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Little Grothendieck's theorem for sublinear operators
J. Math. Anal. Appl. 296 (2004) 541–552 www.elsevier.com/locate/jmaa
Little Grothendieck’s theorem for sublinear operators D. Achour ∗ , L. Mezrag Department of Mathematics, M’sila University, P.O. Pox 166, Ichbilia, 28105, M’sila, Algeria Received 13 December 2003 Available online 2 July 2004 Submitted by J. Diestel
Abstract Let SB(X, Y ) be the set of the bounded sublinear operators from a Banach space X into a Banach lattice Y . Consider π2 (X, Y ) the set of 2-summing sublinear operators. We study in this paper a variation of Grothendieck’s theorem in the sublinear operators case. We prove under some conditions that every operator in SB(C(K), H ) is in π2 (C(K), H ) for any compact K and any Hilbert H . In the noncommutative case the problem is still open. 2004 Elsevier Inc. All rights reserved. Keywords: Banach lattice; Sublinear operator; p-summing operator; p-regular operator
0. Introduction The notion of the p-summing operators was introduced by Pietsch in 1967 (see [17]) as a generalization of Grothendieck’s “application semi intégrale à droite” which are actually called 1-summing operator. In this paper, we try to generalize a result which is called “little Grothendieck’s theorem” to sublinear operators, operators that are subadditive and positively homogeneous. In the linear case this result is: every bounded linear operator from C(K) into H is 2-summing (i.e., B(C(K), H ) = π2 (C(K), H )), where C(K) denotes the continuous functions on the compact Hausdorff topological space K and H is a Hilbert space. * Corresponding author.
E-mail addresses: [email protected] (D. Achour), [email protected] (L. Mezrag). 0022-247X/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2004.04.018
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The fundamental theorem of Grothendieck states that: every linear operator u : L1 (µ) → H , where H is a Hilbert space, is 1-summing. In the other words, there is a constant C such that π1 (u) Cu for all u as above. The smallest such constant is called the universal Grothendieck constant and denoted by KG . Its dual form is: there is a constant C such that for any linear operator from L∞ into an L1 -space, we have π2 (u) C u. Originally this theorem was proved by Grothendieck in [4] who called it “the fundamental theorem of the metric theory of tensor products” and presented in the language of 1-summing operator by Lindenstrauss and Pełczy´nski in [9]. In the noncommutative case, the problem is still open [18, Problem 10.2]. Pisier asked: is every completely bounded operator u : B(H ) → OH necessarily (2, oh)-summing (i.e., CB(B(l2 ), OH) = π20 (B(l2 ), OH))? The space OH is the operator Hilbert space. Le Merdy has proved in [7, Theorem 4.2] that CB B(l2 ) , OH = π20 B(l2 ) , OH and finally in [12] we have shown that πl2 B(l2 ), OH = π20 B(l2 ), OH = B B(l2 ), OH . All these notions πp0 -summing, (2, oh)-summing and πlp -summing are generalizations of the classic notion of p- summing operators to the noncommutative case. In Section 1, we briefly recall Banach lattices and some terminologies. We finish by giving some rudimentary properties concerning order bounded operators which we need later. In Section 2, we introduce the notion of sublinear operators and we adapt some definitions relative to the linear case to this notion. We continue by proving a result related to the little Grothendieck’s theorem. In Section 3, we apply our ideas to generalize the Grothendieck–Maurey theorem as follows: let Y and Z be Banach lattices, with Z 2-concave, T : C(K) → Y be a 2-regular sublinear operator and w : Y → Z be a 2-concave positive linear operator. Then wT is 2-summing.
1. Notation and preliminaries We recall that a Banach lattice (respectively a complete Banach lattice) X is an ordered vector space equipped with a lattice (respectively a complete lattice) structure and a Banach space norm satisfying the following compatibility condition: ∀x, y ∈ X,
|x| |y|
⇒
x y,
where |x| = sup{x, −x}. Note that this implies 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+ . For more details see [8,13,16,20]. The dual X∗ of a Banach lattice X is a complete Banach lattice. The space C(K) is a Banach lattice. The Lp (1 p ∞) are complete Banach lattices. Any reflexive Banach lattice is a complete Banach lattice.
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Let n be an integer. For a Banach lattice X and 1 p ∞, we denote by X(lpn ) the space of all sequences x = (x1 , . . . , xn ) of elements of X for which n 1/p p xX(lpn ) = |xi | if 1 p < ∞ 1
and
n ) = sup |xi | xX(l∞ if p = ∞. 1in
The space
X(lpn )
xy
is a Banach lattice equipped with the natural order
⇔
xi yi ,
∀i.
Let now X be a Banach space and 1 p ∞. We denote by lp (X) (respectively lpn (X)) the space of all sequences (xi ) in X with the norm ∞ 1/p p (xi ) = xi lp (X)
1
respectively (xi )1in n
lp (X)
=
n
1/p xi
p
,
1
and by lpω (X) (respectively lpn ω (X)) the space of all sequences (xi ) in X with the norm ∞ 1/p p xi , ξ (xn ) ω = sup lp (X)
ξ X∗ =1
respectively (xn ) n ω
1
lp (X)
=
sup
ξ X∗ =1
n xi , ξ p
1/p .
1
We know (see [3]) that lp (X) = lpω (X) for some 1 p < ∞ iff dim(X) is finite. If ω (X). We have also if 1 < p ∞, l ω (X) ≡ B(l ∗ , X) isop = ∞, we have l∞ (X) = l∞ p p ω metrically and l1 (X) ≡ B(cO , X) isometrically (where p∗ is the conjugate of p, i.e., ∗ 1/p + 1/p∗ = 1). In other words, let v : lp → X be a linear operator such that v(ei ) = xi (namely v = ∞ e ⊗ x , e denotes the unit vector basis of lp ). Then j j 1 j (1.1) v = (xn ) ω . lp (X)
Let now E be a Banach space and X be a complete Banach lattice. An operator u ∈ B(E, X) is called “order bounded ” (see [5,15]) if u(BX ) is an order bounded subset of X. In this case, we put l(u) = sup u(x). x∈BX
We can show that (see [8] or [19]) l is a norm on l(E, X), the space of all order bounded maps from E to X. If u ∈ l(E, X) and w : X → Y (Y a complete Banach lattice) is a positive operator (i.e., w(x) 0 for all x in X+ ), then wu is order bounded. The following simple fact will be needed later.
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Proposition 1.1. Let X be a Banach lattice. Let v : lpn ∗ → X be such that v(ei ) = xi (1 p ∞). We have l(v) = (xi )X(l n ) . (1.2) p
Proof. We have by Krivine [6] n 1/p 1/p∗ n n
∗ p p |xi | = sup λi xi : |λi | =1 i=1 1 1 n
1/p∗ n p∗ λi v(ei ): |λi | =1 . = sup 1
1
Thus, we obtain n 1/p n n 1/p∗
p p∗ |xi | λi ei : |λi | =1 = sup v i=1 1 1 = sup v(x): x = 1 . This ends the proof. 2 In the particular case X = C(K), where C(K) denotes the continuous functions on the compact Hausdorff topological space K, we have Proposition 1.2. In the same situation as in Proposition 1.1, we have l(v) = v. Proof. We have, by (1.1), n
1/p
n 1/p p p xi , ξ xi (t) v = sup , ξ ∈ BX∗ = sup , t ∈K i=1
n 1/p = |xi |p i=1
i=1
.
C(K)
So, we conclude the proof by Proposition 1.1. 2
2. An approach of Grothendieck’s theorem for sublinear 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).
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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 L(X, Y ) = {linear mappings u : X → Y } and by SL(X, Y ) = {sublinear mappings T : X → Y }. We equip SL(X, Y ) with the natural order induced by Y , T1 T2 and
⇔
T1 (x) T2 (x),
∀x ∈ X,
(2.1)
∇T = u ∈ L(X, Y ): u T i.e., ∀x ∈ X, u(x) T (x) .
The set ∇T is not empty by Proposition 2.3 below. As a consequence u ∈ ∇T
⇔
−T (−x) u(x) T (x),
∀x ∈ X,
(2.2)
and λT (x) T (λx).
(2.3)
Now, we will give the following well-known fact and we leave the details to the reader. Let T be a sublinear operator from a Banach space X into Banach lattice Y . The operator T is continuous iff there is a constant C > 0 such that for all x in X, T (x) Cx. In this case we say also that T is bounded and we put T = sup T (x): xBX = 1 . We denote by SB(X, Y ) = {bounded sublinear operators T : X → Y } and by B(X, Y ) = {bounded linear operators u : X → Y }. We will need the following remark. Remark 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 (i.e., u(x) 0 if x ∈ X+ ). 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). The following proposition, will be useful in the sequel for the proof of Theorem 2.9 below.
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Proposition 2.3. Let X be a Banach space and 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 }). Proof. Let x be in X. Consider vx : R.x → Y defined by vx (λx) = λT (x) for all λ in R. We have by (2.3), vx T on R.x. By Hahn–Banach theorem applied to sublinear operators, see, for example, [20, p. 244], there is a linear extension of vx noted ux such that ux (λx) = vx (λx) for all λ in R and ux (y) T (y) for all y in X. This concludes the proof because ux (x) = T (x). 2 As an immediate consequence of Proposition 2.3, we have Corollary 2.4. In the same conditions of the above proposition, we have for all x in X, (i) T (x) supu∈∇T u(x) T (x) + T (−x). (ii) T supu∈∇T u 2T . Corollary 2.5. Let T : X → Y be a sublinear operator between a Banach space X and a complete Banach lattice Y . Then, the following properties are equivalent: (i) T is bounded. (ii) ∀u ∈ ∇T , u is bounded. We continue this notion by recalling the definitions of the p-concavity and the psumming operators. Definition 2.6. Let X, Y be Banach lattices and let 1 p ∞. A sublinear operator T : X → Y is called p-concave if there is a constant C such that, for all n in N the operators X lpn → lpn (Y ), (x1 , . . . , xn ) → T (x1 ), . . . , T (xn ) are uniformly bounded by C. The smallest constant C for which this holds is denoted by Cp (T ). A Banach lattice X is p-concave if idX is p-concave. Remark 2.7. Any sublinear operator p-concave is bounded and T Cp (T ). Every Banach lattice is ∞-concave. If p r, then p-concave Banach lattices are rconcave. For example, Lp for 1 p < ∞ is p-concave, and Cp (Lp ) = 1. In the case of Banach lattices the notions of cotype 2 and 2-concave are equivalent (see [10, Part II, Theorem 1.f.16]). Now, we give the notion absolutely p-summing operators generalized to sublinear operators. See [1,2] for more details.
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ˆ Y their injective tensor product, Given two Banach spaces X, Y . We denote by X ⊗ i.e., the completion of X ⊗ Y under the cross norm n n
xi ⊗ yi = sup xi (ξ ) ⊗ yi (η): ξ X∗ 1, ξ Y ∗ 1 . i=1
i=1
Let T : X → Y be a sublinear operator between a Banach space X and a Banach lattice Y . We will say that T is “absolutely p-summing” or “p-summing,” 0 < p < ∞ (we write T ∈ πp (X, Y )), if there exists a positive constant C such that for every n in N the mappings In ⊗ T : lpn ⊗ X → lpn (Y ), n
ei ⊗ xi → T (xi ) 1in
1
are uniformly bounded by C (i.e., In ⊗ T lpn ⊗ X→lpn (Y ) C). We put πp (T ) = sup In ⊗ T lpn ⊗ X→lpn (Y ) .
(2.4)
n
Remark 2.8. If Y = Lp , we have T ∈ πp (X, Lp ) iff there is a positive constant Csuch that for all n in N and (x1 , . . . , xn ) in X, we have T (xi ) C (xi )l n ω (X) . Lp (l n ) p
p
Theorem 2.9. 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, ui (x) →T (x) U
and πp (ui ) C uniformly. Then, T ∈ πp (X, Y )
and πp (T ) C.
Conversely if πp (T ) C, then πp (u) 2C for all u in ∇T . Proof. Since ui is p-summing, by Pietsch’s theorem extended to sublinear operators (see [2]) then there is a probability λi on BX such that for all x in X, we have 1/p p ui (x) C x, ξ dλi (ξ ) . BX
As we have for all x in X, ui (x) →T (x), U
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thus we obtain for all x in X, 1/p x, ξ p dλi (ξ ) T (x) C lim . U
BX
The unit ball BX is weak∗ compact, hence λi converge weak∗ to a probability λ on BX and consequently 1/p p T (x) C x, ξ dλ(ξ ) BX
for all x in X. This implies πp (T ) C. Conversely, by Corollary 1.4(i) we have for all x in X and all u in ∇T , u(x) T (x) + T (−x). This shows πp (u) 2πp (T ) and concludes the proof. 2 The next corollary is a partial generalization to Maurey’s theorem. Corollary 2.10. Let Y be a 2-concave complete Banach lattice. Let T : C(K) → Y be a sublinear such that there is an index set I , an ultrafilter U on I and {ui }i∈I ⊂ ∇T such that for all x in X, ui (x) →T (x). U
Then, T is 2-summing. Proof. By Maurey’s theorem [11] and Corollary 2.4(ii), there is a constant C such that for any linear operator u from C(K) into Y , we have π2 (u) Cu. We conclude by Theorem 2.9. 2 Remark 2.11. If we replace Y by any Hilbert space, we have the little Grothendieck’s theorem for sublinear operators and this under the supposition that ui (x) →T (x) for all x in X. U
3. Little Grothendieck’s theorem and p-regular operators In this section, we will extend a result of Maurey, that generalizes the ‘little Grothendieck theorem,’ to the sublinear case. As shown by Corollary 3.4 below, there is equivalence between 2-regular and 2-summing sublinear operators, the same as in the linear case. The problem is just this: in the linear case, every bounded linear operator from a Banach space into a Banach lattice is 2-regular; this is simply not so in the sublinear case as is indicated in an example below even if the sublinear operator is positive.
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Let T : X → Y be a bounded sublinear operator between Banach lattices X, Y . We say that T is “p-regular,” 1 p ∞, and we write T ∈ ρp (X, Y ) if there is a positive constant C such that for all finite sequence (xi )1in ⊂ X, we have T (xi ) n C (xi ) n . Y (lp )
X(lp )
The best possible constant will be denoted by ρp (T ). Recall, it was proved by Krivine in [6] see also [10, Part.II.1.f.14 and 1.d.9] that every bounded linear operator is 2-regular and every positive operator is p-regular for 1 p ∞. If p = 2, ρp (w) = KG w and ρp (w) = w if p = 2. The following example (we can see [14]) is pertinent in our work, because in the case of sublinear operators, a bounded sublinear operator is not in general 2-regular. We take as measure space (Ω, µ) the torus T = R/2πZ, equipped with the invariant measure dθ . Let X be the Hilbert space H = L2 (Ω, µ) = L2 (T) and Y be the space L1 (Ω, µ). For all r such that 0 < r π and for all f ∈ L2 (T), we define a function 2π -periodic Sr f 0 by
x+r f (y)2 dy.
1 Sr f (x) = 2r
∀x ∈ R,
x−r
√ We put Tr f = Sr f . For all x, the expression (Tr f )(x) is the L2 -norm of the function 1(x−r,x+r)f , hence the operator Tr is sublinear and the operator T defined by Tf = sup{Tr f : 0 < r < π}, is also sublinear from L2 (Ω, µ) into L1 (Ω, µ). This operator is positive in the sense of (2.1). Tf is the square root of the maximal function Mf 2 (the Hardy–Littlewood maximal operator) of the function f 2 ∈ L1 , we know that Mf 2 is in weak-L1 , therefore Tf is in weak-L2 . Consider n in N. We can partition T in n-intervals with the same length and we take 2 x1 , . . . , xn in H = L2 (Ω, µ), the characteristic functions. √ We have xi = 2π/n for all i = 1, . . . , n, but every function T (xi ) worth at least C/ 1 + (i − j ) on the support of xi for all j = 1, . . . , n with C = (4π)−1 , hence it results that 1/2
n 1 1 T (xi )(ω)2 dµ(ω) C 1 + + · · · + C log n 2 n Ω
and n i=1
i=1
n 2 xi = |xi | 2
i=1
= 2π.
L2 (Ω,µ)
In conclusion, we have (4π)−2 log n 2π . This is impossible when n is large enough. The question which we pose now is that: is every bounded sublinear operator from C(K) into a Hilbert space H is 2-regular? The following proposition will be useful in the sequel. Proposition 3.1. Let E, F, X, Y be Banach lattices and 1 p ∞. Consider a p-regular T ∈ SB(X, Y ), a bounded linear operator u : E → X and a positive bounded linear operator v : Y → F . Then,
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(i) The operator vT is p-regular and ρp (vT ) vρp (T ). (ii) If u is positive we have ρp (vT u) vρp (T )u. (iii) For p = 2, vT u is 2-regular and ρ2 (vT u) vρ2 (T )uKG . Proof. (i) The operator vT is sublinear by Remark 2.2(i). Using [10, Part II. 1.d.9] or [4], we have n n 1/p 1/p p p vT (xi ) T (xi ) v i=1 i=1 n 1/p p |xi | vρp (T ) . i=1
(ii) The operator vT u is sublinear by Remark 2.2(ii). As (i) we have n n 1/p 1/p p p vT u(xi ) T u(xi ) v i=1 i=1 n 1/p p u(xi ) ρp (T ) i=1 n 1/p p vρp (T )u |xi | . i=1
(iii) For p = 2, the same as (ii) but we take u an arbitrary operator because every linear operator is 2-regular. 2 Proposition 3.2. Let T be a sublinear operator between a Banach lattice X and a complete Banach lattice Y , C be a positive constant and 1 p ∞. Then, the operator T is in ρp (X, Y ) and ρp (X, Y ) C iff for any linear operator v : lpn ∗ → X, we have n 1/p p T v(ei ) C.l(v). i=1
Proof. Immediately by (1.2). 2 In particular, we have obviously Corollary 3.3. If X = C(K) we have T ∈ ρp C(K), Y
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iff there is a positive constant C such that for all n in N and (x1 , . . . , xn ) in C(K), we have T (xi ) n C (xi ) n ω . Y (l ) l (C(K)) p
p
Proof. By Proposition 1.2 we have l(v) = v. We conclude by (1.1) ((xi )lpn ω (C(K)) = v). 2 Corollary 3.4. Let T : C(K) → H be a 2-regular sublinear operator from C(K) into a Hilbert space H . Then T is 2-summing and conversely. Proposition 3.5. Let X, Y, Z be Banach lattices; T : X → Y be a p-regular sublinear operator and w : Y → Z be a positive p-concave linear operator. Then, the composition wT is p-concave. Proof. The operator wT is sublinear, because by Remark 2.2(ii), the composition of a positive linear operator with a sublinear operator is sublinear. Let (xi ) be a finite sequence in X. Then wT (xi ) n Cp (w) T (xi ) n Cp (w)ρp (T )(xi ) n lp (Z)
Y (lp )
X(lp )
which proves the proposition. 2 Now we are ready to present the Grothendieck–Maurey theorem in the sublinear case. Theorem 3.6. Let Y be a Banach lattice and Z be a 2-concave Banach lattice. Let T : C(K) → Y be a 2-regular sublinear operator and w : Y → Z be a positive 2-concave linear operator. Then wT is 2-summing. Proof. Let (xi ) be a finite sequence in C(K). Since w is 2-concave, then wT is also 2-concave by Proposition 3.5 and C2 (wT ) C2 (w)ρ2 (T ). Thus we have 1/2 1/2 wT (xi )2 C2 (w)ρ2 (T ) |xi |2 1/2 xi (t)2 C2 (w)ρ2 (T ) sup t ∈C(K)
C2 (w)ρ2 (T )
sup
λC(K)∗ =1
n λ, xi 2
1/2 .
1
This yields that wT is 2-summing and π2 (ωT ) · C2 (ω)ρ2 (T ).
2
Remark 3.7. Is the assumption of 2-regularity in Theorem 3.5 necessary?
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Acknowledgment The authors are very grateful to the referee for several valuable suggestions and comments which improved the paper.
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Little Grothendieck’s theorem for sublinear operators D. Achour ∗ , L. Mezrag Department of Mathematics, M’sila University, P.O. Pox 166, Ichbilia, 28105, M’sila, Algeria Received 13 December 2003 Available online 2 July 2004 Submitted by J. Diestel
Abstract Let SB(X, Y ) be the set of the bounded sublinear operators from a Banach space X into a Banach lattice Y . Consider π2 (X, Y ) the set of 2-summing sublinear operators. We study in this paper a variation of Grothendieck’s theorem in the sublinear operators case. We prove under some conditions that every operator in SB(C(K), H ) is in π2 (C(K), H ) for any compact K and any Hilbert H . In the noncommutative case the problem is still open. 2004 Elsevier Inc. All rights reserved. Keywords: Banach lattice; Sublinear operator; p-summing operator; p-regular operator
0. Introduction The notion of the p-summing operators was introduced by Pietsch in 1967 (see [17]) as a generalization of Grothendieck’s “application semi intégrale à droite” which are actually called 1-summing operator. In this paper, we try to generalize a result which is called “little Grothendieck’s theorem” to sublinear operators, operators that are subadditive and positively homogeneous. In the linear case this result is: every bounded linear operator from C(K) into H is 2-summing (i.e., B(C(K), H ) = π2 (C(K), H )), where C(K) denotes the continuous functions on the compact Hausdorff topological space K and H is a Hilbert space. * Corresponding author.
E-mail addresses: [email protected] (D. Achour), [email protected] (L. Mezrag). 0022-247X/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2004.04.018
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The fundamental theorem of Grothendieck states that: every linear operator u : L1 (µ) → H , where H is a Hilbert space, is 1-summing. In the other words, there is a constant C such that π1 (u) Cu for all u as above. The smallest such constant is called the universal Grothendieck constant and denoted by KG . Its dual form is: there is a constant C such that for any linear operator from L∞ into an L1 -space, we have π2 (u) C u. Originally this theorem was proved by Grothendieck in [4] who called it “the fundamental theorem of the metric theory of tensor products” and presented in the language of 1-summing operator by Lindenstrauss and Pełczy´nski in [9]. In the noncommutative case, the problem is still open [18, Problem 10.2]. Pisier asked: is every completely bounded operator u : B(H ) → OH necessarily (2, oh)-summing (i.e., CB(B(l2 ), OH) = π20 (B(l2 ), OH))? The space OH is the operator Hilbert space. Le Merdy has proved in [7, Theorem 4.2] that CB B(l2 ) , OH = π20 B(l2 ) , OH and finally in [12] we have shown that πl2 B(l2 ), OH = π20 B(l2 ), OH = B B(l2 ), OH . All these notions πp0 -summing, (2, oh)-summing and πlp -summing are generalizations of the classic notion of p- summing operators to the noncommutative case. In Section 1, we briefly recall Banach lattices and some terminologies. We finish by giving some rudimentary properties concerning order bounded operators which we need later. In Section 2, we introduce the notion of sublinear operators and we adapt some definitions relative to the linear case to this notion. We continue by proving a result related to the little Grothendieck’s theorem. In Section 3, we apply our ideas to generalize the Grothendieck–Maurey theorem as follows: let Y and Z be Banach lattices, with Z 2-concave, T : C(K) → Y be a 2-regular sublinear operator and w : Y → Z be a 2-concave positive linear operator. Then wT is 2-summing.
1. Notation and preliminaries We recall that a Banach lattice (respectively a complete Banach lattice) X is an ordered vector space equipped with a lattice (respectively a complete lattice) structure and a Banach space norm satisfying the following compatibility condition: ∀x, y ∈ X,
|x| |y|
⇒
x y,
where |x| = sup{x, −x}. Note that this implies 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+ . For more details see [8,13,16,20]. The dual X∗ of a Banach lattice X is a complete Banach lattice. The space C(K) is a Banach lattice. The Lp (1 p ∞) are complete Banach lattices. Any reflexive Banach lattice is a complete Banach lattice.
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Let n be an integer. For a Banach lattice X and 1 p ∞, we denote by X(lpn ) the space of all sequences x = (x1 , . . . , xn ) of elements of X for which n 1/p p xX(lpn ) = |xi | if 1 p < ∞ 1
and
n ) = sup |xi | xX(l∞ if p = ∞. 1in
The space
X(lpn )
xy
is a Banach lattice equipped with the natural order
⇔
xi yi ,
∀i.
Let now X be a Banach space and 1 p ∞. We denote by lp (X) (respectively lpn (X)) the space of all sequences (xi ) in X with the norm ∞ 1/p p (xi ) = xi lp (X)
1
respectively (xi )1in n
lp (X)
=
n
1/p xi
p
,
1
and by lpω (X) (respectively lpn ω (X)) the space of all sequences (xi ) in X with the norm ∞ 1/p p xi , ξ (xn ) ω = sup lp (X)
ξ X∗ =1
respectively (xn ) n ω
1
lp (X)
=
sup
ξ X∗ =1
n xi , ξ p
1/p .
1
We know (see [3]) that lp (X) = lpω (X) for some 1 p < ∞ iff dim(X) is finite. If ω (X). We have also if 1 < p ∞, l ω (X) ≡ B(l ∗ , X) isop = ∞, we have l∞ (X) = l∞ p p ω metrically and l1 (X) ≡ B(cO , X) isometrically (where p∗ is the conjugate of p, i.e., ∗ 1/p + 1/p∗ = 1). In other words, let v : lp → X be a linear operator such that v(ei ) = xi (namely v = ∞ e ⊗ x , e denotes the unit vector basis of lp ). Then j j 1 j (1.1) v = (xn ) ω . lp (X)
Let now E be a Banach space and X be a complete Banach lattice. An operator u ∈ B(E, X) is called “order bounded ” (see [5,15]) if u(BX ) is an order bounded subset of X. In this case, we put l(u) = sup u(x). x∈BX
We can show that (see [8] or [19]) l is a norm on l(E, X), the space of all order bounded maps from E to X. If u ∈ l(E, X) and w : X → Y (Y a complete Banach lattice) is a positive operator (i.e., w(x) 0 for all x in X+ ), then wu is order bounded. The following simple fact will be needed later.
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Proposition 1.1. Let X be a Banach lattice. Let v : lpn ∗ → X be such that v(ei ) = xi (1 p ∞). We have l(v) = (xi )X(l n ) . (1.2) p
Proof. We have by Krivine [6] n 1/p 1/p∗ n n
∗ p p |xi | = sup λi xi : |λi | =1 i=1 1 1 n
1/p∗ n p∗ λi v(ei ): |λi | =1 . = sup 1
1
Thus, we obtain n 1/p n n 1/p∗
p p∗ |xi | λi ei : |λi | =1 = sup v i=1 1 1 = sup v(x): x = 1 . This ends the proof. 2 In the particular case X = C(K), where C(K) denotes the continuous functions on the compact Hausdorff topological space K, we have Proposition 1.2. In the same situation as in Proposition 1.1, we have l(v) = v. Proof. We have, by (1.1), n
1/p
n 1/p p p xi , ξ xi (t) v = sup , ξ ∈ BX∗ = sup , t ∈K i=1
n 1/p = |xi |p i=1
i=1
.
C(K)
So, we conclude the proof by Proposition 1.1. 2
2. An approach of Grothendieck’s theorem for sublinear 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).
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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 L(X, Y ) = {linear mappings u : X → Y } and by SL(X, Y ) = {sublinear mappings T : X → Y }. We equip SL(X, Y ) with the natural order induced by Y , T1 T2 and
⇔
T1 (x) T2 (x),
∀x ∈ X,
(2.1)
∇T = u ∈ L(X, Y ): u T i.e., ∀x ∈ X, u(x) T (x) .
The set ∇T is not empty by Proposition 2.3 below. As a consequence u ∈ ∇T
⇔
−T (−x) u(x) T (x),
∀x ∈ X,
(2.2)
and λT (x) T (λx).
(2.3)
Now, we will give the following well-known fact and we leave the details to the reader. Let T be a sublinear operator from a Banach space X into Banach lattice Y . The operator T is continuous iff there is a constant C > 0 such that for all x in X, T (x) Cx. In this case we say also that T is bounded and we put T = sup T (x): xBX = 1 . We denote by SB(X, Y ) = {bounded sublinear operators T : X → Y } and by B(X, Y ) = {bounded linear operators u : X → Y }. We will need the following remark. Remark 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 (i.e., u(x) 0 if x ∈ X+ ). 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). The following proposition, will be useful in the sequel for the proof of Theorem 2.9 below.
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Proposition 2.3. Let X be a Banach space and 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 }). Proof. Let x be in X. Consider vx : R.x → Y defined by vx (λx) = λT (x) for all λ in R. We have by (2.3), vx T on R.x. By Hahn–Banach theorem applied to sublinear operators, see, for example, [20, p. 244], there is a linear extension of vx noted ux such that ux (λx) = vx (λx) for all λ in R and ux (y) T (y) for all y in X. This concludes the proof because ux (x) = T (x). 2 As an immediate consequence of Proposition 2.3, we have Corollary 2.4. In the same conditions of the above proposition, we have for all x in X, (i) T (x) supu∈∇T u(x) T (x) + T (−x). (ii) T supu∈∇T u 2T . Corollary 2.5. Let T : X → Y be a sublinear operator between a Banach space X and a complete Banach lattice Y . Then, the following properties are equivalent: (i) T is bounded. (ii) ∀u ∈ ∇T , u is bounded. We continue this notion by recalling the definitions of the p-concavity and the psumming operators. Definition 2.6. Let X, Y be Banach lattices and let 1 p ∞. A sublinear operator T : X → Y is called p-concave if there is a constant C such that, for all n in N the operators X lpn → lpn (Y ), (x1 , . . . , xn ) → T (x1 ), . . . , T (xn ) are uniformly bounded by C. The smallest constant C for which this holds is denoted by Cp (T ). A Banach lattice X is p-concave if idX is p-concave. Remark 2.7. Any sublinear operator p-concave is bounded and T Cp (T ). Every Banach lattice is ∞-concave. If p r, then p-concave Banach lattices are rconcave. For example, Lp for 1 p < ∞ is p-concave, and Cp (Lp ) = 1. In the case of Banach lattices the notions of cotype 2 and 2-concave are equivalent (see [10, Part II, Theorem 1.f.16]). Now, we give the notion absolutely p-summing operators generalized to sublinear operators. See [1,2] for more details.
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ˆ Y their injective tensor product, Given two Banach spaces X, Y . We denote by X ⊗ i.e., the completion of X ⊗ Y under the cross norm n n
xi ⊗ yi = sup xi (ξ ) ⊗ yi (η): ξ X∗ 1, ξ Y ∗ 1 . i=1
i=1
Let T : X → Y be a sublinear operator between a Banach space X and a Banach lattice Y . We will say that T is “absolutely p-summing” or “p-summing,” 0 < p < ∞ (we write T ∈ πp (X, Y )), if there exists a positive constant C such that for every n in N the mappings In ⊗ T : lpn ⊗ X → lpn (Y ), n
ei ⊗ xi → T (xi ) 1in
1
are uniformly bounded by C (i.e., In ⊗ T lpn ⊗ X→lpn (Y ) C). We put πp (T ) = sup In ⊗ T lpn ⊗ X→lpn (Y ) .
(2.4)
n
Remark 2.8. If Y = Lp , we have T ∈ πp (X, Lp ) iff there is a positive constant Csuch that for all n in N and (x1 , . . . , xn ) in X, we have T (xi ) C (xi )l n ω (X) . Lp (l n ) p
p
Theorem 2.9. 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, ui (x) →T (x) U
and πp (ui ) C uniformly. Then, T ∈ πp (X, Y )
and πp (T ) C.
Conversely if πp (T ) C, then πp (u) 2C for all u in ∇T . Proof. Since ui is p-summing, by Pietsch’s theorem extended to sublinear operators (see [2]) then there is a probability λi on BX such that for all x in X, we have 1/p p ui (x) C x, ξ dλi (ξ ) . BX
As we have for all x in X, ui (x) →T (x), U
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thus we obtain for all x in X, 1/p x, ξ p dλi (ξ ) T (x) C lim . U
BX
The unit ball BX is weak∗ compact, hence λi converge weak∗ to a probability λ on BX and consequently 1/p p T (x) C x, ξ dλ(ξ ) BX
for all x in X. This implies πp (T ) C. Conversely, by Corollary 1.4(i) we have for all x in X and all u in ∇T , u(x) T (x) + T (−x). This shows πp (u) 2πp (T ) and concludes the proof. 2 The next corollary is a partial generalization to Maurey’s theorem. Corollary 2.10. Let Y be a 2-concave complete Banach lattice. Let T : C(K) → Y be a sublinear such that there is an index set I , an ultrafilter U on I and {ui }i∈I ⊂ ∇T such that for all x in X, ui (x) →T (x). U
Then, T is 2-summing. Proof. By Maurey’s theorem [11] and Corollary 2.4(ii), there is a constant C such that for any linear operator u from C(K) into Y , we have π2 (u) Cu. We conclude by Theorem 2.9. 2 Remark 2.11. If we replace Y by any Hilbert space, we have the little Grothendieck’s theorem for sublinear operators and this under the supposition that ui (x) →T (x) for all x in X. U
3. Little Grothendieck’s theorem and p-regular operators In this section, we will extend a result of Maurey, that generalizes the ‘little Grothendieck theorem,’ to the sublinear case. As shown by Corollary 3.4 below, there is equivalence between 2-regular and 2-summing sublinear operators, the same as in the linear case. The problem is just this: in the linear case, every bounded linear operator from a Banach space into a Banach lattice is 2-regular; this is simply not so in the sublinear case as is indicated in an example below even if the sublinear operator is positive.
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Let T : X → Y be a bounded sublinear operator between Banach lattices X, Y . We say that T is “p-regular,” 1 p ∞, and we write T ∈ ρp (X, Y ) if there is a positive constant C such that for all finite sequence (xi )1in ⊂ X, we have T (xi ) n C (xi ) n . Y (lp )
X(lp )
The best possible constant will be denoted by ρp (T ). Recall, it was proved by Krivine in [6] see also [10, Part.II.1.f.14 and 1.d.9] that every bounded linear operator is 2-regular and every positive operator is p-regular for 1 p ∞. If p = 2, ρp (w) = KG w and ρp (w) = w if p = 2. The following example (we can see [14]) is pertinent in our work, because in the case of sublinear operators, a bounded sublinear operator is not in general 2-regular. We take as measure space (Ω, µ) the torus T = R/2πZ, equipped with the invariant measure dθ . Let X be the Hilbert space H = L2 (Ω, µ) = L2 (T) and Y be the space L1 (Ω, µ). For all r such that 0 < r π and for all f ∈ L2 (T), we define a function 2π -periodic Sr f 0 by
x+r f (y)2 dy.
1 Sr f (x) = 2r
∀x ∈ R,
x−r
√ We put Tr f = Sr f . For all x, the expression (Tr f )(x) is the L2 -norm of the function 1(x−r,x+r)f , hence the operator Tr is sublinear and the operator T defined by Tf = sup{Tr f : 0 < r < π}, is also sublinear from L2 (Ω, µ) into L1 (Ω, µ). This operator is positive in the sense of (2.1). Tf is the square root of the maximal function Mf 2 (the Hardy–Littlewood maximal operator) of the function f 2 ∈ L1 , we know that Mf 2 is in weak-L1 , therefore Tf is in weak-L2 . Consider n in N. We can partition T in n-intervals with the same length and we take 2 x1 , . . . , xn in H = L2 (Ω, µ), the characteristic functions. √ We have xi = 2π/n for all i = 1, . . . , n, but every function T (xi ) worth at least C/ 1 + (i − j ) on the support of xi for all j = 1, . . . , n with C = (4π)−1 , hence it results that 1/2
n 1 1 T (xi )(ω)2 dµ(ω) C 1 + + · · · + C log n 2 n Ω
and n i=1
i=1
n 2 xi = |xi | 2
i=1
= 2π.
L2 (Ω,µ)
In conclusion, we have (4π)−2 log n 2π . This is impossible when n is large enough. The question which we pose now is that: is every bounded sublinear operator from C(K) into a Hilbert space H is 2-regular? The following proposition will be useful in the sequel. Proposition 3.1. Let E, F, X, Y be Banach lattices and 1 p ∞. Consider a p-regular T ∈ SB(X, Y ), a bounded linear operator u : E → X and a positive bounded linear operator v : Y → F . Then,
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(i) The operator vT is p-regular and ρp (vT ) vρp (T ). (ii) If u is positive we have ρp (vT u) vρp (T )u. (iii) For p = 2, vT u is 2-regular and ρ2 (vT u) vρ2 (T )uKG . Proof. (i) The operator vT is sublinear by Remark 2.2(i). Using [10, Part II. 1.d.9] or [4], we have n n 1/p 1/p p p vT (xi ) T (xi ) v i=1 i=1 n 1/p p |xi | vρp (T ) . i=1
(ii) The operator vT u is sublinear by Remark 2.2(ii). As (i) we have n n 1/p 1/p p p vT u(xi ) T u(xi ) v i=1 i=1 n 1/p p u(xi ) ρp (T ) i=1 n 1/p p vρp (T )u |xi | . i=1
(iii) For p = 2, the same as (ii) but we take u an arbitrary operator because every linear operator is 2-regular. 2 Proposition 3.2. Let T be a sublinear operator between a Banach lattice X and a complete Banach lattice Y , C be a positive constant and 1 p ∞. Then, the operator T is in ρp (X, Y ) and ρp (X, Y ) C iff for any linear operator v : lpn ∗ → X, we have n 1/p p T v(ei ) C.l(v). i=1
Proof. Immediately by (1.2). 2 In particular, we have obviously Corollary 3.3. If X = C(K) we have T ∈ ρp C(K), Y
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iff there is a positive constant C such that for all n in N and (x1 , . . . , xn ) in C(K), we have T (xi ) n C (xi ) n ω . Y (l ) l (C(K)) p
p
Proof. By Proposition 1.2 we have l(v) = v. We conclude by (1.1) ((xi )lpn ω (C(K)) = v). 2 Corollary 3.4. Let T : C(K) → H be a 2-regular sublinear operator from C(K) into a Hilbert space H . Then T is 2-summing and conversely. Proposition 3.5. Let X, Y, Z be Banach lattices; T : X → Y be a p-regular sublinear operator and w : Y → Z be a positive p-concave linear operator. Then, the composition wT is p-concave. Proof. The operator wT is sublinear, because by Remark 2.2(ii), the composition of a positive linear operator with a sublinear operator is sublinear. Let (xi ) be a finite sequence in X. Then wT (xi ) n Cp (w) T (xi ) n Cp (w)ρp (T )(xi ) n lp (Z)
Y (lp )
X(lp )
which proves the proposition. 2 Now we are ready to present the Grothendieck–Maurey theorem in the sublinear case. Theorem 3.6. Let Y be a Banach lattice and Z be a 2-concave Banach lattice. Let T : C(K) → Y be a 2-regular sublinear operator and w : Y → Z be a positive 2-concave linear operator. Then wT is 2-summing. Proof. Let (xi ) be a finite sequence in C(K). Since w is 2-concave, then wT is also 2-concave by Proposition 3.5 and C2 (wT ) C2 (w)ρ2 (T ). Thus we have 1/2 1/2 wT (xi )2 C2 (w)ρ2 (T ) |xi |2 1/2 xi (t)2 C2 (w)ρ2 (T ) sup t ∈C(K)
C2 (w)ρ2 (T )
sup
λC(K)∗ =1
n λ, xi 2
1/2 .
1
This yields that wT is 2-summing and π2 (ωT ) · C2 (ω)ρ2 (T ).
2
Remark 3.7. Is the assumption of 2-regularity in Theorem 3.5 necessary?
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Acknowledgment The authors are very grateful to the referee for several valuable suggestions and comments which improved the paper.
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