We give an analogue to “Pietsch’s domination theorem” and we study some properties concerning this notion

ON THE COHEN p-NUCLEAR SUBLINEAR OPERATORS

ACHOUR DAHMANE, MEZRAG LAHCÈNE, AND SAADI KHALIL LABORATOIRE DEMATHÉMATIQUESPURES ETAPPLIQUÉES

UNIVERSITÉ DEM’SILA

ICHBILIA, M’SILA, 28000, ALGÉRIE

dachourdz@yahoo.fr lmezrag@yahoo.fr kh_saadi@yahoo.fr

Received 29 January, 2009; accepted 22 March, 2009 Communicated by C.P. Niculescu

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.

Key words and phrases: Banach lattice, Cohenp-nuclear operators, Pietsch’s domination theorem, Stronglyp-summing oper- ators, Sublinear operators.

2000 Mathematics Subject Classification. 46B42, 46B40, 47B46, 47B65.

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 constantC such that for alln∈N;x1, ..., xn∈X andy^∗₁, ..., y_n^∗ ∈Y^∗we have

n

X

i=1

hu(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∗.

The smallest constant C which is noted by n_p(u), such that the above inequality holds, is called the Cohenp-nuclear norm on the spaceN_p(X, Y)of all Cohenp-nuclear operators from X intoY which is a Banach space. Forp= 1andp=∞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, 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

026-09

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 positivep-summing operators introduced by Blasco [5, 6] and we present the notion of stronglyp-summing sublinear operators initiated in [3].

In Section 3, 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 Section 4, by studying some relations between the different classes of sublin- ear operators (p-nuclear, stronglyp-summing andp-summing). We study also the relation be- tweenT 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 positivep-nuclear and consequentlyu^∗is positivep^∗-summing. For the converse, we add one condition concerning T.

We start by recalling the abstract definition of Banach lattices. LetX be a Banach space. If X is a vector lattice andkxk ≤ kykwhenever|x| ≤ |y|(|x| = sup{x,−x}) we say thatX 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 anyx∈ X the elementsxand|x|have the same norm. We denote byX₊ ={x∈X :x≥0}. An elementxofX is positive ifx∈X₊.

The dualX^∗ 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 thatsup{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 dualX^∗∗ is an order isometry fromX onto a sub- lattice ofX^∗∗,see [9, Proposition 1.a.2]. If we considerX as a sublattice ofX^∗∗ 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. LetX be a Banach space and1≤ p≤ ∞.

We denote byl_p(X)(resp. l_pⁿ(X)) the space of all sequences(x_i)inXwith the norm

k(xi)k_lp(X)=

∞

X

1

kxik^p

!¹_p

<∞



resp.

(x_i)_1≤i≤n

lⁿ_p(X) =

n

X

1

kx_ik^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

!¹_p

<∞



resp.k(x_n)k_ln ω

p (X)= sup

kξk_X∗=1 n

X

1

|hx_i, ξi|^p

!_p¹



whereX^∗ denotes the dual (topological) ofX and BX denotes the closed unit ball of X. We know (see [8]) thatlp(X) = l_p^ω(X)for some1 ≤ p < ∞iff dim (X)is finite. Ifp = ∞, we havel∞(X) =l_∞^ω (X). We have also if1< p≤ ∞, l^ω_p (X)≡B(l_p^∗, X)isometrically (where p^∗is the conjugate ofp, i.e., ¹_p +_p¹∗ = 1). 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). 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 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 multiplication 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 2.2 below.

Consequently,

(2.2) u≤T ⇐⇒ −T(−x)≤u(x)≤T(x), ∀x∈X.

LetT be sublinear from a Banach spaceXinto a Banach latticeY. Then we have,

• 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 intoY.

We say that a sublinear operatorT is positive if for allxinX,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 operator u∈ L(X, Y)is positive ifu(x)≥0forx≥0).

We will need the following obvious properties.

Proposition 2.1. LetXbe an arbitrary Banach space. LetY, Z be Banach lattices.

(i) ConsiderT in SL(X, Y)anduin L(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, Propo- sition 2.3].

Proposition 2.2. LetX be a Banach space and let Y be a complete Banach lattice. Let T ∈ 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}).

We have thus that∇T is not empty ifY is a complete Banach lattice. IfY is simply a Banach lattice then∇T is empty in general (see [10]).

As an immediate consequence of Proposition 2.2, we have:

• the operatorT is bounded if and only if for allu∈ ∇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 2.2. Let X be a Banach space and Y be a Banach lattice. A sublinear operator T : X −→ Y is stronglyp-summing (1 < p < ∞), if there is a positive constantC such that for anyn∈N;x₁, ..., x_n∈X andy^∗₁, ..., y_n^∗ ∈Y^∗we have

(2.3)

n

X

i=1

|hT (xi), y_i^∗i| ≤Ck(xi)k_ln

p(X) sup

y∈BY

k(y_i^∗(y))k_ln ω p∗ .

We denote byD_p(X, Y)the class of all stronglyp-summing sublinear operators fromXinto Y and byd_p(T)the smallest constantCsuch that the inequality (2.3) holds. Forp= 1, we have D₁(X, Y) =SB(X, Y).

Theorem 2.3 ([3]). Let X be a Banach space andY be a Banach lattice. An operator T ∈ SB(X, Y)is stronglyp-summing (1 < p < ∞), if and only if, there exists a positive constant C >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

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

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 2.3. LetX, Y be Banach lattices. LetT :X −→Y be a sublinear operator. We will say thatT is “positive p-summing” (0 ≤ p ≤ ∞) (we writeT ∈ π_p⁺(X, Y)), if there exists a positive constantCsuch 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 2.4. A sublinear operator between Banach latticesX,Y is positivep-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 haveπ_p⁺(u)≤2π_p⁺(T). We do not know if the converse is true.

3. COHENp−NUCLEARSUBLINEAROPERATORS

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 Cohenp-nuclear (1 < p < ∞), if there is a positive constantC such that for anyn ∈N,x₁, ..., x_n∈X andy^∗₁, ..., y_n^∗ ∈Y^∗,we have

(3.1)

n

X

i=1

hT(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∗.

We denote byN_p(X, Y)the class of all Cohenp-nuclear sublinear operators fromX intoY and byn_p(T)the smallest constantCsuch that the inequality (3.1) holds. For the definition of positive Cohenp-nuclear, we replaceY^∗ byY₊^∗ andn_p(T)byn⁺_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, 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 andY, Z be two Banach lattices. Consider T in SB(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 andn_p(u◦T)≤ kukn_p(T).

(ii) IfT is a Cohenp-nuclear sublinear operator, thenT◦Sis a Cohenp-nuclear sublinear operator andnp(T ◦S)≤ kSknp(T).

Proof. (i) Letn∈N;x₁, ..., x_n∈X andz₁^∗, ..., 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 −→lⁿ_p∗ 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

≤np(T) sup

x^∗∈B_X∗

k(x^∗(xi))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

ei.

This implies that

kwk ≤ kuk sup

y∈B_Y

(z_i^∗(z))_1≤i≤n

≤ kuk kvk.

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

n

X

i=1

hT ◦S(e_i), y^∗_ii

≤n_p(T) sup

x^∗∈B_X∗

n

X

i=1

|hS(e_i), 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 andnp(T ◦S)≤ kSknp(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 ofE. Let M 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.

Theorem 3.3. LetX be a Banach space andY be a Banach lattice. ConsiderT ∈ 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 on B_X^∗ andµ₂ onB_Y^∗∗, such that for all x∈X andy^∗ ∈Y^∗, we have

(3.3) |hT (x), y^∗i| ≤C Z

B_X∗

|x(x^∗)|^pdµ₁(x^∗) ¹_p Z

B_Y∗∗

|y^∗(y^∗∗)|^p∗dµ₂(y^∗∗) _p¹∗

.

Moreover, in this case

n_p(T) = inf{C > 0 : for allCverifying 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∈B_Y

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 (x_i), y_i^∗i| ≤n_p(T)k(x_i)k_ln ω

p (X) sup

y∈B_Y

k(y_i^∗(y))k_ln p∗.

To prove that (2) implies (3). We consider the setsP(B_X^∗)andP(B_Y^∗∗)of probability measures in C(B_X^∗)^∗ and C(B_Y^∗∗)^∗, respectively. These are convex sets which are compact when we endow C(B_X^∗)^∗ andC(B_Y^∗∗)^∗ with their weak∗topologies. We are going to apply Ky Fan’s Lemma withE =C(BX^∗)^∗×C(BY^∗∗)^∗andC =P(BX^∗)×P(BY^∗∗).

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

n

X

i=1

|hT(xi), y^∗_ii| −C 1 p

n

X

i=1

Z

B_X∗

|xi(x^∗)|^pdµ1(x^∗)

+ 1 p^∗

n

X

i=1

Z

BY∗∗

|y^∗_i(y^∗∗)|^p∗dµ₂(y^∗∗)

! ,

wherex₁, ..., x_n∈X andy₁^∗, ..., 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, gbe inM andα∈[0,1]such that

f((^x0i)^,(^y0∗i ))(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^0∗_i , y^∗∗i|^p∗dµ₂(y^∗∗)

# ,

and

g((^x00i)^,(^y00∗i ))(µ₁, µ₂) =

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^∗∗)

# .

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

BY∗∗

|hy^0∗_i , y^∗∗i|^p∗dµ₂(y^∗∗)

!#

=

k

X

i=1

D

T α^1px⁰_i

, α^p1∗y_i^0∗E

−C 1 p

k

X

i=1

Z

B_X∗

D

α^1px⁰_i, x^∗E

p

dµ₁(x^∗)

+1 p^∗

k

X

i=1

Z

B_Y∗∗

D

α^p1∗y^0∗_i , y^∗∗E

p^∗

dµ₂(y^∗∗)

!

=f

α^1px⁰_i

,

α

1 p∗

y_i^0∗

(µ1, µ2), and

f+g =

k

X

i=1

|hT(x⁰_i), y^0∗_i i| −C 1 p

k

X

i=1

Z

BX∗

|hx⁰_i, x^∗i|^pdµ₁(x^∗)

+ 1 p^∗

k

X

i=1

Z

B_Y∗∗

|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

BX∗

|hx⁰⁰_i, x^∗i|^pdµ₁(x^∗) + 1 p^∗

l

X

i=k+1

|hy_i^00∗, y^∗∗i|^p∗dµ₂(y^∗∗)

!

=

k

X

i=1

|hT(x⁰_i), y^0∗_i 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^∗_ii| −C 1 p

n

X

i=1

Z

B_X∗

|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), sinceB_X^∗ andB_Y^∗∗ are weak∗compact and norming sets, there exist for f ∈M two elements,x^∗₀ ∈B_X^∗ andy₀ ∈B_Y^∗∗ such that

sup

x^∗∈B_X∗

n

X

i=1

|hx_i, x^∗i|^p =

n

X

i=1

|hx_i, x^∗₀i|^p

and

sup

y∈B_Y

k(y_i^∗(y))k^p_ln^∗ p∗ =

n

X

i=1

|hy_i^∗, y0i|^p∗. 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∈BY

k(y_i^∗(y))k_ln p∗

and= 1, then f δ_x^∗

0, δ_y0

=

n

X

i=1

|hT(x_i), y_i^∗i| −C

p sup

x^∗∈B_X∗

n

X

i=1

|hx_i, x^∗i|^p

!

− C p^∗ sup

y∈B_Y

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 takingr = 0. By Ky Fan’s Lemma, there is(µ₁, µ₂)∈ Cwithf(µ₁, µ₂)≤ 0for all f ∈M. Then, iff is generated by the single elementsx∈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(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

B_Y∗∗

|hy^∗, y^∗∗i|^p∗dµ2(y^∗∗)

_p¹∗#p^∗





≤C Z

B_X∗

|hx, x^∗i|^pdµ₁(x^∗) ¹_p 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

B_X∗

|x_i(x^∗)|^pdµ₁(x^∗) ¹_p Z

B_Y∗∗

|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_i^∗i|

≤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∗

≤C sup

x^∗∈B_X∗

n

X

i=1

|x_i(x^∗)|^p

!¹_p sup

y∈BY

(y^∗_i(y))_1≤i≤n lⁿ_p∗.

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

4. RELATIONSHIPSBETWEEN π_p(X, Y), D_p(X, Y)ANDN_p(X, Y)

In this section we investigate the relationships between the various classes of sublinear oper- ators discussed in Section 2 and 4. We also give a relation betweenT and∇T concerning the notion of Cohenp-nuclear.

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

(1) N_p(X, Y)⊆ D_p(X, Y)andd_p(T)≤n_p(T).

(2) N_p(X, Y)⊆π_p(X, Y)andπ_p(T)≤n_p(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

BY∗∗

|y^∗(y^∗∗)|^p∗dµ₂(y^∗∗) _p¹∗

≤n_p(T)kxk Z

B_Y∗∗

|y^∗(y^∗∗)|^p∗dµ₂(y^∗∗) _p¹∗

so

|hT(x), y^∗i| ≤np(T)kxk Z

B_Y∗∗

|y^∗(y^∗∗)|^p∗dµ2(y^∗∗) _p¹∗

. Then, by Theorem 2.3,T is stronglyp-summing andd_p(T)≤n_p(T).

(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

B_X∗

|x^∗(x)|^pdµ₁(x^∗) ¹_p Z

B_Y∗∗

|y^∗(y^∗∗)|^p∗dµ₂(y^∗∗) _p¹∗

≤n_p(T) Z

B_X∗

|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 2.4,T isp-summing andπ_p(T)≤n_p(T).The proof is complete.

Theorem 4.2. LetXbe Banach space andY, Z be two Banach lattices. Let1< p <∞.

(1) LetT ∈ SB(X, Y)andL∈ SB(Y, Z). Assume thatLis increasing. IfLis a strongly p-summing sublinear operator, andT is ap-summing sublinear operator, thenL◦T is a Cohenp-nuclear sublinear operator andn_p(L◦T)≤d_p(L)π_p(T).

(2) ConsideruinB(Z, X)ap-summing operator andT inSB(X, Y)a stronglyp-summing one. Then,T ◦uis a Cohenp-nuclear sublinear operator andn_p(T◦u)≤d_p(T)π_p(u).

(3) ConsiderT inSB(X, Y)ap-summing operator andvinB(Y, Z)a stronglyp-summing one. Assume thatvis positive. Then,v◦T is a Cohenp-nuclear sublinear operator and n_p(v◦T)≤d_p(v)π_p(T).

Proof. (1) The operatorL◦T is sublinear by Proposition 2.1(iii). Letx∈ Xandz^∗ ∈Z^∗. By Theorem 2.3, we have

|hL◦T (x), z^∗i|=|hL(T (x)), z^∗i|

≤d_p(L)kT (x)k Z

BZ∗∗

|z^∗(z^∗∗)|^p∗dλ(z^∗∗) _p¹∗

and by Theorem 2.4

≤dp(L)πp(T) Z

B_X∗

|x(x^∗)|^pdµ(x^∗) ¹_p Z

B_Z∗∗

|z^∗(z^∗∗)|^p∗dλ(z^∗∗) _p¹∗

, so

|hL◦T(x), z^∗i| ≤d_p(L)π_p(T) Z

BX∗

|x(x^∗)|^pdµ(x^∗) ¹_p Z

BZ∗∗

|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 Proposition 2.1(ii), Theorem 2.3 and Theorem 2.4.

(3) The operatorv◦T is sublinear by Proposition 2.1(i). Lettingx∈X andz^∗ ∈Z^∗, we have

|hv(T (x)), z^∗i|=|hT(x), v^∗(z^∗)i|

≤ kT (x)k kv^∗(z^∗)k

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

B_X^∗

|x^∗(x)|^pdµ₁(x^∗) ¹_p Z

B_Z^∗∗

|z^∗∗(z^∗)|^p∗dµ₂(z^∗∗) _p¹∗

.

This implies thatv◦T ∈ N_p(X, Z)andnp(v◦T)≤dp(v)πp(T).

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

Example 4.1. Let1 ≤ p < ∞andn, N ∈ N. Letube a linear operator fromlⁿ₂ intol^N_p such thatS(x) = |u(x)|.Letvbe a linear operator fromL_q(µ)(1≤q <∞) intolⁿ₂.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],v is2-summing. Then by Theorem 4.2 part (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. Let T be a bounded sublinear operator from X into Y. Suppose that T is positive Cohen p-nuclear (1< p <∞). Then for allS ∈ SB(X, Y)such thatS ≤T,S is positive Cohenp-nuclear.

Proof. Lettingx_i ∈X₁andy_i^∗ ∈Y₊^∗,by (1.2), we have hS(x_i), y_i^∗i ≤ hT(x_i), y^∗_ii and consequently, by (2.2),

− hS(x_i), y^∗_ii ≤ hT(−x_i), y_i^∗i for all1≤i≤n. This implies that

n

X

i=1

|hS(x_i), y_i^∗i| ≤

n

X

i=1

sup{hT(x), y_i^∗i,hT(−x), y_i^∗i}

≤

n

X

i=1

sup{|hT(x), y_i^∗i|,|hT(−x), y_i^∗i|}

≤

n

X

i=1

|hT(x), y^∗_ii|+

n

X

i=1

|hT(−x), y_i^∗i|

and hence

n

X

i=1

|hS(x_i), y_i^∗i| ≤2n⁺_p(T) sup

x^∗∈B_X∗

k(x^∗(x_i))k_ln p sup

y∈B_Y⁺

k(y_i^∗(y))k_ln p∗.

Thus the operatorSis positive Cohenp-nuclear andn⁺_p (S)≤2n⁺_p(T).

Remark 1. IfS, T are any sublinear operators, we have no answer.

Corollary 4.4. IfT is positive Cohenp-nuclear (1< p <∞), then for allu∈ ∇T,uis positive Cohenp-nuclear and consequentlyu^∗is positivep^∗-summing.

Proof. Let T be a positive Cohen p-nuclear sublinear operator. Then for all u ∈ ∇T, u is positive Cohenp-nuclear (replacingSbyuin Proposition 4.3). Ifuis positive Cohenp-nuclear (by Theorem 4.1, uis positive stronglyp-summing), thenu^∗ 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. LetXbe Banach space andY be a complete Banach lattice. LetT :X →Y be a sublinear operator. Suppose that there is a constantC >0, a setI, an ultrafilterU onIand {u_i}i∈I ⊂ ∇T such that for allxinX andy^∗ inY^∗,

|hu_i(x), y^∗i| −→

U |hT(x), y^∗i|

andn_p(u_i)≤Cuniformly. Then,T ∈ N_p(X, Y)andn_p(T)≤C.

Proof. Since u_i is Cohen p-nuclear, by Theorem 3.3 there is a Radon probability measure (µ_i, ν_i)onK =B_X^∗ ×B_Y^∗∗ such that for allx∈Xandy^∗ inY^∗, we have

|hui(x), y^∗i| ≤np(ui) Z

B_X^∗

|x(x^∗)|^pdµi

¹_pZ

B_Y^∗∗

|y^∗(y^∗∗)|^p∗dνi

_p¹∗

.

As we have for allxinX and y^∗ ∈Y^∗,

|hui(x), y^∗i| −→

U |hT(x), y^∗i|

thus we obtain that for allxinX and y^∗ ∈Y^∗,

|hT(x), y^∗i| ≤lim

U n_p(u_i) Z

B_X∗

|x(x^∗)|^pdµ_i ¹_pZ

B_Y∗∗

|y^∗(y^∗∗)|^p∗dν_{i p}¹∗

.

The setK = B_X^∗ ×B_Y^∗∗ is weak^∗ compact, hence (µ_i, ν_i) converge weak^∗ to a probability (µ, ν)onK =B_X^∗ ×B_Y^∗∗ and consequently, for allxinXand y^∗ ∈Y^∗

|hT(x), y^∗i| ≤C Z

BX∗

|x(x^∗)|^pdµ ¹_pZ

BY∗∗

|y^∗(y^∗∗)|^p∗dν _p¹∗

.

This implies thatn_p(T)≤C.

REFERENCES

[1] D. ACHOUR ANDL. 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 L_q1(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, Positivep-summing operators fromL_p-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. JARCHOWANDA. TONGE, Absolutely Summing Operators, Cambridge Univer- sity 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.

[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.