµ₂(y^∗∗)

!#

=

k

X

i=1

D

T α¹^px⁰_i

, α^p¹^∗y_i^0∗E

−C 1 p

k

X

i=1

Z

B_X∗

D

α¹^px⁰_i, x^∗E

p

dµ₁(x^∗)

+1 p^∗

k

X

i=1

Z

B_Y∗∗

D

α^p¹^∗y_i^0∗, y^∗∗E

p^∗

dµ₂(y^∗∗)

!

=f

α¹^px⁰_i

,

α

1 p∗

y_i^0∗

(µ₁, µ₂), and

f+g =

k

X

i=1

|hT(x⁰_i), y_i^0∗i| −C 1 p

k

X

i=1

Z

BX∗

|hx⁰_i, x^∗i|^pdµ₁(x^∗)

+ 1 p^∗

k

X

i=1

Z

BY∗∗

|hy_i^0∗, y^∗∗i|^p^∗dµ₂(y^∗∗)

! +

l

X

i=k+1

|hT(x⁰⁰_i), y_i^00∗i|

−C 1 p

l

X

i=k+1

Z

B_X∗

|hx⁰⁰_i, x^∗i|^pdµ₁(x^∗) + 1 p^∗

l

X

i=k+1

|hy_i^00∗, y^∗∗i|^p^∗dµ₂(y^∗∗)

!

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=

k

X

i=1

|hT(x⁰_i), y_i^0∗i|+

l

X

i=k+1

|hT(x⁰⁰_i), y_i^00∗i| −C 1 p

n

X

i=1

Z

B_X∗

|hx_i, x^∗i|^pdµ₁(x^∗)

+1 p^∗

n

X

i=1

Z

B_Y^∗∗

|hy_i^∗, y^∗∗i|^p^∗dµ2(y^∗∗)

!

=

n

X

i=1

|hT(x_i), y_i^∗i| −C 1 p

n

X

i=1

Z

BX∗

|hx_i, x^∗i|^pdµ₁(x^∗)

+ 1 p^∗

n

X

i=1

Z

BY∗∗

|hy_i^∗, y^∗∗i|^p^∗dµ₂(y^∗∗)

!

withn =k+l,

x_i =

( x⁰_i if 1≤i≤k, x⁰⁰_i if k+ 1≤i≤l and

y^∗_i =

( y^0∗_i if 1≤i≤k, y^00∗_i if k+ 1≤i≤l.

For the condition (c), since BX^∗ and BY^∗∗ are weak∗ compact and norming sets, there exist forf ∈M two elements,x^∗₀ ∈BX^∗ andy0 ∈BY^∗∗ such that

sup

x^∗∈B_X∗

n

X

i=1

|hxi, x^∗i|^p =

n

X

i=1

|hxi, x^∗₀i|^p and

sup

y∈BY

k(y_i^∗(y))k^p_ln^∗ p∗ =

n

X

i=1

|hy_i^∗, y0i|^p^∗.

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Using the elementary identity

(3.5) ∀α, β ∈R^∗+ αβ = inf

>0

1 p

α

p

+ 1

p^∗ (β)^p^∗

,

taking

α = sup

x^∗∈B_X∗

n

X

i=1

|hx_i, x^∗i|^p

!¹_p

, β = sup

y∈B_Y

k(y_i^∗(y))k_ln p∗

and= 1, then f δx^∗₀, δy0

=

n

X

i=1

|hT(xi), y_i^∗i| − C

p sup

x^∗∈B_X∗

n

X

i=1

|hxi, x^∗i|^p

!

− C p^∗ sup

y∈BY

k(y_i^∗(y))k^p_ln^∗ p∗

≤

n

X

i=1

|hT(x_i), y_i^∗i| −C sup

x^∗∈B_X∗

n

X

i=1

|hx_i, x^∗i|^p

!¹_p sup

y∈BY

k(y^∗_i(y))k_ln 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∈Xandy^∗ ∈Y^∗,

|hT (x), y^∗i| ≤ C p

Z

B_X∗

|hx, x^∗i|^pdµ₁(x^∗) + C p^∗

Z

B_Y∗∗

|hy^∗, y^∗∗i|^p^∗dµ₂(y^∗∗).

Fix > 0. Replacingxby 1

x,andy^∗ byy^∗ and taking the infimum over all > 0

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(using the elementary identity (3.5)), we find

|hT (x), y^∗i| ≤C (1

p

"

1

Z

B_X∗

|hx, x^∗i|^pdµ₁(x^∗) ¹_p#p

+1 p^∗

"

Z

BY∗∗

|hy^∗, y^∗∗i|^p^∗dµ₂(y^∗∗)

_p¹∗#p^∗





≤C Z

B_X∗

|hx, x^∗i|^pdµ₁(x^∗) _p¹ Z

B_Y∗∗

|hy^∗, y^∗∗i|^p^∗dµ₂(y^∗∗) _p¹∗

.

To prove that (3)=⇒(1), letx₁, ..., x_n ∈Xandy₁^∗, ..., y^∗_n∈Y^∗. We have by (3.3)

|hT(x_i), y^∗_ii| ≤C Z

BX∗

|x_i(x^∗)|^pdµ₁(x^∗) ¹_p Z

BY∗∗

|y^∗_i(y^∗∗)|^p^∗dµ₂(y^∗∗) _p¹∗

for all1≤i≤n. Thus we obtain by using Hölder’s inequality

n

X

i=1

hT (x_i), y_i^∗i

≤

n

X

i=1

|hT (x_i), y^∗_ii|

≤C

n

X

i=1

Z

B_X∗

|x_i(x^∗)|^pdµ₁(x^∗) ¹_p Z

B_Y∗∗

|y^∗_i(y^∗∗)|^p^∗dµ₂(y^∗∗) _p¹∗

≤C Z

B_X∗

n

X

i=1

|x_i(x^∗)|^pdµ₁(x^∗)

!¹_p _n X

i=1

Z

B_Y∗∗

|y^∗_i(y^∗∗)|^p^∗dµ₂(y^∗∗)

!_p¹∗

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≤C sup

x^∗∈B_X∗

n

X

i=1

|x_i(x^∗)|^p

!¹_p sup

y∈B_Y

(y^∗_i(y))_1≤i≤n lⁿ_p∗.

This implies thatT ∈ N_p(X, Y)andn_p(T)≤Cand this concludes the proof.

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4. Relationships Between π

_p

(X, Y ), D

_p

(X, Y ) and N

_p

(X, Y )

In this section we investigate the relationships between the various classes of sublinear operators discussed in Section2and 4. We also give a relation betweenT and

∇T concerning the notion of Cohenp-nuclear.

Theorem 4.1. LetX be a Banach space andY be a Banach lattice. We have:

1. Np(X, Y)⊆ Dp(X, Y)anddp(T)≤np(T).

2. Np(X, Y)⊆πp(X, Y)andπp(T)≤np(T).

Proof. (1) LetT ∈ N_p(X, Y).Letx∈X andy^∗ ∈Y^∗. We have by (3.3)

|hT(x), y^∗i| ≤n_p(T) Z

B_X∗

|x^∗(x)|^pdµ₁(x^∗) ¹_p Z

B_Y∗∗

|y^∗(y^∗∗)|^p^∗dµ₂(y^∗∗) _p¹∗

≤n_p(T) sup

x^∗∈B_X∗

|x^∗(x)|

Z

B_Y∗∗

|y^∗(y^∗∗)|^p^∗dµ₂(y^∗∗) _p¹∗

≤n_p(T)kxk Z

B_Y∗∗

|y^∗(y^∗∗)|^p^∗dµ₂(y^∗∗) _p¹∗

so

|hT(x), y^∗i| ≤n_p(T)kxk Z

B_Y∗∗

|y^∗(y^∗∗)|^p^∗dµ₂(y^∗∗) _p¹∗

.

Then, by Theorem2.5,T is stronglyp-summing andd_p(T)≤n_p(T).

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(2) LetT be an operator inN_p(X, Y) kT(x)k= sup

y^∗∈B_Y∗

|hT (x), y^∗i|

≤ sup

y^∗∈B_Y∗

n_p(T) Z

BX∗

|x^∗(x)|^pdµ₁(x^∗) ¹_p Z

BY∗∗

|y^∗(y^∗∗)|^p^∗dµ₂(y^∗∗) _p¹∗

≤n_p(T) Z

BX∗

|x^∗(x)|^pdµ₁(x^∗) ¹_p

sup

y^∗∈B_Y∗

ky^∗k. Then

kT(x)k ≤n_p(T) Z

B_X∗

|x^∗(x)|^pdµ₁(x^∗) ¹_p

and by Theorem 2.7, T isp-summing and πp(T) ≤ np(T). The proof is complete.

Theorem 4.2. Let X be Banach space andY, 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, thenL◦T is a Cohenp-nuclear sublinear operator andn_p(L◦T)≤ d_p(L)π_p(T).

2. Consideru inB(Z, X) ap-summing operator andT 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 andv in B(Y, Z) a strongly p-summing one. Assume thatv is positive. Then, v ◦T is a Cohenp-nuclear sublinear operator andn_p(v◦T)≤d_p(v)π_p(T).

Proof. (1) The operator L◦T is sublinear by Proposition 2.2(iii). Let x ∈ X and z^∗ ∈Z^∗. By Theorem2.5, we have

|hL◦T (x), z^∗i|=|hL(T(x)), z^∗i|

≤d_p(L)kT (x)k Z

B_Z∗∗

|z^∗(z^∗∗)|^p^∗dλ(z^∗∗) _p¹∗

and by Theorem2.7

≤d_p(L)π_p(T) Z

BX∗

|x(x^∗)|^pdµ(x^∗) ¹_p Z

BZ∗∗

|z^∗(z^∗∗)|^p^∗dλ(z^∗∗) _p¹∗

,

so

|hL◦T(x), z^∗i|

≤d_p(L)π_p(T) Z

B_X∗

|x(x^∗)|^pdµ(x^∗) ¹_p Z

B_Z∗∗

|z^∗(z^∗∗)|^p^∗dλ(z^∗∗) _p¹∗

.

This implies thatL◦T ∈ N_p(X, Y)andn_p(L◦T)≤d_p(L)π_p(T).

(2) Follows immediately by using Proposition2.2(ii), Theorem2.5and Theorem2.7.

(3) The operatorv◦T is sublinear by Proposition2.2(i). Lettingx∈Xandz^∗ ∈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 stronglyp−summing iffv^∗ isp^∗−summing andd_p(v) =π_p^∗(v^∗)(see [7, Theorem 2.2.1 part(ii)]), so

kT (x)k kv^∗(z^∗)k

≤d_p(v)kT (x)k Z

B_Z^∗∗

|z^∗∗(z^∗)|^p^∗dµ₂(z^∗∗) _p¹∗

≤π_p(T)d_p(v) Z

BX∗

|x^∗(x)|^pdµ₁(x^∗) ¹_p Z

BZ∗∗

|z^∗∗(z^∗)|^p^∗dµ₂(z^∗∗) _p¹∗

.

This implies thatv◦T ∈ N_p(X, Z)andn_p(v◦T)≤d_p(v)π_p(T).

We now present an example of Cohenp-nuclear sublinear operators.

Example 4.1. Let1≤p <∞andn, N ∈N. Letube a linear operator froml₂ⁿinto l^N_p such that S(x) = |u(x)|.Let v be a linear operator fromL_q(µ)(1 ≤ q < ∞) intol₂ⁿ.ThenT =S◦v is a Cohen2-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],vis2-summing. Then by Theorem4.2part (2),T =S◦v is a Cohen2-nuclear sublinear operator.

Proposition 4.3. Let X be a Banach lattice and Y be a complete Banach lattice.

LetT be a bounded sublinear operator fromX intoY. Suppose thatT is positive Cohenp-nuclear (1< p < ∞). Then for allS ∈ SB(X, Y)such thatS ≤ T,S is positive Cohenp-nuclear.

Proof. Lettingx_i ∈X₁ andy_i^∗ ∈Y₊^∗,by (1.2), we have hS(x_i), y_i^∗i ≤ hT(x_i), y^∗_ii