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Regular Operators Between Banach Lattices

arXiv:math/9306207v1 [math.FA] 7 Jun 1993Complex InterpolationandRegular Operators Between Banach LatticesbyGilles Pisier*Texas A. and M. UniversityCollege Station, TX 77843, U. S. A.andUniversit´e Paris 6Equipe d’Analyse, Boˆıte 186, 75252 Paris Cedex 05, FranceAbstract.We study certain interpolation and extension properties of the space ofregular operators between two Banach lattices. Let Rp be the space of all the regular (orequivalently order bounded) operators on Lp equipped with the regular norm.

We provethe isometric identity Rp = (R∞, R1)θ if θ = 1/p, which shows that the spaces (Rp) forman interpolation scale relative to Calder´on’s interpolation method. We also prove thatif S ⊂Lp is a subspace, every regular operator u : S →Lp admits a regular extension˜u : Lp →Lp with the same regular norm.

This extends a result due to Mireille L´evy inthe case p = 1. Finally, we apply these ideas to the Hardy space Hp viewed as a subspaceof Lp on the circle.

We show that the space of regular operators from Hp to Lp possessesa similar interpolation property as the spaces Rp defined above. * Supported in part by N.S.F.

grant DMS 90035501

In a recent paper [HeP] we have observed that the real interpolation spaces associatedto the couples(B(c0, ℓ∞),B(ℓ1, ℓ1))and(B(L∞, L∞),B(L1, L1))can be described and an equivalent of the Kt-functional can be given (cf. [HeP]).

It isnatural to wonder whether analogous results hold for the complex interpolation methodand this is the subject of the present paper.Let X0, X1 be Banach lattices of measurable functions defined on some set S (we aredeliberately vague, see e.g. [LT] for a detailed theory).

We will denote byXθ = X1−θ0Xθ1the space of all measurable functions f on the set S such that there are f0 ∈X0, f1 ∈X1satisfying |f| ≤|f0|1−θ|f1|θ and we let∥f∥Xθ = inf{∥f0∥1−θX0 ∥f1∥θX1}where the infimum runs over all possible decompositions of f.We will denote by (X0, X1)θ and (X0, X1)θ the complex interpolation spaces as definedfor examples in [BL]. Recall the fundamental identity (due to Calder´on)X1−θ0Xθ1 = (X0, X1)θwith identical norms, which is valid under the assumption that the unit ball of X1−θ0Xθ1 isclosed in X0 + X1 (see [Ca]).

Moreover, if either X0 or X1 is reflexive (cf. also [HP] and[B]) we have(X0, X1)θ = (X0, X1)θ,with identical norms.

In particular when X0, X1 are finite dimensional spaces, there isno need to distinguish (X0, X1)θ and (X0, X1)θ. We will use this fact repeatedly in thesequel.

Let c0 (resp. ℓ∞) be the space of all sequences of complex scalars tending to zero2

(resp. bounded) at infinity equipped with the usual norm, and let ℓ1 denote the usual dualspace of absolutely summable sequences.

Recall ℓ1 = (c0)∗and ℓ∞= (ℓ1)∗. Given Banachspaces X, Y we denote by B(X, Y ) the space of all bounded operators u: X →Y equippedwith the usual operator norm.

We will always identify an operator on a sequence spacewith a matrix in the usual way. We will denote by A0 (resp.

A1) the space of all complexmatrices (aij) such thatsupiXj|aij| < ∞ resp. supjXi|aij| < ∞!equipped with the norm∥(aij)∥A0 = supiXj|aij|.resp.∥(aij)∥A1 = supjXi|aij|.We will need to work with complex spaces, so we recall that if E is a real Banach lattice itscomplexification E + iE can be naturally equipped with a norm so that for all f = a + ibin E +iE we have ∥a+ib∥= ∥(|a|2+|b|2)1/2∥E.

We will call the resulting complex Banachspace a complex Banach lattice.Let E, F be real or complex Banach lattices. We will denote by Br(E, F) the spaceof all operators u: E →F for which there is a constant C such that for all finite sequences(xi)i≤m in E we have(1)∥supi≤m|u(xi)| ∥F ≤C∥supi≤m|xi| ∥E.We will denote by ∥u∥r the smallest constant C for which this holds, i.e.

we set ∥u∥r =inf{C}. It is known, under the assumption that F is Dedekind complete in the sense of[MN] (sometimes also called order complete), that in the real (resp.

complex) case, everyregular operator u: E →F is of the form u = u+ −u−(resp. u = a+ −a−+ i(b+ −b−))where u+, u−(resp.

a+, a−, b+, b−) are bounded positive operators from E to F, cf. [MN]or [S] p.233.

Since the converse is obvious this gives a very clear description of the spaceBr(E, F). Here of course “positive” means positivity preserving.

Let E, F be real (resp.3

complex) Banach lattices. Under the same assumption on F (cf.

e.g. [MN] p.27) it isknown that Br(E, F) equipped with the usual ordering is a real (resp.

complex) Banachlattice in such a way that we have∀T ∈Br(E, F)∥T∥r = ∥|T| ∥B(E,F ).We refer to [MN] for more information on the spaces Br(E, F). We will only use thefollowing elementary particular cases.If A is in Br(ℓp, ℓp) with associated matrix (aij), let us denote by |A| the operator admitting(|aij|) as its associated matrix.

Then we have(2)∥A∥r = ∥|A| ∥B(ℓp,ℓp).By a well known result (going back, I believe, to Grothendieck) for any measure spaces(Ω, µ), (Ω′, µ′) we have an isometric identity(3)B(L1(µ), L1(µ′)) = Br(L1(µ), L1(µ′)).On the other hand, we have trivially (isometrically)(4)B(L∞(µ), L∞(µ′)) = Br(L∞(µ), L∞(µ′)).The next result is known to many people in some slightly different form (in particularsee [W]), I believe that our formulation is useful and hope to demonstrate this in the restof this note. Motivated by the results in [HeP], I suspected that this result was knownand I asked F. Lust-Piquard whether she knew a reference for this, she did not but sheimmediately showed me the following proof.Theorem 1.

Let A0, A1 be as above. For any fixed integer n, let An0 ⊂A0, An1 ⊂A1be the subspace of all matrices (aij) which are supported by the upper left n × n corner,so that the elements of An0 or An1 can be viewed as n × n matrices.

Note the elementaryidentificationsAn0 = B(ℓn∞, ℓn∞)andAn1 = B(ℓn1, ℓn1)A0 = B(c0, ℓ∞)andA1 = B(ℓ1, ℓ1).4

We have then for all 0 < θ < 1 the following isometric identities where p = 1/θ:(i) (An0, An1)θ = Br(ℓnp, ℓnp). (ii) Aθ = (A0, A1)θ = Br(ℓp, ℓp).Remark.

Note that A0 and A1 (resp. An0 and An1) are isometric as Banach spaces.

This isa special case of the fact that the transposition induces an isometric isomorphism betweenthe spaces B(X, Y ∗) and B(Y, X∗) when X and Y are Banach spaces.Proof of Theorem 1. (The main point was shown to me by F.

Lust-Piquard.) Wewill prove (i) only.

The second part (ii) follows easily from (i) by a weak-∗compactnessargument which we leave to the reader.Now let (X0, X1) and (Y0, Y1) be compatible couples of finite dimensional complex Banachspaces and let Xθ = (X0, X1)θ, Yθ = (Y0, Y1)θ. It is well known and easy to check fromthe definitions that we have a norm 1 inclusion(B(X0, Y0), B(X1, Y1))θ ⊂B(Xθ, Yθ).Applying this to the spaces X0 = Y0 = ℓn∞(ℓm∞) and X1 = Y1 = ℓn1(ℓm∞) with m an arbitraryinteger, we obtain the norm 1 inclusion(An0, An1)θ →Br(ℓnp, ℓnp).To show the converse, consider the spacesBn0 = An∗0andBn1 = An∗1 .For any n × n matrix (bij) we have∥(bij)∥Bn0 =nXi=1supj≤n|bij|and∥(bij)∥Bn1 =nXj=1supi≤n|bij|.We claim that for all (bij) in the unit ball of Bnθ = (Bn0 )1−θ(Bn1 )θ = (Bn0 , Bn1 )θ we have(5)∀A ∈Br(ℓnp, ℓnp)Xi,j|aijbij| ≤∥A∥r.5

With this claim, we conclude easily since (5) yields a norm one inclusion Br(ℓnp, ℓnp) ⊂(Bnθ )∗, and (Bnθ )∗= (An∗0 , An∗1 )∗θ = Anθ . Therefore it suffices to prove the claim (5).

Since(bij) is assumed in the unit ball of (Bn0 )1−θ(Bn1 )θ there are n × n matrices (b0ij) and (b1ij)such that |bij| = |b0ij| · |b1ij| and such that Pisupj|b0ij|p′ ≤1, Pjsupi|b1ij|p ≤1, with p = 1/θand p′ = 1/(1 −θ).Now let βi = supj|b0ij| and αj = supi|b1ij|. ThenX|aijbij| ≤Xi,jβi|aij|αjhence by (2)≤∥A∥r.This proves our claim and concludes the proof.It is then routine to deduce the following extension.Corollary 2.

Let (Ω, µ) and (Ω′, µ′) be arbitrary measure spaces. Consider the coupleX0 = B(L∞(µ), L∞(µ′)),X1 = B(L1(µ), L1(µ′)).We will identify (for the purpose of interpolation) elements in X0 or X1 with linear op-erators from the space of integrable step functions into L1(µ′) + L∞(µ′).

We have thenisometrically(6)X1−θ0Xθ1 = (X0, X1)θ = Br(Lp(µ), Lp(µ′)).Remark. Recalling (3) and (4), we can rewrite (6) as follows(Br(L∞(µ), L∞(µ′)), Br(L1(µ), L1(µ′)))θ = Br(Lp(µ), Lp(µ′)).By [Be], it follows that the space (X0, X1)θ coincides with the closure in Br(Lp(µ), Lp(µ′))of the subspace of all the operators which are simultaneously bounded from L1(µ) to L1(µ′)and from L∞(µ) to L∞(µ′).We will now consider operators defined on a subspace S of a Banach lattice E andtaking values in a Banach lattice F. Let u : S →F be such an operator.

We will again6

say that u is regular if there is a constant C such that u satisfies (1) for all finite sequencesx1, ..., xm in E. We again denote by ∥u∥r the smallest constant C for which this holds.Clearly the restriction to S of a regular operator defined on E is regular. Conversely, ingeneral a regular operator on S is not necessarily the restriction of a regular operator onE: for instance if E is L1, if S is the closed span of a sequence of standard independentGaussian random variables and if u : S →L2 is the natural inclusion map, then u is regular(this is a well known result of Fernique, see e.g.

[LeT] p. 60) but does not extend to anybounded map from L1 into L2 since by a weakening of Grothendieck’s theorem (cf. [P4]p. 57), the identity of S would then be 2-absolutely summing, which is absurd since S isinfinite dimensional (cf.

e.g. [P4] p. 14).Nevertheless, it turns out that in several interesting cases, conversely every regularoperator on S is the restriction of a regular operator on E with the same regular norm.In particular, the next statement is an extension theorem for regular operators whichgeneralizes a result due to M. L´evy [L´e] in the case p = 1.

We will proveTheorem 3. Let 1 ≤p ≤∞.Let (Ω, µ), (Ω′, µ′) be arbitrary measure spaces.LetS ⊂Lp(µ) be any closed subspace.

Then every regular operator u: S →Lp(µ′) admits aregular extension ˜u: Lp(µ) →Lp(µ′) such that ∥˜u∥r = ∥u∥r.Actually this will be a consequence of the following more general result. (We refer thereader to [LT] for the notions of p-convexity and p-concavity.

)Theorem 4. Let L, Λ be Banach lattices and let S ⊂Λ be a closed subspace.

Assumethat L is a dual space, or merely that there is a regular projection P: L∗∗→L with∥P∥r ≤1. Assume moreover that for some 1 ≤p ≤∞Λ is p-convex and L p-concave.Then every regular operator u: S →L extends to a regular operator ˜u: Λ →L with∥˜u∥r = ∥u∥r.Remark.

Note that by known results (cf. [K] or [LT]) in the above situation every positiveoperator u: Λ →L factors through an Lp-space, i.e.

there is a measure space (Ω, µ) andoperators B: Λ →Lp(µ) and A: Lp(µ) →L such that U = AB and ∥A∥· ∥B∥=∥u∥. Actually for this conclusion to hold, it suffices to assume that u can be written asthe composition of first a p−convex operator with constant ≤1 followed by a p-concave7

operator with constant ≤1. Therefore, since every regular operator with regular norm≤1 on a p-convex Banach lattice clearly is itself p-convex with constant ≤1, every regularu: Λ →L factors through an Lp-space with factorization constant at most 1.

Actually itis easy to modify Krivine’s argument to prove that, in the same situation as in Theorem4, every regular u: Λ →L can be written as u = AB as above but with A, B regular andsuch that ∥A∥r∥B∥r = ∥u∥r.Proof of Theorem 4. By a standard ultraproduct argument it is enough to considerthe case when L is finite dimensional with an unconditional basis (e1, .

. ., en).

As usualin extension problems, we will use the Hahn-Banach theorem. We need to introduce aBanach space X such that X∗= Br(Λ, L).

The space X is defined as the tensor productL∗⊗Λ equipped with the following norm, for all v = Pn1 αke∗k ⊗xk with αi scalar andxi ∈Λ we define∥v∥X = infXn1 αieiL∗∥supi≤n|xi| ∥Λ.The only assumption needed for our extension theorem is that ∥∥X is a norm (see theremark below). This follows from the p′-convexity of L∗and the p-convexity of Λ.To check this we assume as we may that Λ is included in a space of measurablefunctions L0(µ) on some measure space.

Let Y0 be the space of n-tuples of measurablefunctions y1, . .

., yn in L∞(µ) equipped with the norm∥(yi)∥Y0 =nX1∥yi∥1p′∞e∗ip′L∗.That this is indeed a norm follows from the p′-convexity of L∗.Let Y1 be the space of n-tuples of measurable functions y1, . .

., yn is L0(µ) such that|yi|1p ∈Λ equipped with the norm∥(yi)∥Y1 = ∥supi≤n|yi|1p ∥pΛ.Again this is a norm by the p-convexity of Λ. But now if we consider the unit ball of thespaceY 1−θ0Y θ1withθ = 1p8

we find exactly the set C. This shows that C is convex as claimed above. We will nowcheck that X∗= Br(Λ, L) isometrically.Consider u: Λ →L.

We have(7)∥u∥r = sup(nXk=1supi≤m|⟨u(xi), e∗k⟩|ekL)where the supremum runs over all m and all m-tuples (x1, . .

., xm) in Λ such that∥supi≤m|xi| ∥Λ ≤1. Let us denote by β the unit ball of L∗.

Then (7) can be rewritten(8)∥u∥r = sup(nXk=1αk⟨u(xik), e∗k⟩)where the supremum runs over all integers m, all choices i1, . .

. , in in {1, .

. ., m}, all ele-ments α =nP1αke∗k in β and all m-tuples x1, .

. ., xm in Λ with ∥sup |xi| ∥Λ ≤1.

But forsuch elements clearly v =nPk=1αke∗k ⊗xik is in the set C which is the unit ball of X, hence(8) yields∥u∥r = sup{|⟨u, v⟩| | v ∈BX}.This proves the announced claim that X∗= Br(Λ, L) isometrically.We can then complete the proof by a well known application of the Hahn-Banachtheorem.Consider the subspace M ⊂X formed by all the v = Pn1 αke∗k⊗xk such that αkxk ∈Sfor all k = 1, . .

., n. If u: S →L is regular we clearly have for all v in M|⟨u, v⟩| =nX1⟨u(αkxk), e∗k⟩ ≤∥u∥r∥v∥Xhence we can find a Hahn-Banach extension of the linear form v ∈M →⟨u, v⟩definedon the whole of X and still with norm ≤∥u∥r. Clearly we can write the extension in theform v ∈X →⟨˜u, v⟩for some operator ˜u: Λ →L and sincesup∥v∥X≤1|⟨˜u, v⟩| ≤∥u∥r, we have∥˜u∥r ≤∥u∥r as announced.Remark.

Assume again L finite dimensional as above. The assumption “Λ p-convex, Lp′-concave” can be replaced by the property that in L∗⊗Λ the setC =nXn1 αie∗i ⊗xi | αi ∈Cxi ∈ΛXn1 αie∗iL∗≤1,∥sup |xi|∥Λ ≤1o9

is a convex set.As the preceding proof shows this is true is L∗is p′-convex and Λ p-convex. However,it is clearly true also in other cases.

For instance if L∗= ℓn∞then C is just the unit ball ofΛ(ℓn∞) which is clearly convex for all Λ. Moreover if Λ = L∞(µ) for some measure µ thenC is the unit ball of L∗(L∞(µ)) which is convex for all L. More generally, what we reallyuse (and which is then equivalent to the extension theorem, by a reasoning well known tomany Banach space specialists) is that the closed convex hull of the set C, satisfiesconv(C) ∩M = conv(C ∩M),where M denotes as above the subspace M = L∗⊗S ⊂L∗⊗Λ.We now give some applications to Hp-spaces, mainly motivated by our paper [P2].Let 1 ≤p ≤∞.

Let Hp be the usual Hp-space of functions on the torus T equipped withits normalized Haar measure m(dt) =dt2π. We denote simply Lp = Lp(T, m).

Given afinite dimensional normed space E we denoteHp(E) = {f ∈Lp(m; E) | ˆf(n) = 0∀n < 0}.By a result of P. Jones [J] (see also [BX, P1, X] for a discussion of the vector valued case)we have isomorphically and with isomorphism constants independent of E(9)Hp(E) = (H∞(E), H1(E))θifθ = 1/p.More precisely, there is a constant Cp such that for all f in Hp(E) we have(10)∥f∥(H∞(E),H1(E))θ ≤Cp∥f∥Hp(E).We will prove the following extension of Corollary 2.Theorem 5. Let (Ω, µ) be an arbitrary measure space.

LetB0 = Br(H∞, L∞(µ))B1 = Br(H1, L1(µ)).Then (isomorphically) (B0, B1)θ = Br(Hp, Lp(µ)) with θ = 1/p.Proof. By Theorem 4, if u: Hp →Lp(µ) is such that ∥u∥r < 1, then ∃˜u: Lp →Lp(µ) extending u such that ∥˜u∥r < 1.

By Corollary 2, ˜u is of norm < 1 in the space10

(Br(L∞, L∞(µ)), Br(L1, L1(µ)))θ hence by restriction u is of norm < 1 in (B0, B1)θ. Con-versely, assume that u is in the unit ball of (B0, B1)θ. Consider then f in the unit ball ofHp(ℓn∞), or equivalently consider an n-tuple (f1, .

. ., f1) in Hp such thatRsupk≤n|fk|pdm ≤1.By P. Jones’s theorem (10) we have∥f∥(H∞(ℓn∞),H1(ℓn∞))θ ≤Cp,hence since ∥u∥(B0,B1)θ ≤1 by assumption, it is easy to deduce∥u(f)∥(L∞(ℓn∞),L1(ℓn∞))θ ≤Cpor equivalently since Lp(ℓn∞) = (L∞(ℓn∞), L1(ℓn∞))θZsupk≤n|u(fk)|pdm ≤Cpp.By homogeneity we conclude that ∥u∥Br(Hp,Lp(µ)) ≤Cp.Remark.

Once again by [Be], the space (B0, B1)θ coincides with the closure in Br(Hp, Lp(µ))of the operators which are simultaneously regular from H1 to L1(µ) and from H∞toL∞(µ).Remarks. (i) For a version of Theorem 1 and Corollary 2 in the case of noncommutative Lp-spaces,we refer the reader to [P3].

(ii) Of course Theorem 5 and its proof remain valid with the couple (H∞, H1) replacedby any couple of subspaces of (L∞, L1) for which (10) holds.References[B] J. Bergh. On the relation between the two complex methods of interpolation.

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