Interpolation of Banach Spaces by the Γ-Method

INTERPOLATION OF BANACH SPACES BY THE γ-METHOD

JESUS´ SUAREZ´ AND LUTZ WEIS

Abstract. In this note we study some basic properties of the γ- interpolation method introduced in [6] e.g. interpolation of Bochner spaces and interpolation of analytic operator valued functions in the sense of Stein. As applications we consider the interpolation of almost summing operators (in particular γ-radonifying and Hilbert-Schmidt operators) and of γ-Sobolev spaces as introduced in [6]. We also compare this new interpolation method with the real interpolation method.

1. Introduction In [6] a new method of interpolation was introduced, which in contrast to the classical complex and real interpolation method is based on almost summing sequences and gaussian averages. This becomes necessary in order to characterize the boundedness of the holomorphic functional calculus of sectorial operators in terms of the interpolation of the domains of its fractional powers (see [6, 7]). This is not possible using the real or complex interpolation method. Some information on this new method can be found in [6, 7]. In particular, it has equivalent formulations modelled after the complex, the real and discrete interpolation methods. Furthermore, in a Hilbert space, it agrees with the complex method. In this note we provide some basic information concerning the γ-interpolation method. We show that interpolating Bochner spaces we obtain the ’right’ result (see theorem 3.1) and that Stein’s interpolation schema for analytic families of operators is still applicable (see theorem 4.2). With these tools we interpolate spaces γ(Hi,Xi) of almost summing operators (or radonifying operators), with respect to the complex and the γ-method, where Hi are Hilbert spaces and Xi are B-convex Banach spaces. As a corollary we obtain an improvement of a result of Cobos and Garc´ıa-Dav´ıa[3] on the interpolation of Hilbert-Schmidt operators (see corollary 5.1) and show that the γ-Sobolev spaces introduced in [6] form an interpolation scale. Finally we compare the γ-method with the real interpolation method: the possible inclusions between

The work of the ﬁrst author was supported in part by a Marie Curie grant HPMT- GH-01-00286-04 at Karlsruhe University under the direction of Prof. L. Weis and in part during a visit to the IMUB at Barcelona University. Keywords: Interpolation of spaces of vector valued functions, γ-radonifying operators. AMS-classiﬁcation: 46B70, 47B07, 46E40. 1 2 interpolation spaces depend on the geometry of the Banach spaces, namely their type and cotype properties.

2. Definitions and Basic Properties We will always work with an infinite dimensional separable Hilbert space and will reserve the letter H for this. In what follows, F(Y,X) will denote the space of all finite rank operators from Y into X. Given (ej) an orthonormal basis in H, we define the γ-norm of a finite rank operator u ∈ F(H,X) given Pn Pn by u = j=1 xj ⊗ ej and acting as u(h) = j=1(h, ej)xj, in the form

  1 Xn Z Xn 2  2  kukγ := Ek gjxjkX = k gj(ω)xjkX dP (ω) , j=1 Ω j=1 where {gn} is a sequence of standard Gaussian variables on a fixed probability space (Ω, σ, P ). To extend this notation to operators of possibly infinite rank we proceed as follows. If v : H → X is any bounded operator, we define

kvkγ := sup {kvukγ : u ∈ F(H,H), kuk ≤ 1} and then form the spaces

γ+(H,X) := {v ∈ L(H,X): kvkγ < ∞} and γ γ(H,X) := F(H,X) We call γ(H,X) the space of radonifying operators. The following result can be found in [6]:

Proposition 2.1. Let be X a Banach space, then γ(H,X) = γ+(H,X) if and only if X contains no copy of c0. Now we sketch the basic ideas of the γ-method as developed in [6]. Assume that X = (X0,X1) is a compatible pair of Banach spaces.

First, we introduce admissible classes A and A+ of operators. −2t Let u : L2(dt) + L2(e dt) → X0 + X1 be a bounded operator, we will say −2jt u ∈ A iﬀ u ∈ γ(L2(e dt),Xj) for j = 0, 1, and set

kuk := max kujkγ(L (e−2jtdt),X ). A j=0,1 2 j

−2jt We will say u ∈ A+ iﬀ u ∈ γ+(L2(e dt),Xj) for j = 0, 1, and set

kuk := max kujkγ (L (e−2jtdt),X ), A+ j=0,1 + 2 j

−2jt where in all cases uj denotes the operator u on L2(e dt) for j = 0, 1. These spaces are complete under their corresponding norm. 3

γ As in [6] we define the interpolation space (X0,X1)θ , which we will briefly γ call Xθ , as follows : if x ∈ X0 + X1 we introduce the norm γ kxkθ = inf {kukA : u ∈ A, u(eθ) = x} , θt γ where eθ denotes the function eθ(t) = e . Now (X0,X1)θ is the space of all γ γ x ∈ X0 + X1 such that kxkθ < ∞ equipped with k · kθ . The same definitions work by replacing γ by γ+ and by interchanging the roles of A and A+, respectively. The resulting interpolation space we denote γ+ by Xθ .

Proposition 2.2. Suppose that (X0,X1) and (Y0,Y1) are compatible pairs of Banach spaces. Suppose that S : X0 + X1 → Y0 + Y1 is a bounded operator γ γ such that S(X0) ⊂ Y0 and S(X1) ⊂ Y1. Then S :(X0,X1)θ → (Y0,Y1)θ is a bounded operator with 1−θ θ kSk γ γ ≤ kSk kSk . Xθ →Yθ X0 X1

The same holds true replacing γ by γ+. See [6] for a detailed proof.

It is important for us that the interpolation we have described has an alternative formulation as a complex method. Denote by S the strip {z : 0 < <(z) < 1} and consider the space A(X0,X1) of all analytic functions zt F : S → X0 + X1 which are of the form F(z) = u(e ) where u ∈ A. We deﬁne a norm on this space by kFkA = kukA. Then we have the formula γ kxkθ = inf {kFkA : F(θ) = x} . Most of the proofs involved in the paper use density arguments, so it is interesting to observe that: Lemma 2.1. The set of ﬁnite-rank operators is dense in A. As a consequence we get: γ Corollary 2.1. For θ ∈ (0, 1), X0 ∩ X1 is dense in Xθ . See a proof of this facts in [6]. Closer to the ”complex spirit” is the following lemma:

Lemma 2.2. Suppose F : S → X0 + X1 is a bounded analytic function such that the boundary values F(j + it) = limξ→j F(ξ + it) exists t-a.e. and are in Xj for j = 0, 1. Suppose the functions Fj(it) := F(j + it) are Xj strongly −2jt measurable and Fj ∈ γj(Xj) := γ(L2(e dt),Xj) for j = 0, 1. Then F ∈ A and 1 kFkA = (2π) 2 max kFjkγ . j=0,1 j

−2jt By Fj ∈ γj(Xj) we mean that the operator uj : L2(e dt) → Xj deﬁned by Z

uj(h) := h(t)Fj(t)dt R 4

−2jt for h ∈ L2(e dt) belongs to γj(Xj) and kFjkγj = kujkγj . See [6] for a detailed proof. Lemma 2.1 guarantees that the space of functions © zt ª A0(X0,X1) := F ∈ A(X0,X1): F(z) = u(e ), u ∈ A, rank(u) < ∞ is dense in A(X0,X1). This is also true for

A00(X0,X1) := {F ∈ A(X0,X1): Range(F) ⊆ X0 ∩ X1} .

3. Interpolation of Lp spaces

The interpolation of Lp spaces by the γ-method follows basically the arguments of the proof for the complex method. Two technical lemmas are needed.

Lemma 3.1. For 1 ≤ p < ∞, γ(H,Lp(Ω,A)) and Lp(Ω, γ(H,A)) are isomorphic Banach spaces. Pn Proof. A family of ﬁnite dimensional maps T (ω) = j=1 fj(ω) ⊗ en with fj ∈ Lp(Ω,A) can be thought of as an element of both spaces, Lp(Ω, γ(H,A)) and γ(H,Lp(Ω,A)). With Kahane’s inequalities [4] we obtain: Z kT kp = kT ωkp dµ(ω) Lp(Ω,γ(H,A)) γ(H,A) Ω Z µZ ° ° ¶ °X °p ∼ ° T ω(en)gn(s)° dP (s) dµ(ω) Ω S A Z µZ ° ° ¶ °X °p = ° T ω(en)gn(s)° dµ(ω) dP (s) S Ω A Z ° ° °X °p = ° T ω(en)gn(s)° dP (s) S Lp(Ω,A) ∼ kT kp , γ(H,Lp(Ω,A)) where ∼ stands for ”equivalent norms”. Since such elements T are dense in both spaces, the claim follows. ¤ The second lemma is essentially Hadamard’s three line lemma.

Lemma 3.2. For F ∈ A0(X0,X1), we have γ 1 2 1−θ θ kF(θ)kθ ≤ (2π) kF0kγ0 kF1kγ1 . Proof. Note that the analytic function

Φ(z) = (log kF0kγ0 ) (1 − z) + (log kF1kγ1 )z has real part (1 − θ) log kF0kγ0 + θlogkF1kγ1 for z = θ + it. Let θ ∈ (0, 1), −Φ(z) then by lemma 2.2 e F(z) is a function in A(X0,X1) and h i 1 − −Φ(θ) γ − log kFj kγ (2π) 2 ke F(θ)k ≤ max ke j Fjkγ θ j=0,1 j h i − log kFj kγ ≤ max e j kFjkγ ≤ 1. j=0,1 j 5

With all this we have 1 γ − 2 log k(2π) F(θ)kθ ≤ <Φ(θ) ≤ (1 − θ) log kF0kγ0 + θ log kF1kγ1 . This gives the desired inequality. ¤

Corollary 3.1. For x ∈ X0 ∩ X1, γ 1 2 1−θ θ kxkθ = (2π) inf{kF0kγ0 kF1kγ1 : F(θ) = x}.

Theorem 3.1. Assume that A0 and A1 are Banach spaces and that 1 ≤ p0 < 1 1−θ θ ∞, 1 ≤ p1 < ∞, θ ∈ (0, 1), if = + then p p0 p1 γ γ (Lp0 (Ω,A0),Lp1 (Ω,A1))θ = Lp(Ω, (A0,A1)θ ) with equivalent norms.

Proof. If we denote S the space of simple functions with values in A0∩A1, then γ S is dense in Lp0 (Ω,A0) ∩ Lp1 (Ω,A1) and thus in (Lp0 (Ω,A0),Lp1 (Ω,A1))θ γ γ and then also in Lp(Ω, (A0,A1)θ ) simply because A0∩A1 is dense in (A0,A1)θ . So it is enough to consider functions a in S.

1 Let us see that kak γ ≤ (2π) 2 Kkak γ where the (Lp0 (Ω,A0),Lp1 (Ω,A1))θ Lp(Ω,Aθ ) constant K comes from Kahane’s inequality [4]. Given a simple function, a ∈ S, and given ε > 0 there are functions g(·, ω) ∈ A0(A0,A1) for ω ∈ Ω

(which are also steps functions with respect to ω) such that kg(·, ω)kA0 ≤ (1 + ε)ka(ω)k γ with x ∈ Ω and g(θ, ω) = a(ω). Aθ

We set Ã !p( 1 − 1 )(θ−z) p0 p1 ka(ω)kAγ F(z, ω) := g(z, ω) θ kakL (Ω,Aγ ) p θ . Using Lemma 3.1, we have that Z kF(i·, ·)kp0 ∼ kF(i·, ω)kp0 dµ(ω) γ(Lp0 (Ω,A0)) γ(A0) ZΩ ≤ kg(i·, ω)kp0 A(ω)dµ(ω) γ(A0) Ω p0 p0 ≤ (1 + ε) kak γ , Lp(Ω,Aθ )

1 1 1 1 p0p( − )θ −p0p( − )θ p0 p1 p0 p1 where A(ω) = ka(ω)k γ kak γ . Aθ Lp(Ω,Aθ ) The same idea works to show that

kF(1 + i·, ·)k ≤ K(1 + ε)kak γ ; γ(Lp1 (Ω,A1)) Lp(Ω,Aθ ) thus, it follows that:

1 kak γ ≤ (2π) 2 Kkak γ . (Lp0 (Ω,A0),Lp1 (Ω,A1))θ Lp(Ω,Aθ ) 6

Conversely, since a ∈ S there are functions F(·, ω) ∈ A0(A0,A1) with F(θ, ω) = a(ω) for ω ∈ Ω (which are step functions with respect to ω) and

µZ ¶ 1 µZ ¶ 1 p p p p kak γ = ka(ω)k γ dµ(ω) ≤ kF(θ, ω)k γ dµ(ω) Lp(Ω,Aθ ) A A Ω θ Ω θ

Using Lemma 3.2 and H¨older’sinequality with 1−θ + θ = 1, we obtain: pp0 pp1

µZ ¶ 1 p 1 (1−θ)p θp ≤ (2π) 2 kF(·, ω)k kF(·, ω)k dµ(ω) γ(A0) γ(A1) Ω µZ ¶ 1−θ µZ ¶ θ p p 1 p 0 p 1 ≤ (2π) 2 kF(·, ω)k 0 dµ(ω) kF(·, ω)k 1 dµ(ω) γ(A0) γ(A1) Ω Ω 1 2 1−θ θ ≤ (2π) KkFk kFkγ(L (Ω,A )) γ(Lp0 (Ω,A0)) p1 1 1 ≤ (2π) 2 KkFk , A(Lp0 (Ω,A0)Lp1 (Ω,A1))

−2jt where we applied Lemma 3.2 to the operators in γ(L2(e dt),Lpj (Ω,Aj)) defined by F. This finishes the proof. ¤ Following [4], we call by Rad(X) the completion of the space of finite sequences, (xn) ⊆ X under the norm Ã ! 1 X∞ Z 1 X∞ 2 2 k(xn)k := Ek rnxnkX = k rn(t)xnkX dt . n=1 0 n=1

As usually rn(·) denotes the nth-Rademacher function.

Corollary 3.2. If Xi is B-convex for i = 0, 1, then

γ γ (Rad(X0), Rad(X1))θ = Rad(Xθ ) with equivalent norms.

Proof. B-convexity implies that the canonical projections P : L2([0, 1],Xi) −→ Rad(Xi) are bounded for i = 0, 1, by a result of G. Pisier (see [4]) for dyadic simple functions f : [0, 1] → X0 ∩ X1 the projection P is defined by X µZ 1 ¶ Pf := rn(u)f(u)du rn. n 0 γ By [13, Th. 1.2.4], P defines a projection of L = (L2([0, 1],X0),L2([0, 1],X1))θ onto the closed span R of the finite Rademacher sequences in L. Furthermore, γ R is isomorph to (Rad(X0), Rad(X1))θ . By theorem 3.1 the norms of L and γ γ L2([0, 1],Xθ ) are equivalent on R and therefore R is isomorphic to Rad(Xθ ). ¤ 7

1 Corollary 3.3. If Xi has type pi > 1 and cotype qi for i = 0, 1 and p = 1−θ θ 1 1−θ θ γ + , = + then (X0,X1) has type p and cotype q. p0 p1 q q0 q1 θ

Proof. That Xi has type pi means that the natural mapping lpi (Xi) −→ Rad(Xi) is bounded, so we have that the following operator is bounded too:

γ γ γ γ lp(Xθ ) = (lp0 (X0), lp1 (X1))θ −→ (Rad(X0), Rad(X1))θ = Rad(Xθ ) A similar argument works for the cotype. ¤

Theorem 3.1 admits the following generalization:

Theorem 3.2. Let A0, A1 be Banach spaces and let 1 ≤ p0 < ∞, 1 ≤ p1 < (1−θ)p θp ∞, θ ∈ (0, 1), if 1 = 1−θ + θ and µ = µ p0 µ p1 then p p0 p1 0 1 γ γ (Lp0 (Ω(µ0),A0),Lp1 (Ω(µ1),A1))θ = Lp(Ω(µ), (A0,A1)θ ) with equivalent norms.

Proof. For a given F ∈ A(Lp0 (Ω(µ0),A0),Lp1 (Ω(µ1),A1)) we set

(1−z)p zp Fe(z, ω) = µ0(ω) p0 µ1(ω) p1 F(z, ω). The mapping F → Fe is clearly an isomorphism between

A(Lp0 (Ω(µ0),A0),Lp1 (Ω(µ1),A1)) and A(Lp0 (Ω(µ),A0),Lp1 (Ω(µ),A1)). Now the arguments in the preceding theorem with obvious modiﬁcations go through. ¤

4. Abstract Stein interpolation

In this section (X0,X1) and (Y0,Y1) are interpolation couples. Let us recall the following theorem that can be found in [14]:

Theorem 4.1. Let {Tz : z ∈ S} be a family of linear mappings Tz : X0∩X1 → Y0 + Y1 with the following properties:

(1) ∀x ∈ X0 ∩X1 the function T(·)x : S → Y0 +Y1 is continuous, bounded, and analytic on S. (2) For j = 0, 1, x ∈ X0 ∩ X1, the function R 3 s → Tj+isx ∈ Yj is continuous and

Mj := sup{kTj+isxkYj : s ∈ R, x ∈ X0 ∩ X1, kxkXj ≤ 1} < ∞.

Then for all t ∈ [0, 1], Tt(X0 ∩ X1) ⊂ Y[t] and

1−t t kTtxk[t] ≤ M0 M1kxk[t] for all x ∈ X0 ∩ X1. In order to formulate the analogous Stein interpolation result for the γ- method, we need the notion of γ-boundedness. A set of operators τ ⊆ B(X,Y ) 8 is γ-bounded, if there is a constant C such that for all T1, ..., Tn ∈ τ and x1, ..., xn ∈ X we have: Xn Xn Ek gkTkxkk ≤ CEk gkxkk. k=1 k=1 The smallest constant C possible in this inequality is denoted by R(τ).

Theorem 4.2. Let {Tz : z ∈ S} be a family of linear mappings Tz : X0∩X1 → Y0 + Y1 with the following properties:

(1) ∀x ∈ X0 ∩X1, the function T(·)x : S → Y0 +Y1 is continuous, bounded and analytic on S. (2) For j = 0, 1, x ∈ X0 ∩ X1, the function s 7→ Tj+isx ∈ Yj is strongly measurable, continuous and the operators Tj+is ∈ B(Xj,Yj) thereby deﬁned on Yj satisfy for j = 0, 1:

Mj = R ({Tj+it : t ∈ R}) < ∞. Then, for every θ ∈ (0, 1), we have: γ (1) Tθ (X0 ∩ X1) ⊆ Yθ . 1−θ θ (2) kTθxk γ ≤ M M1 kxk γ for all x ∈ X0 ∩ X1. Yθ 0 Xθ

Proof. It is enough to consider M0 = M1 = 1, otherwise replacing T by z−1 −z M0 M1 T (z). We deﬁne Tθ : A00(X0,X1) → A(Y0,Y1) by F 7→ T(·)F(·). Indeed , by the γ-boundedness of {Tj+it : t ∈ R} and an estimate in [6] we have

kT F(j + i·)k ≤ kF(j + i·)k j+i· A(Y0,Y1) A00(X0,X1) for j = 0, 1. The set {F ∈ A00(X0,X1): F(θ) = 0} is dense in {F ∈ A(X0,X1): F(θ) = 0} (compare [14, Th. 1.1]) and T(·) maps the first set into {F ∈ A(X0,X1): F(θ) = 0}. Therefore for x ∈ X0 ∩ X1 and F ∈ A00(X0,X1) with F(θ) = x we have: ° ° ° ° kT (θ)xkY γ ≤ T(·)F(·) ≤ kFk θ A(Y0,Y1) A00(X0,X1) and taking infimum over such all F gives the result. ¤ Remark: Let us state a special case where the γ-boundedness assumption of theorem 4.2 can be checked directly. If the functions s → Tj+is ∈ B(Xj,Yj) are differentiable and satisfy Mf = kT k j Zj p 2 j 2 −s + e kTj+iskB(Xj ,Yj ) j + 4s e ds ZR j d −s2 + e k Tj+iskB(Xj ,Yj )e ds < ∞ , R ds for j = 0, 1 then, for x ∈ X0 ∩ X1 we still can conclude: 1−θ θ kTθxk γ ≤ Mf Mf kxk γ . Yθ 0 1 Xθ 9

−θ2 z2 Proof. Consider the analytic function Sz = e e Tz with Sθ = Tθ. Our assumption implies that Z d f kSjk + k Tj+iskB(Xj ,Yj )ds ≤ Mj. R ds

By [8, Sect. 3] we conclude that R({Sj+it : t ∈ R}) ≤ Mfj and we may apply the theorem. ¤

5. Interpolation of γ(H,X) We are ready to obtain our stability results for the class of γ-radonifying operators.

Theorem 5.1. Let H0,H1 be separable Hilbert spaces with H0 ∩ H1 dense in Hi for i = 0, 1. If X0, X1 are B-convex, then

(γ(H0,X0), γ(H1,X1))[θ] = γ((H0,H1)[θ], (X0,X1)[θ]) with equivalent norms.

Proof. Step 1 : Representation of (H0,H1)[θ].

Let (en) be an orthonormal basis of H0 ∩ H1 with respect to the inner product (h, g) = (h, g)0 +(h, g)1 where (·, ·)j is the inner product of Hj so that (en) is also a complete orthogonal system in Hj for j = 0, 1. To construct such (en) inductively, assume that e1, ...eN are already chosen. Then pick some h0 2N in the kernel of the map h ∈ H0 ∩ H1 → {(h, en)j}n=1,...N;j=0,1 ∈ C and normalize h0 with respect to (·, ·) to obtain eN+1. Then clearly ( ) X X 2 2 Hj = βnen : |βn| αj,n < ∞ , n n where αj,n = kenkHj and we get ( ) X X 2 2(1−θ) 2θ (H0,H1)[θ] = βnen : |βn| α0,n α1,n < ∞ n n with orthonormal basis (α(1−θ)αθ )−1e , n ∈ N. To see this, identify H ∩H 0,nP 1,n n 0 1 with l2 via the maps βn → βnen and use e.g. [1, Sect. 5.6].

Step 2 : Operators of dimension m.

m m By H0 we denote the subspace of H0 spanned by the e1, ..., em. Let m Pm Rad (Xj) be the span of the n=1 rnxn with x1, ..., xm ∈ Xj. Note that γ(H0,X0)∩γ(H1,X1) and γ(H0,X0)+γ(H1,X1) can be seen as continuously embedded subspaces of γ(H0∩H1,X0+X1) (due to the operator ideal property of γ) and Rad(X0)∩Rad(X1) and Rad(X0)+Rad(X1) are naturally embedded into L2([0, 1],X0 + X1). For a ﬁxed m ∈ N and all z ∈ S we deﬁne maps m m m m m Tz : γ(H0 ,X0) ∩ γ(H1 ,X1) −→ Rad (X0) + Rad (X1) 10

¡ ¢m by T m(S) := α−1+zα−z S(e ) . For z = j + it with j ∈ [0, 1] and t ∈ R z 0,n 1,n n n=1 m and all S ∈ γ(Hj ,Xj) we obtain by the contraction principle Xm m −1+j+it −j+it m kTj+it(S)kRad (Xj ) ≤ Ek rnα0,n α1,n S(en)kXj n=1 Xm −1+j −j ≤ 2Ek r α α S(e )k = 2kSk m . n 0,n 1,n n Xj γ(Hj ,Xj ) n=1 The remaining assumptions of theorem 4.2 follow directly from the special m form of Tz . Hence for θ ∈ (0, 1) there is a map m m m m m Tθ :(γ(H0 ,X0), γ(H1 ,X1))[θ] → (Rad (X0), Rad (X1))[θ] m with kTθ k ≤ 2. By interpolating in a similar way the ’inverse’ functions m −1 m m m m (Tz ) : Rad (X0) ∩ Rad (X1) → γ(H0 ,X0) + γ(H1 ,X1) m −1 which assign to (x1, ..., xm) the operator S = (Tz ) (xn) given by S(en) = 1−z z m α0,n α1,nxn with n = 1, ..., m it follows that Tθ is an isomorphism with m −1 k(Tθ ) k ≤ 2 for all θ ∈ (0, 1).

Step 3 : Inﬁnite dimensional operators.

Since X0 and X1 are B-convex Banach spaces, we have

(Rad(X0), Rad(X1))[θ] = Rad(X[θ]) with equivalent norms (see the argument in the proof of Corollary 3.2). Notice that we have consistent retractions m Pj : Rad(Xj) → Rad (Xj) m given by (xn)n∈N → (xn)n=1 with kPjk = 1 and hence by [13, Sect. 1.2.4] there is a constant M which depends on X but not on m or θ so that for x1, ..., xm ∈ X0 ∩ X1 −1 m m m m M k(xn)kRad (X[θ]) ≤ k(xn)k(Rad (X0),Rad (X1))[θ] ≤ Mk(xn)kRad (X[θ]).

Next we observe that the consistent retractions Qj : γ(Hj,Xj) → m γ(Hj ,Xj) given by S → S|span[e1,...,em] have norm 1 and, again by m m [13, Sect. 1.2.4], ensure that the inclusion (γ(H0 ,X0), γ(H1 ,X1))[θ] ⊆ (γ(H0,X0), γ(H1,X1))[θ] becomes an isometric embedding. m −1 For x1, ..., xm ∈ X0 ∩ X1 and the associated operator S = (Tθ ) (xn) given 1−θ θ by S(en) = α0,n α1,nxn with xk = 0 for k = m + 1, m + 2, ... we obtain kSk = kSk m m (γ(H0,X0),γ(H1,X1))[θ] (γ(H0 ,X0),γ(H1 ,X1))[θ] Xm M m ∼ k(xn)kRad (X[θ]) = Ek rnxnkX[θ] n=1 Xm θ−1 −θ = Ek rnα0,n α1,nS(en)kX[θ] = kSkγ(H[θ],X[θ]). n=1 11

Hence the norms of γ(H[θ],X[θ]) and (γ(H0,X0), γ(H1,X1))[θ] are equivalent on all ﬁnite dimensional operators H0 ∩ H1 → X0 ∩ X1. Since these operators are dense in both spaces the result follows. ¤

For the γ-method this results holds under an additional assumption: Deﬁnition 1. A Banach space has Pisier’s property (α) e.g. [8] if for all αi,j ∈ C we have X X 0 0 0 0 E Ek αijrirjxijk ≤ sup |αij |E Ek rirjxijk i,j i,j i,j 0 where (ri) and (rj) are two independent Bernoulli sequences.

Theorem 5.2. Let H0,H1 be separable Hilbert spaces with H0 ∩ H1 dense in Hi for i = 0, 1. If X0 and X1 are B-convex and have property (α) then γ γ γ (γ(H0,X0), γ(H1,X1))θ = γ((H0,H1)θ , (X0,X1)θ ) with equivalent norms. γ Proof. First we recall that (H0,H1)θ = (H0,H1)[θ] with equivalent norms for Hilbert spaces Hj, j = 0, 1. Using theorem 4.2 in place of theorem 4.1 we can now repeat the proof of theorem 5.1. Property (α) comes in m since we have to check the γ-boundedness of the sets {Tj+it : t ∈ R} in m m B(γ(Hj ,Xj), Rad (Xj)) with a constant independent of m. Indeed for m S1, ..., SN ∈ γ(Hj ,Xj) and t1, ..., tN ∈ R we have

XN Xm XN 0 0 −1+j+itn −j−itn m Ek rnTj+itn (Sn)kRad (Xj ) = EE k rlrnα0,n α1,n Sn(el)k n=1 l=1 n=1 Xm XN 0 0 −1+j −j ≤ CEE k rlrnα0,n α1,nSn(el)k l=1 n=1 XN ≤ CEk r S k m , n n γ(Hj ,Xj ) n=1 where the constant C only depends on the property (α) constant of X [8, Section 4]. ¤

Corollary 5.1. Let H0,H1 be separable Hilbert spaces with H0 ∩ H1 dense in Hi for i = 0, 1. IfX0, X1 are Hilbert spaces and S2 denotes the Hilbert- Schmidt class, then:

(S2(H0,X0), S2(H1,X1))[θ] = S2((H0,H1)[θ], (X0,X1)[θ]),

γ γ γ (S2(H0,X0), S2(H1,X1))θ = S2((H0,H1)θ , (X0,X1)θ ). This corollary improves a result of [3, Prop. 5.1.] which states: 12

Let H1 ⊂ H0, X1 ⊂ X0 be Hilbert spaces with H1 dense in H0, X1 dense in X0 and the inclusions being compact, then ¡ ¢ (S2(H0,X0), S2(H1,X1))[θ] = S2 (H0,H1)[θ], (X0,X1)[θ] . with equal norms.

We recall that we obtain isometries in Corollary 5.1 since all the constants appearing in the proofs of theorems 5.1 and 5.2 derive from the B-convexity constants and Kahane’s contraction principle and will be 1 in the Hilbert case. For a more direct proof of Corollary 5.1 see [12].

In [6] ”Sobolev spaces” of the following kind were introduced: γs(Rn,X) := γ(Hs,X) where Hs is the completion of S(Rn), the Schwartz class, in the norm −1 2 s 2 b n kfks = kF [(1 + | · | ) f(·)]kL2(R ). Theorems 5.1 and 5.2 allow us to extend the well known interpolation results for Hs to γs(Rn,X):

Corollary 5.2. Let X0, X1 be B-convex Banach spaces. If s = (1−θ)s0 +θs1 for sj ∈ R, j = 0, 1, then

s0 n s1 n s n (γ (R ,X0), γ (R ,X1))[θ] = γ (R ,X[θ]) with equivalent norms. The same is true for γ-interpolation method if X0 and X1 have in addition property (α). The same is of course true if we replace the Hs in the deﬁnition of γs(Rn,X) by another scale of Sobolev spaces, e.g. the scale deﬁned by Riesz potentials instead of Bessel potentials. 13

6. Comparison with the real interpolation method

If (X0,X1) is an interpolation couple so that X0 and X1 have ﬁnite cotype, γ then the interpolation space Xθ has also a discrete description (cf. [6, 7]): γ P Xθ consists of all x ∈ X0+X1 which can be represented as a sum x = n∈Z xn convergent in X0 + X1 with xn ∈ X0 ∩ X1 satisfying ( ) X kxkθ = inf k(xn)k : x = xn, xn ∈ X0 ∩ X1 < ∞ n∈Z where X n(j−θ) k(xn)k = max Ek rn2 xnkX (∗) j=0,1 j n∈Z γ deﬁnes an equivalent norm on Xθ . If we replace in (∗) the Rademacher averages with:

Ã ! 1 X pj n(j−θ) pj max k2 xnk j=0,1 Xj n∈Z we obtain a well known characterization of the Lions-Peetre interpolation space (X0,X1)θ,p0,p1 , [9, Chapitre 2]. By using the well known equality 1 1−θ θ (X0,X1)θ,p ,p = (X0,X1)θ,p with = + , see [11], and the deﬁni- 0 1 p p0 p1 tion of type and cotype we obtain immediately the following observation:

Theorem 6.1. Let (X0,X1) be an interpolation couple so that Xj has 1 1−θ θ Rademacher type pj ≥ 1and cotype qj < ∞ for j = 0, 1, then for = + p p0 p1 , 1 = 1−θ + θ with θ ∈ (0, 1) we have continuous embeddings q q0 q1 γ (X0,X1)θ,p ⊂ (X0,X1)θ ⊂ (X0,X1)θ,q. A corresponding result for the complex method making the stronger assumption of Fourier type is due to Peetre [10]. Acknowledgement: The ﬁrst author want to express his sincere gratitude to his colleagues in Karlsruhe, specially to B. Haak and J. Zimmerschied for endless hours of mathematics and help. To L. Weis for the opportunity to stay in Karlsruhe and particularly for sharing his ideas with me. To all of them thanks for such a nice atmosphere where to work. To M.J. Carro for some fruitful comments about the text. Finally, special thanks to my advisor J.M.F. Castillo for helping me with some mistakes and encourage me to ﬁnish it.

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Departamento de Matematicas,´ Universidad de Extremadura, Avenida de Elvas s/n, 06071 Badajoz, Spain. Mathematisches Institut I, Universitat¨ Karlsruhe, 76128 Karlsruhe, Germany. [email protected] [email protected]