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Interpolation of Banach Spaces by the Γ-Method

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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 op- erators) 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¶³a-Dav¶³a[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 ¯rst 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-classi¯cation: 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 in¯nite dimensional separable Hilbert space and will reserve the letter H for this. In what follows, F(Y; X) will denote the space of all ¯nite rank operators from Y into X. Given (ej) an orthonormal basis in H, we de¯ne the γ-norm of a ¯nite rank operator u 2 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 0 1 1 Xn Z Xn 2 @ 2 A kukγ := Ek gjxjkX = k gj(!)xjkX dP (!) ; j=1 ­ j=1 where fgng is a sequence of standard Gaussian variables on a ¯xed probability space (­; σ; P ). To extend this notation to operators of possibly in¯nite rank we proceed as follows. If v : H ! X is any bounded operator, we de¯ne kvkγ := sup fkvukγ : u 2 F(H; H); kuk · 1g and then form the spaces γ+(H; X) := fv 2 L(H; X): kvkγ < 1g 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 2 A i® u 2 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 2 A+ i® u 2 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 de¯ne the interpolation space (X0;X1)θ , which we will briefly γ call Xθ , as follows : if x 2 X0 + X1 we introduce the norm γ kxkθ = inf fkukA : u 2 A; u(eθ) = xg ; θt γ where eθ denotes the function eθ(t) = e . Now (X0;X1)θ is the space of all γ γ x 2 X0 + X1 such that kxkθ < 1 equipped with k ¢ kθ . The same de¯nitions 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) ½ Y0 and S(X1) ½ 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 fz : 0 < <(z) < 1g 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 2 A. We de¯ne a norm on this space by kFkA = kukA. Then we have the formula γ kxkθ = inf fkFkA : F(θ) = xg : Most of the proofs involved in the paper use density arguments, so it is interesting to observe that: Lemma 2.1. The set of ¯nite-rank operators is dense in A. As a consequence we get: γ Corollary 2.1. For θ 2 (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) = lim»!j F(» + 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 2 γj(Xj) := γ(L2(e dt);Xj) for j = 0; 1. Then F 2 A and 1 kFkA = (2¼) 2 max kFjkγ : j=0;1 j ¡2jt By Fj 2 γj(Xj) we mean that the operator uj : L2(e dt) ! Xj de¯ned by Z uj(h) := h(t)Fj(t)dt R 4 ¡2jt for h 2 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 2 A(X0;X1): F(z) = u(e ); u 2 A; rank(u) < 1 is dense in A(X0;X1). This is also true for A00(X0;X1) := fF 2 A(X0;X1): Range(F) ⊆ X0 \ X1g : 3. Interpolation of Lp spaces The interpolation of Lp spaces by the γ-method follows basically the ar- guments of the proof for the complex method. Two technical lemmas are needed. Lemma 3.1. For 1 · p < 1, γ(H; Lp(­;A)) and Lp(­; γ(H; A)) are iso- morphic Banach spaces. Pn Proof. A family of ¯nite dimensional maps T (!) = j=1 fj(!) ­ en with fj 2 Lp(­;A) can be thought of as an element of both spaces, Lp(­; γ(H; A)) and γ(H; Lp(­;A)). With Kahane's inequalities [4] we obtain: Z kT kp = kT!kp d¹(!) Lp(­,γ(H;A)) γ(H;A) ­ Z µZ ° ° ¶ °X °p » ° T!(en)gn(s)° dP (s) d¹(!) ­ S A Z µZ ° ° ¶ °X °p = ° T!(en)gn(s)° d¹(!) dP (s) S ­ A Z ° ° °X °p = ° T!(en)gn(s)° dP (s) S Lp(­;A) » kT kp ; γ(H;Lp(­;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 2 A0(X0;X1), we have γ 1 2 1¡θ θ kF(θ)kθ · (2¼) kF0kγ0 kF1kγ1 : Proof. Note that the analytic function ©(z) = (log kF0kγ0 ) (1 ¡ z) + (log kF1kγ1 )z has real part (1 ¡ θ) log kF0kγ0 + θlogkF1kγ1 for z = θ + it. Let θ 2 (0; 1), ¡©(z) then by lemma 2.2 e F(z) is a function in A(X0;X1) and h i 1 ¡ ¡©(θ) γ ¡ log kFj kγ (2¼) 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(2¼) F(θ)kθ · <©(θ) · (1 ¡ θ) log kF0kγ0 + θ log kF1kγ1 : This gives the desired inequality.
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