Complex Interpolation of Operators and Optimal Domains

Complex Interpolation of Operators and Optimal Domains Ricardo del Campo, Antonio Fern´andez, Orlando Galdames, Fernando Mayoral and Francisco Naranjo Abstract. Let X0and X1be two order continuous Banach function spaces on a finite measure space, (E0,E 1) a Banach space interpolation pair, and T:X0+X1→E0+E1an admissible operator between the pairs (X0,X 1)and(E0,E 1). If Tθ:[X0,X 1][θ]→[E0,E 1][θ]is the interpolated operator by the first complex method of Calder´on and m0,m 1 and mθare the vector measures coming from T|X0and T|X1and Tθ, respectively, then we study the relationship between the optimal domain L1(mθ)ofTθand the complex interpolation space [L1(m0),L 1(m1)][θ] of the optimal domains of T|X0and T|X1. Then, we apply the obtained result to study interpolation of p-th power factorable and bidual (p, q)- power-concave operators. Mathematics Subject Classification (2010). Primary 46E30; Secondary 47B38, 46B42, 46B28. Keywords. Banach function space, optimal domain, interpolation space, vector measure, factorable operator, bidual concave operator. 1. Introduction Let T:X→Ebe a (continuous and linear) operator from a Banach function space X, over a measure space (Ω,Σ,μ), into a Banach space E. Under some mild conditions, a vector measure mT:Σ→Eis defined by mT(A):= T(χA). The space L1(mT) of scalar integrable functions with respect to mT is the optimal domain of Tin the sense that it is the largest space among all order continuous Banach function spaces (based on (Ω,Σ,μ)) into which Xis continuously embedded and to which Tadmits an E-valued continuous linear extension (see [5]or[12, Theorem 4.14]). Let (X0,X 1) be an interpolation pair of Banach function spaces, (E0,E 1) an interpolation pair of Banach spaces and Tan admissible operator This work has been supported by the Ministerio de Econom´ıa y Competitividad (Spain) and FEDER, under project MTM2012-36740, and by the Junta de Andaluc´ıa, FQM-133. between the pairs (X0,X 1) and (E0,E 1), that is, an operator T:X0+X1→ E0+E1such that its restrictions T0:= T|X0:X0→E0and T1:= T|X1:X1→E1are continuous. Moreover, let Tθ:[X0,X 1][θ]→[E0,E 1][θ] be the interpolated operator for 0 <θ<1, where [·,·][θ]denotes Calder´on’s first complex interpolation method, and set mθ:= mTθfor all 0 ≤θ≤1. Mastylo asked one of the authors whether the equality between the interpolated space [L1(m0),L 1(m1)][θ]of the optimal domains of T0and T1,and the optimal domain L1(mθ) of the interpolated operator Tθholds. In general, the answer to this question lies in the negative (see Remark 3.3). However, in the present paper we prove that the inclusion [L1(m0),L 1(m1)][θ]⊆L1(mθ) holds. This is carried out in Sect. 3(Theorem 3.1), after having established the necessary preliminaries in Sect. 2. Finally, in Sect. 4, we apply the obtained results to study the complex interpolation of certain classes of operators characterized by some extension or Maurey–Rosenthal factorization property. In concrete, we study the complex interpolation of p-th power factorable operators (Theorem 4.4), bidual (p, q)-power-concave operators (Theorem 4.9)and q-concave operators (Theorem 4.12). 2. Preliminaries and Notation Let (Ω,Σ,μ) be a finite measure space. A Banach function space Xover μ (B.f.s. for short) is an ideal of the space of (equivalence classes of) measurable functions L0(μ) endowed with a complete norm · Xthat is compatible with the μ-a.e. order and such that L∞(μ)⊆X⊆L1(μ)(see[9, p. 28]). The unit ball of Xwill be denoted by B(X). The Banach space of all integral functionals on Xis the K¨othe dual space and is denoted by X. The topological dual is denoted by X∗. A B.f.s. Xis order continuous if for every sequence (fn)nin Xsuch that 0 ≤fn↓0 pointwise we have that fnX↓0. In [9, p. 29-30] we can find the following characterization: Xis order continuous if and only if X=X∗. Given 0 <p<∞,thep-th power space of a B.f.s. Xwith norm · Xis the space X[p]:= f∈L0(μ):|f|1 p∈Xwhich is a quasi-Banach function space with quasi-norm fX[p]:= |f|1 pp X,f∈X[p]. This quasi-norm is equivalent to a norm if and only if Xis p-convex, and for 0<p<1, it is in fact a norm (see [12, Proposition 2.23]). Therefore, X[p]is a B.f.s. for 0 <p<1. If p≥1 then X⊆X[p]. For a given pair (X0,X 1) of B.f.s. over μand 0 <θ<1, the Calder´on– Lozanovskii’s product space X1−θ 0Xθ 1(see [3]) is the Banach space consisting of all f∈L0(μ) such that there exist f0∈B(X0), f1∈B(X1)andλ>0for which |f(w)|≤λ|f0(w)|1−θ|f1(w)|θ,w∈Ω(μ-a.e.),(2.1) endowed with the norm xX1−θ 0Xθ 1=infλ, where the infimum is taken over all λsatisfying (2.1). The Calder´on–Lozanovskii’s product space has the following relationships with the first Calder´on’s complex interpolation method (see [1]or[2]): X0∩X1⊆[X0,X 1][θ]⊆X1−θ 0Xθ 1⊆X0+X1with x[X0,X1][θ]=xX1−θ 0Xθ 1for all x∈[X0,X 1][θ]. Furthermore, if X0or X1is order continuous, then [X0,X 1][θ]=X1−θ 0Xθ 1. Now we present the essential definitions and results about integration with respect to vector measures. Let m:Σ→Ebe a vector measure defined onaσ–algebra Σ of subsets of a nonempty set Ω. This will always mean that mis countably additive on Σ with values in a real Banach space E. The semivariation of mis the subadditive set function m:Σ→[0,∞) defined by m(A):=sup{|m, x∗| (A):x∗∈B(E∗)},A∈Σ, where |m, x∗| is the total variation measure of the scalar measure m, x∗ given by m, x∗(A):=m(A),x ∗,for all A∈Σ.Note that for a positive scalar (finite) measure m, the semivariation of mand the measure mcoincide. A set A∈Σ is called m–null if m(A)=0.Let L0(m) be the space of all R–valued Σ–measurable functions on Ω.Two functions f,g ∈L0(m) are identified if they are equal m–a.e., that is, if {w∈Ω:f(w)=g(w)}is an m–null set. A function f∈L0(m) is called integrable (with respect to m)if f∈L1(|m, x∗|) for all x∗∈E∗and for each A∈Σthereexistsanelement Afdm ∈E(called the integral of fover A) such that Afdm,x∗= Afdm, x∗for all x∗∈E∗(see [10], [8]). The space L1(m) of all (equivalence classes of) integrable functions becomes a Banach lattice when it is endowed with the natural order m–a.e., and the norm fL1(m):= sup ⎧ ⎨ ⎩ Ω |f|d|m, x∗| :x∗∈B(E∗)⎫ ⎬ ⎭ ,f∈L1(m). The integration map Im:L1(m)→E, given by Im(f):=Ωfdm for all f∈L1(m), is an operator. Let 1 ≤p<∞. A measurable function f:Ω→Ris called p–integrable (with respect to m)if|f|p∈L1(m). We denote by Lp(m) the space of (equivalence classes of) p–integrable functions. The natural norm for this space is given by fLp(m):= sup ⎧ ⎪ ⎨ ⎪ ⎩⎛ ⎝ Ω |f|pd|m, x∗|⎞ ⎠ 1 p :x∗∈B(E∗)⎫ ⎪ ⎬ ⎪ ⎭ ,f∈Lp(m). Ameasure|m, x∗| that is equivalent to m(in the sense that they have both the same null sets) is called a Rybakov control measure for m. Such a measure always exists. We refer to [6] for this notion and basic results on vector measures and to [7] for spaces Lp(m). If μis a Rybakov control measure for mthen Lp(m) is a B.f.s. on (Ω,Σ,μ) for all p≥1. Moreover, Lp(m)is order continuous and it can be easily checked that Lp(m)=(L1(m))[1 p]. 3. Interpolation and Optimal Domains Let T:X→Ebe a Banach-space valued operator on an order continuous B.f.s. X. Under these conditions the expression mT(A):=T(χA) defines a vector measure mT:Σ→Ewhich is called the vector measure associated to T. The operator Tis said to be μ-determined if the measures μand mThave exactly the same null sets. When Tis μ-determined, the space L1(mT)isan order continuous B.f.s. on (Ω,Σ,μ), Xis continuously embedded into L1(mT) via the natural inclusion JT:X→L1(mT) and the integration operator ImT:L1(mT)→Eis the unique continuous linear extension of Tsatisfying T=ImT◦JT(see [5]or[12, Proposition 4.4]). Therefore, if Yis another order continuous B.f.s such that X⊆Y⊆ L0(μ)andT:Y→Eis a continuous linear extension of Tthen Y⊆L1(mT). In this sense, it is said that L1(mT) is the (order continuous) optimal domain for the operator T. From now on, X0and X1will be order continuous B.f.s. on the same finite measure space (Ω,Σ,μ), (E0,E 1) a Banach space pair and Tan admissible μ-determined operator. In this situation we have the optimal domains L1(m0), L1(m1)andL1(mθ) corresponding to the restricted μ-determined operators T0:X0→E0,T1:X1→E1and Tθ:[X0,X 1][θ]→[E0,E 1][θ]. The following result relates the interpolation space [L1(m0),L 1(m1)][θ]of the optimal domains L1(m0)andL1(m1) with the optimal domain L1(mθ) of the interpolated operator Tθ. Theorem 3.1. [L1(m0),L 1(m1)][θ]is continuously embedded into L1(mθ). Proof. For i=0,1, the space Xiis continuously embedded into the space L1(mi) and there exists a unique extension Imiof Tito L1(mi)(see[5]or [12, Theorem 4.14]). By the order continuity of Xiand L1(mi)wehavethe following chain of inclusions [X0,X 1][θ]=X1−θ 0Xθ 1⊆L1(m0)1−θL1(m1)θ=[L1(m0),L 1(m1)][θ]. Note that if f∈L1(m0)∩L1(m1) then Im0(f)=Im1(f). In fact, if 0≤f∈L1(m0)∩L1(m1) then we can take a sequence of simple functions 0≤ϕn↑fpointwise. Since L1(mi) has order continuous norm we have ϕn→fin L1(mi). By the continuity of Imiwe obtain Imi(f) = lim Imi(ϕn), but ϕn∈Xi,soImi(ϕn)=Ti(ϕn)=T(ϕn) and hence Imi(f) = lim T(ϕn). Now we define the operator ˆ T:L1(m0)+L1(m1)→E0+E1by ˆ T(f0+f1):=Im0(f0)+Im1(f1),f 0+f1∈L1(m0)+L1(m1). ˆ Tis well–defiend since Im0(f)=Im1(f) for every f∈L1(m0)∩L1(m1). Moreover, ˆ T|L1(m0)=Im0,ˆ T|L1(m1)=Im1and ˆ T|X0+X1=T. Therefore, we have that the interpolated operator ˆ Tθ:[L1(m0),L 1(m1)][θ]→[E0,E 1][θ] satisfies ˆ Tθ|[X0,X1][θ]=Tθ. Thus, ˆ Tθis a continuous linear extension of Tθ:[X0,X 1][θ]→[E0,E 1][θ] to the order continuous Banach space [L1(m0),L 1(m1)][θ]. By the optimality of the domain L1(mθ) for the operator Tθ(see again [5]or[12, Theorem 4.14]) we conclude that [L1(m0),L 1(m1)][θ]is continuously embedded into L1(mθ).  Remark 3.2.When (E0,E 1) is a pair of B.f.s. on the same finite measure space and the operator Tis positive, the vector measure mθcoincides with the interpolated measure [m0,m 1]θintroduced and studied in [4] and then Theorem 3.1 is a generalization of [4, Corollary 2.7]. Remark 3.3.(see the interesting paper [11] for details). In general we can not expect the equality between the spaces involved in Theorem 3.1 as the Fourier transform map F:L1(T)→∞(Z) shows. Recall that Fassigns to each f∈L1(T) its Fourier transform ˆ f:Z→Cgiven by ˆ f(n):= 1 2π 2π  0 f(x)e−inxdx, n ∈Z. It is known that L1(m0)=L1(T)andL1(m1)=L2(T) for the vector measures m0(A)=F0(χA)andm1(A)=F1(χA) associated to F0:L1(T)→ ∞(Z)andF1:L2(T)→2(Z). Therefore, for 0 <θ<1, [L1(m0),L 1(m1)][θ]=[L1(T),L 2(T)][θ]=Lp(T), where 1 p=1−θ 2. However, the optimal domain L1(mθ) for the interpolated operator Tθ:Lp(T)→2 θcan be described in this way L1(mθ)=Fp(T):=f∈L1(T):F(fχA)∈2 θ(Z),∀A∈B(T), and, what is more interesting and deeper, the inclusion Lp(T)⊆Fp(T)is proper (see [11, Theorem 1.4]). 4. Interpolation of p-th Power Factorable Operators and Bidual (p,q)-Power-Concave Operators A classical question on interpolation theory is the following: if we have an admissible operator Tbetween the pairs (X0,X 1) and (E0,E 1) such that both T0:X0→E0and T1:X1→E1have a certain property Pthen does the operator Tθ:[X0,X 1][θ]→[E0,E 1][θ]between the interpolated spaces satisfy the same property P? We are going to apply our result about optimal domains to answer this question in the affirmative for the properties of being p-th power factorable operator and being bidual (p, q)-power-concave operator. We begin by recalling the definition and properties of p-th power factorable operators introduced in [12]. Definition 4.1. Let 1 ≤p<∞,Xan order continuous Banach function space, and Ea Banach space. An operator T:X→Eis said to be p-th power factorable if there exists an operator T[p]:X[p]→E, which equals T over X⊆X[p]. When we consider a μ-determined operator T:X→Eon an order continuous B.f.s. Xit is easy to see that the restriction Ip mT:Lp(mT)→Eof the integration operator ImT:L1(mT)→Eis μ-determined and p-th power factorable. The characterization of p-th power factorable operator is given by the following result (see [12, Theorem 5.7] for more equivalences). Theorem 4.2. Given 1≤p<∞, an order continuous B.f.s. X,aBanach space Eand a μ-determined operator T:X→E, the following assertions are equivalent: (i) Tis p-th power factorable. (ii) X⊆Lp(mT)with a continuous inclusion. Remark 4.3.If an operator Tis p-th power factorable then it is q-th power factorable for every 1 ≤q<p. This is due to the facts that Lp(mT)is continuously embedded in Lq(mT)for1≤q<pand that the composition of ap-th power factorable operator with an operator is a p-th power factorable operator (see [12, Lemma 5.4]). Now we are going to prove that the property of being p-th power factorable is preserved by the first Calder´on’s complex interpolation method. Recall that from now on (X0,X 1) is a pair of order continuous B.f.s on the same measure space (Ω,Σ,μ), (E0,E 1) is a Banach space interpolation pair and Tis an admissible μ-determined operator with restrictions T0:= T|X0:X0→E0and T1=: T|X1:X1→E1. Theorem 4.4. If T0and T1are p-th power factorable then the interpolated operator Tθis p-th power factorable. Proof. By Theorem 4.2 we have to prove that [X0,X 1][θ]⊆Lp(mθ). Since X0and X1have order continuous norm and T0and T1are p-th factorable we have [X0,X 1][θ]=X1−θ 0Xθ 1⊆Lp(m0)1−θLp(m1)θ. So, we only need to check that Lp(m0)1−θLp(m1)θ⊆Lp(mθ), but this condition is equivalent to prove that L1(m0)1−θL1(m1)θ⊆L1(mθ) and this follows from Theorem 3.1 since L1(m0) (or L1(m1)) has order continuous norm.  What can we say if we consider an admissible operator Tsuch that T0 is p0-th power factorable and T1is p1-th power factorable? The next result sheds some light on this issue: Theorem 4.5. Let 0<θ<1≤p0≤p1<∞.IfT0is p0-th power factorable and T1is p1-th power factorable then Tθ,α :[X0,X 1][θ]→[E0,E 1][α]is welldefined and it is a p-th power factorable restriction of T,wherepis given by 1 p:= 1−θ p0+θ p1and α:= θp p1. Proof. First, observe that 0 <α<1. During the proof we will check that [X0,X 1][θ]⊆Lp(mα), where mαis the measure associated to the operator Tα:[X0,X 1][α]→[E0,E 1][α]. Thus, in particular we have [X0,X 1][θ]⊆ L1(mα) which allows us to consider the operator Tθ,α :[X0,X 1][θ]→ [E0,E 1][α]given by Tθ,α := Imα|[X0,X1][θ]. Note that mTθ,α =mα. Therefore, keeping in mind Theorem 4.2, we only have to prove that [X0,X 1][θ]⊆ Lp(mα). Since X0is p0-th factorable and X1is p1-th factorable we have [X0,X 1][θ]=X1−θ 0Xθ 1⊆Lp0(m0)1−θLp1(m1)θ. Thus, if we check that Lp0(m0)1−θLp1(m1)θ⊆Lp(mα) then the proof will be completed. So, let 0 ≤f0∈Lp0(m0), 0 ≤f1∈Lp1(m1) and take g0:= fp0 0∈L1(m0)andg1:= fp1 1∈L1(m1). By Theorem 3.1 we have L1(m0)1−αL1(m1)α⊆L1(mα). Hence, f1−θ 0fθ 1p=f(1−θ)p 0fθp 1=(fp0 0)(1−θ)p p0(fp1 1)θp p1=g1−α 0gα 1∈L1(mα), that is, f1−θ 0fθ 1∈Lp(mα).  Observe that if p0=p1in Theorem 4.5 then α=θand we recover Theorem 4.4. Furthermore, if p0<p 1,T0is p0-th power factorable and T1is p1-th power factorable then T1is also p0-th power factorable and so the operator Tθ:[X0,X 1][θ]→[E0,E 1][θ]is p0-th power factorable by Theorem 4.4. Theorem 4.5 provides a better result in the sense that the operator Tθ,α is even p-th power factorable (with p0<p). However, the range space is not [E0,E 1][θ]but [E0,E 1][α].In particular when the pair of Banach spaces (E0,E 1) satisfies that E0⊆E1we have the following result. Corollary 4.6. Let 1≤p0<p 1<∞and E0⊆E1.IfT0is p0-th power factorable and T1is p1-th power factorable then Tθis p-th power factorable, where pis given by 1 p:= 1−θ p0+θ p1. Proof. If p0<p 1then p0<p<p 1and so α=θp p1<θ.FromE0⊆E1and [2, Theorem 4.2.1] it follows that [E0,E 1][α]⊆[E0,E 1][θ]. By Theorem 4.5,the operator Tθ,α :[X0,X 1][θ]→[E0,E 1][α]is p-th power factorable and hence Tθ=Tθ,θ :[X0,X 1][θ]→[E0,E 1][θ]is p-th power factorable.  In connection with the Maurey–Rosenthal factorization theory, Okada, Ricker and S´anchez-P´erez introduced in [12] the concept of bidual (p, q)- power-concave operator. Definition 4.7. Let 1 ≤p, q < ∞,Xan order continuous Banach function space, and Ea Banach space. An operator T:X→Eis said to be bidual (p, q)-power-concave if there exists a constant C>0 such that n  j=1 T(fj) q p E≤Csup ⎧ ⎨ ⎩n  j=1 |fj|q p,ξ :ξ∈B(X∗ [q])⎫ ⎬ ⎭ for all n∈Nand f1,...,f n∈X. A bidual (1,q)-power-concave operator is called simply bidual q-concave. The following theorem characterizes bidual (p, q)-power-concave operators (see [12, Theorem 6.9] for other equivalences). Theorem 4.8. Let T:X→Eaμ-determined operator. The following assertions are equivalent: (i) Tis bidual (p, q)-power-concave operator. (ii) There exists 0<g∈L0(μ)such that the inclusions X⊆Lq(gdμ)⊆Lp(mT) hold and are continuous. Now we can obtain a result about interpolation of bidual (p, q)-power- concave operators. Theorem 4.9. If T0and T1are bidual (p, q)-power-concave then Tθis bidual (p, q)-power-concave. Proof. By Theorem 4.8 there exists 0 <g i∈L0(μ) such that Xi⊆ Lq(gidμ)⊆Lp(mi)fori=0,1. Therefore, [X0,X 1][θ]⊆[Lq(g0dμ),L q(g1dμ)][θ]⊆[Lp(m0),L p(m1)][θ]. According to [2, Theorem 5.5.3] we have [Lq(g0dμ),L q(g1dμ)][θ]=Lq(gdμ), where g:= g1−θ 0gθ 1>0. On the other hand, by Theorem 3.1 it follows that L1(m0)1−θL1(m1)θ⊆L1(mθ) and hence Lp(m0)1−θLp(m1)θ⊆Lp(mθ). Thus, [X0,X 1][θ]⊆Lq(gdμ)⊆[Lp(m0),L p(m1)][θ]⊆Lp(mθ) and so the operator Tθis bidual (p, q)-power-concave (again by Theorem 4.8).  We can obtain similar results to Theorem 4.5 and Proposition 4.6 for bidual (p, q)-power-concave operators. Theorem 4.10. Let 1≤p0≤p1<∞,1≤q0≤q1<∞,0<θ<1. If T0is bidual (p0,q 0)-power-concave and T1is bidual (p1,q 1)-power-concave then Tθ,α :[X0,X 1][θ]→[E0,E 1][α]is bidual (p, q)-power-concave, where 1 p:= 1−θ p0+θ p1,1 q:= 1−θ q0+θ q1and α:= θp p1. Proof. Again there exists 0 <g i∈L0(μ) such that Xi⊆Lqi(gidμ)⊆ Lpi(mi), for i=0,1 and hence [X0,X 1][θ]⊆[Lq(g0dμ),L q(g1dμ)][θ]=Lq(gdμ)⊆[Lp(m0),L p(m1)][θ], with g=g1−θ 0gθ 1>0. Now we can repeat the same argument as in Theorem 4.5 to prove that Lp0(m0)1−θLp1(m1)θ⊆Lp(mα) and this completes the proof.  Observe that if p0=p1and q0=q1then Theorem 4.10 reduces to Theorem 4.9. Corollary 4.11. Let 1≤p0<p 1<∞,1≤q<∞and E0⊆E1.IfT0is bidual (p0,q)-power-concave and T1is bidual (p1,q)-power-concave then Tθ is bidual (p, q)-power-concave, where pis given by 1 p:= 1−θ p0+θ p1. Proof. If p0<p 1then p0<p<p 1and so α=θp p1<θ.From E0⊆E1and [2, Theorem 4.2.1] we deduce that [E0,E 1][α]⊆[E0,E 1][θ]. By Theorem 4.10, the operator Tθ,α :[X0,X 1][θ]→[E0,E 1][α]is bidual (p, q)-power-concave and thus Tθ=Tθ,θ :[X0,X 1][θ]→[E0,E 1][θ]is bidual (p, q)-power-concave (see [12, Proposition 6.2 (vi)]).  Finally we obtain a new result about interpolation of q-concave operators. Recall that a bidual q-concave operator is in particular q-concave (see [12, Proposition 6.2 (i)] with p= 1). Corollary 4.12. Let (X0,X 1)be a pair of q-convex order continuous B.f.s and Tan admissible operator. 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