On a result of Peetre about interpolation of operator spaces

Publicacions Matem`atiques, Vol. 44 (2000), 457–481 ON A RESULT OF PEETRE ABOUT INTERPOLATION OF OPERATOR SPACES Fernando Cobos∗and Teresa Signes∗∗ Dedicated to Professor Jaak Peetre with affection and deep admiration for his useful and creative work during the last 40 years Abstract We establish interpolation formulæ for operator spaces that are components of a given quasi-normed operator ideal. Sometimes we assume that one of the couples involved is quasi-linearizable, some other times we assume injectivity or surjectivity in the ideal. We also show the necessity of these suppositions. 1. Introduction Let B=(B0,B 1) be a Banach couple. The Peetre K-functional is defined by K(t, b) = inf{b0B0+tb1B1:b=b0+b1,b i∈Bi} where t>0 and b∈B0+B1. This function plays a major role in Interpolation Theory (see e.g. [15], [1], [21]); it can be computed explicitly in many concrete cases, and sometimes it coincides with well-known objects in Analysis. For example, in the case of the couple of Lebesgue spaces (L1,L ∞), one can prove that K(t, f)=t 0 f∗(s)ds where f∗is the non-increasing rearrangement of fon (0,∞). For the couple (C0,C1), formed by the space C0of bounded uniformly continuous functions on the real line and the space C1of functions with 1991 Mathematics Subject Classification. 46B70, 47D50. ∗Supported in part by DGES (PB97-0254). ∗∗ Supported by grant FP-95 of Ministerio de Educaci´on y Ciencia and by DGES (PB97-0254). 458 F. Cobos, T. Signes derivatives in C0, it turns out that K(t, f)≈w(t, f), that is to say, K(t, f) is equivalent to the modulus of continuity w(t, f) of f. A Banach couple B=(B0,B 1) is said to be quasi-linearizable if for any b∈B0+B1one can get in a linear way an optimum decomposition of bfor the K-functional. We give the precise definition and examples in Section 2. This notion was introduced by Peetre in [16] and [17]. It is rather restrictive, but many interesting couples satisfy it (see [21]). Moreover, the idea of quasi-linearization is important for the computation of the K-functional in other special instances (see, for example, [12]). Peetre proved in [16] that if Bis quasi-linearizable then for any Banach space Aand any positive measurable function f=f(t), the inclusion L(A, Bf,∞;K)→(L(A, B0),L(A, B1))f,∞;K (1.1) is valid, with embedding constant depending only on B. Here the space Bf,∞;Kis defined similarly to the real interpolation space Bθ,∞ realized as a K-space, but replacing the function tθby a more general function f(t) (see Section 2 for more details). The interest of formula (1.1) is due to its being a converse inclusion to the one given by the interpolation property of the real method, namely (L(A, B0),L(A, B1))θ,1→L(A, Bθ,1). It turns out that quasi-linearizable couples Bcan be characterized as those for which (1.1) holds (see [16]). The aim of this paper is to continue the research on embedding (1.1) in several directions. After reviewing some basic notions in Section 2, we show in Section 3 that (1.1) remains valid if we replace the class of bounded linear operators by any quasi-normed operator ideal J.Itis also possible to substitute Bf,∞;Kby any intermediate space Bwith respect to B. In other words, it holds J(A, B)→(J(A, B0),J(A, B1))ψ,∞;K.(1.2) Here ψ=ψ(t) is a function associated in a natural way to Band B.We study then the “dual” situation in which the couple Bis in the front, and we prove that J(B,A)→(J(B0,A),J(B1,A))ρ∗,∞;K (1.3) On A Result of Peetre 459 where ρ=ρ(t) is another function naturally associated to Band B and ρ∗(t)=1/ρ(t−1). Moreover, the validity of (1.3) implies the quasilinearizability of B. Other embeddings that arise interpolating by the J-method are established in Section 3 as well. In Section 4 we combine the previous results with some arguments of bilinear type to identify some interpolation spaces between spaces of bounded operators, spaces of nuclear operators and spaces of tensor products. Previous results in this direction are due to Kouba [11] and to Ovchinnikov [13] and [14]. We also establish some interpolation formulæ for approximation numbers and some embeddings for operator ideals defined by approximation numbers. Finally, in Section 5, we return to the J-formulæ, and we show that it is possible to dispense with the quasi-linearizability of Bprovided the operator ideal Jis injective or surjective, and that these conditions on Jare essential for the results. 2. Preliminaries Let B=(B0,B 1) be a Banach couple, that is, B0and B1are Banach spaces continuously embedded in some Hausdorff topological vector space. We equip B0+B1[respectively B0∩B1] with the norm K(1,·) [respectively J(1,·)] where for t>0 K(t, ·)=K(t, ·;B0,B 1) and J(t, ·)=J(t, ·;B0,B 1) are the Peetre functionals, defined by K(t, b) = inf{b0B0+tb1B1:b=b0+b1,b i∈Bi} and J(t, b) = max{bB0,tbB1}. A Banach couple B=(B0,B 1) is said to be quasi-linearizable if there exist two families of operators Vj(t)∈L(B0+B1,B j), j=0,1, 0 <t< ∞and a constant ksuch that V0(t)+V1(t)=I(identity mapping in B0+B1),(2.1) V0(t)bB0≤ktjbBj,b∈Bj(j=0,1),(2.2) V1(t)bB1≤ktj−1bBj,b∈Bj(j=0,1),(2.3) (see [16]or[17]). For instance, the couples (Lp(w0),L p(w1)) of weighted Lp-spaces, or (Lp(Rn),W p(Rn)) satisfy this condition. More generally, if Ais a Banach space and D(Λ) is the domain of the infinitesimal generator 460 F. Cobos, T. Signes of a strongly continuous semi-group of operators in A, then the couple (A, D(Λ)) is quasi-linearizable. Another example is the couple of Besov spaces (Bs0 p,q(Rn),Bs1 p,q(Rn)). On the other hand, if p0=p1, the couple (Lp0,L p1) is not quasi-linearizable. See [16] and [21] for more details. We say that a Banach space Bis intermediate with respect to the couple B=(B0,B 1)ifB0∩B1→B→B0+B1, with continuous inclusions. The position of Bwith respect to Bcan be described by the functions ψ(t)=ψ(t, B;B) = sup{K(t, b):bB=1}(2.4) and ρ(t)=ρ(t, B;B) = inf{J(t, b):b∈B0∩B1,bB=1}.(2.5) These functions have been introduced by Cobos, Cwikel and Matos in [2] and they are variants of functions studied by Dmitriev [5] and Pustylnik [20]. We have 0 <ρ(t), ψ(t)<∞for all t>0; the functions ψ(t) and ρ(t) being non-decreasing, while ψ(t)/t,ρ(t)/t are non-increasing. As an example, let Bbe a rearrangement invariant space on a σ-finite measure space (Ω,µ) and let ϕBbe its fundamental function, i.e. ϕB(t)=χEB where E⊆Ω with µ(E)=t. Then Bis an intermediate space with respect to the couple (L1,L ∞) and ψ(t, B;L1,L ∞)=ρ(t, B;L1,L ∞)=t/ϕB(t) (see [20]). A class Jof bounded linear operators between Banach spaces is said to be an operator ideal if each component J∩L(A, B)=J(A, B)isa linear subspace of L(A, B) that contains the finite rank operators and satisfies that STR ∈J(E,F) whenever R∈L(E,A), T∈J(A, B) and S∈L(B,F). A non-negative function τ:J−→[0,∞) is called a quasi-norm on J if τhas the following properties: τ(h⊗b)=hA∗bBfor h∈A∗and b∈B.(2.6) τ(S+T)≤c(τ(S)+τ(T)) for S, T ∈J(A, B),(2.7) where c=cτ≥1 is a constant. τ(STR)≤SB,F τ(T)RE,A for R∈L(E,A),(2.8) T∈J(A, B) and S∈L(B,F). On A Result of Peetre 461 It follows from (2.8) that τ(λT)=|λ|τ(T), and (2.6) and (2.8) yield that T≤τ(T) (see [18]). Sometimes we write τ(T)=τ(TA,B)to emphasize that Tis considered as acting from Ainto B. A quasi-normed operator ideal is an operator ideal Jequipped with a quasi-norm τso that all components J(A, B) are complete with respect to the induced metric. If cτ= 1, then [J,τ] is said to be a normed operator ideal. Clearly, the class of all bounded linear operators with the operator norm [L,·] is an example of a normed operator ideal. Another example is the ideal of nuclear operators [N,· 1]. Recall that T∈L(A, B)is said to be nuclear if there are sequences (hn)⊂A∗and (yn)⊂Bsuch that Tx =∞ n=1 hn(x)ynand ∞ n=1 hnA∗ynB<∞. The norm on Nis given by T1= inf ∞  n=1 hnA∗ynB where the infimum is taken over all representations T=∞ n=1 hn⊗yn as above. More examples will be given in Sections 4 and 5. We refer to [18] and [4] for other details on operator ideals. In successive sections we shall interpolate operator spaces that are components of a given ideal, so they are quasi-Banach spaces. The concept of quasi-Banach couple X=(X0,X 1) is analogous to the Banach case, but replacing Banach spaces by quasi-Banach spaces. The Kand J-functionals are defined in the same way. Now X0∩X1and X0+X1 are only quasi-Banach spaces. If X=(X0,X 1) is a quasi-Banach couple, 0 <θ<1 and 0 <q≤∞, the real interpolation space Xθ,q =(X0,X 1)θ,q consists of all x∈X0+X1 which have a finite quasi-norm xθ,q =∞ 0 (t−θK(t, x))qdt t1/q if 0 <q<∞, xθ,q = sup t>0 {t−θK(t, x)}if q=∞ (see [1]or[21]). For some results we shall need to replace in the former definition the function tθby a general positive function g:(0,∞)−→ (0,∞). The space Xg,∞;K=(X0,X 1)g,∞;Kis formed of all x∈X0+X1which have a finite quasi-norm xg,∞;K= sup t>0K(t, x) g(t). 462 F. Cobos, T. Signes It is clear that Xg,∞;K→X0+X1, but it might happen that Xg,∞;K= {0}. Indeed, for any x∈X0+X1, it follows from min{1,t}xX0+X1≤ K(t, x) that sup t>0min{1,t} g(t)xX0+X1≤xg,∞;K. Consequently, the space Xg,∞;Kreduces to {0}if supt>0min{1,t} g(t)= ∞. Note that Peetre’s embedding (1.1) is trivial in this case. For this reason we shall assume in the following that the function gsatisfies sup t>0min{1,t} g(t)<∞.(2.9) This condition guarantees that X0∩X1→Xg,∞;Kbecause xg,∞;K= sup t>0K(t, x) g(t)≤sup t>0min{1,t} g(t)xX0∩X1. So (X0,X 1)g,∞;Kis an intermediate space with respect to X. Note that the function ψ(see (2.4)) associated to (X0,X 1)g,∞;Kand Xsatisfies ψ(t)=ψ(t, Xg,∞;K;X)≤g(t),t>0.(2.10) For g(t)=tθit holds Xg,∞;K=Xθ,∞. We shall also work with J-spaces. Let f:(0,∞)−→ (0,∞)beany positive function. Assume that the constant in the triangle inequality for Xjis cj. Put c= max{c0,c 1}and define pby the equation (2c)p=2. If 0 <q≤p, the space Xf,q;J=(X0,X 1)f,q;Jconsists of all sums x=∞ m=−∞ xm(convergence in X0+X1) where (xm)⊂X0∩X1and ∞  m=−∞ J(2m,x m) f(2m)q1/q <∞. We put xf,q;J= inf   ∞  m=−∞ J(2m,x m) f(2m)q1 q :x= ∞  m=−∞ xm   . The difficulty now is that the functional · f,q;Jis, in general, only a semi-quasi-norm. Indeed, for any x∈X0∩X1and any m∈Z,wehave xf,q;J≤J(2m,x) f(2m)≤max{1,2m} f(2m)xX0∩X1. On A Result of Peetre 463 Hence xf,q;J≤1 sup m∈Zf(2m) min{1,1 2m}xX0∩X1. In other words, if supt>0{f(t) min{1,1 t}} =∞then xf,q;J= 0 for every x∈X0∩X1. To avoid this obstruction we shall assume in the following that fsatisfies sup t>0f(t) min{1,1 t}<∞.(2.11) Then it is clear that X0∩X1is densely and continuously embedded in Xf,q;J. On the other hand, taking into account that the triangle inequality is valid in X0+X1with the constant c,ifx=∞ m=−∞ xmit follows from [1, Lemma 3.10.2], that xX0+X1≤21/p ∞  m=−∞ xmp X0+X11 p ≤21/p ∞  m=−∞ K(1,x m)q1 q ≤21/p ∞  m=−∞ min{1,1 2m}J(2m,x m)q1 q ≤21/p sup m∈Zf(2m) min{1,1 2m}∞  m=−∞ J(2m,x m) f(2m)q1 q . Since the supremum is finite by (2.11) we get that Xf,q;J→X0+X1. Let now ρ(t)=ρ(t, Xf,q;J;X) be the function ρassociated to the J-space Xf,q;Jand the couple X(see (2.5)). One can check easily that f(2m)≤ρ(2m),m∈Z.(2.12) Moreover, if fis non-decreasing, it follows that f(t)≤2ρ(t),t>0.(2.13) Let us also point out that if f(t)=tθthen the equivalence theorem (see [1]or[21]) yields that (X0,X 1)f,q;J=(X0,X 1)θ,q with equivalence of quasi-norms. The reason for the restriction on qin the definition of Xf,q;Jis that we are working with very general functions f. If we impose 464 F. Cobos, T. Signes stronger conditions on f, then Xf,q;Jmakes sense for 0 <q≤∞and still is an intermediate space with respect to X. The same happens with the K-spaces Xg,q;K(see [15]). We observe that Xg,∞;Kand Xf,q;Jare complete. Given any positive function f, we put f∗(t)=1/f(t−1). It turns out that fsatisfies (2.9) (resp. (2.11)) if and only if f∗satisfies (2.11) (resp. (2.9)). If f(t)=tθ, then f∗(t)=f(t)=tθ. Observe also that if Bis an intermediate space with respect to the couple B, then both functions ψ(t, B;B) and ρ(t, B;B) satisfy (2.9) and (2.11). 3. Operator spaces and quasi-linearizable couples We start with an estimate for the norms of the operators Vj(t) associated to a quasi-linearizable couple when they are considered as operators with Bas domain or Bas target space, where Bis any intermediate space. Lemma 3.1. Let B=(B0,B 1)be a quasi-linearizable couple, let Bbe an intermediate space with respect to Band let ψ(t)=ψ(t, B;B)and ρ(t)=ρ(t, B;B)be the functions associated to Band B. Then a) Vj(t)B,Bj≤kt−jψ(t),j=0,1,0<t<∞; b) V0(t)B1,B ≤(1 + k)t ρ(t),0<t<∞, V1(t)B0,B ≤(1 + k)1 ρ(t),0<t<∞. Proof: Since Vj(t)∈L(B0+B1,B j) and B→B0+B1, it is clear that Vj(t)∈L(B,Bj). In order to estimate the norm, first note that by (2.2) and (2.3) Vj(t)bBj≤kt−jK(t, b). So, if b∈Bwith bB=1,Vj(t)bBj≤kt−jψ(t). We go on to establish b). Assume j= 0. By the definition of ρand properties (2.1) to (2.3), for b∈B1we obtain V0(t)bB≤J(t, V0(t)b) ρ(t)=1 ρ(t)max{V0(t)bB0,tb−V1(t)bB1} ≤(1 + k)tbB1 ρ(t). The case j= 1 can be treated analogously. We can now extend the result by Peetre [16, Satz 3.1]. On A Result of Peetre 465 Theorem 3.2. Let B=(B0,B 1)be a quasi-linearizable couple, let B be an intermediate space with respect to Band let ψ(t)be the ψ-function associated to Band B. Assume [J,τ]is a quasi-normed operator ideal and let Abe any Banach space. Then J(A, B)→(J(A, B0),J(A, B1))ψ,∞;K. Moreover, there is a constant ddepending only on Bsuch that Tψ,∞;K≤dτ(TA,B). Proof: Given any T∈J(A, B) and any t>0, we have by (2.1), T= V0(t)T+V1(t)Twith Vj(t)T∈J(A, Bj) due to the ideal property of J. Whence, using Lemma 3.1/a, we derive that Tψ,∞;K= sup t>0K(t, T) ψ(t) ≤sup t>0τ([V0(t)T]A,B0)+tτ([V1(t)T]A,B1) ψ(t) ≤sup t>0τ(TA,B)V0(t)B,B0+tτ(TA,B)V1(t)B,B1 ψ(t) ≤2kτ(TA,B). Observe that (J(A, B0),J(A, B1)) is a quasi-Banach couple because J(A, Bj)→L(A, B0+B1), j=0,1. When the couple Bis in the front, we need to assume that Bis regular, i.e. B0∩B1is dense in B0and in B1, in order to guarantee that (J(B0,A),J(B1,A)) is a quasi-Banach couple. Indeed, under this extra supposition we have J(Bj,A)→L(Bj,A)→L(B0∩B1,A),j=0,1. The next result refers to this “dual” case when Bis in the front. Theorem 3.3. Let B=(B0,B 1)be a regular quasi-linearizable couple, let Bbe an intermediate space with respect to Bwith B0∩B1dense in B, and let ρ(t)be the ρ-function associated to Band B. Assume [J,τ] is a quasi-normed operator ideal and let Abe any Banach space. Then J(B,A)→(J(B0,A),J(B1,A))ρ∗,∞;K, where ρ∗(t)=1/ρ(t−1). Moreover, there is a constant ddepending only on Bsuch that Tρ∗,∞;K≤dτ(TB,A). 472 F. Cobos, T. Signes Since T=∞ n=1 Sn, we conclude that T∈(N(A, B0),N(A, B1))f,1;J with Tf,1;J≤ ∞  n=1 Snf,1;J≤2 ∞  n=1 hnA∗bnf,1;J≤4T1. The proof is complete. Assume next that the function gbelongs to the class P+−(see [10]). This means that g(t) is non-decreasing, g(t)/t is non-increasing with g(t) = sups>0{g(ts)/g(s)}finite for every t>0 and g(t)=o(max{1,t}) as t→0 and t→∞. Then it is well-known that the spaces (B0,B 1)g,q;K and (B0,B 1)g,q;Jcoincide, and that for regular couples the following duality formula holds ((B0,B 1)o g,∞;K)∗=(B∗ 0,B∗ 1)g∗,1;J. Arguing as in Theorem 4.2 but using Corollary 3.9/b, we obtain Theorem 4.3. Let B=(B0,B 1)be a regular quasi-linearizable couple, let Abe a Banach space and assume that g∈P +−. Then N((B0,B 1)o g,∞;K,A)=(N(B0,A),N(B1,A))g∗,1;J (equivalent norms). Previous results in this direction are due to Ovchinnikov [13] and [14]. His approach works for regular couples of Hilbert spaces (H0,H 1), (G0,G 1) and gives that (L(H0,G 0),L(H1,G 1))θ,∞=L((H0,H 1)θ,1,(G0,G 1)θ,∞), (N(H0,G 0),N(H1,G 1))θ,1=N((H0,H 1)o θ,∞,(G0,G 1)θ,1). Note that each one of the couples (H0,H 1), (G0,G 1) is quasi-linearizable. The reason is that any regular couple of Hilbert spaces is isomorphic to a couple (42(Gm),4 2(2−mGm)) formed by vector valued 42-spaces (see [6]), so the quasi-linearizability follows from the next result. Lemma 4.4. Let 1≤p≤∞and let (Gm)m∈Zbe a sequence of Banach spaces. Then the couple (4p(Gm),4 p(2−mGm)) is quasi-linearizable. On A Result of Peetre 473 Proof: Let x=(xm)m∈Z∈4p(Gm)+4p(2−mGm). Then K(t, x)p≈inf ∞  m=−∞ ymp Gm+tp ∞  m=−∞ 2−mpzmp Gm: x=y+z, y ∈4p(Gm),z∈4p(2−mGm) = ∞  m=−∞ min{1,t p2−mp}xmp Gm = m∈Z1 xmp Gm+tp m∈Z2 2−mpxmp Gm where Z1={m∈Z:2 m≤t}and Z2=Z\Z1. This suggest to define for each t>0 V0(t)x=(vm),V 1(t)x=(wm) where vm=xmif m∈Z1 0 otherwise ,w m=0 if m∈Z1 xmotherwise . A direct computation shows that conditions (2.1), (2.2) and (2.3) are fulfilled with k=1. It is well-known the relationship between N(A, B) and the projective tensor product A∗ˆ ⊗B, so one may guess that the results of Section 3 are also useful to interpolate projective tensor products, or even injective tensor products Aˇ ⊗B. In fact, by means of similar arguments to those described in Theorems 3.2 and 4.1, one can establish Theorem 4.5. Let B=(B0,B 1)be a regular quasi-linearizable couple, let Abe a Banach space and assume that g∈P +−. Then Aˇ ⊗(B0,B 1)o g,∞;K=(Aˇ ⊗B0,Aˇ ⊗B1)o g,∞;K (equivalent norms). The result for the projective tensor product reads Theorem 4.6. Let B=(B0,B 1)be a regular quasi-linearizable couple and let Abe a Banach space such that (Aˆ ⊗B0,Aˆ ⊗B1)is also a Banach couple. If fis any positive function satisfying (2.11), then Aˆ ⊗(B0,B 1)f,1;J=(Aˆ ⊗B0,Aˆ ⊗B1)f,1;J (equivalent norms). 474 F. Cobos, T. Signes The proof follows similar lines to those in Theorem 4.2. Previous results on interpolation of tensor products are due to Kouba [11]. He found conditions on the regular Banach couples (A0,A 1) and (B0,B 1) so that the following complex interpolation formulæ [A0,A 1]θˇ ⊗[B0,B 1]θ=[A0ˇ ⊗B0,A 1ˇ ⊗B1]θ, [A0,A 1]θˆ ⊗[B0,B 1]θ=[A0ˆ ⊗B0,A 1ˆ ⊗B1]θ are valid. His conditions has nothing to be with the quasi-linearizability, but they refers to the notions of type 2 and 2-convexity. Next we focus our attention on operator ideals defined by approximation numbers. Recall that for T∈L(A, B) and n=1,2,..., the n-th approximation number is given by an(T)=an(TA,B) = inf{T−L:L∈L(A, B),rank L<n}. For 0 <p<∞, the quasi-normed operator ideals [L(a) p,σ p] generated by the approximation numbers and the sequence space 4pare defined by L(a) p(A, B)=   T∈L(A, B):σp(T)=∞  n=1 an(T)p1 p <∞   (see [18] and [19]). It follows from [8, Theorem 3.3.4], that approximation numbers have no unrestricted interpolation properties. However, working with quasilinearizable couples, we can use ideas of Section 3 to establish the following formulæ: Lemma 4.7. Let Abe a Banach space, let B=(B0,B 1)be a quasilinearizable couple, let Bbe an intermediate space with respect to Band let ψ(t)and ρ(t)be the functions associated to Band B. a) If T∈L(A, B0∩B1)and n0,n 1∈N, then an0+n1−1(TA,B)≤2(1 + k)an0(TA,B0)ρ∗an1(TA,B1) an0(TA,B0). b) If Bis regular, T∈L(B0+B1,A)and n0,n 1∈N, then an0+n1−1(TB,A)≤2kan0(TB0,A)ψan1(TB1,A) an0(TB0,A). On A Result of Peetre 475 Proof: Let T∈L(A, B0∩B1). Splitting Tas T=V1(t)T+V0(t)T, using Lemma 3.1 and additivity of approximation numbers, we get that an0+n1−1(TA,B)≤an0([V1(t)T]A,B)+an1([V0(t)T]A,B) ≤V1(t)B0,Ban0(TA,B0)+V0(t)B1,Ban1(TA,B1) ≤(1 + k)an0(TA,B0) ρ(t)+tan1(TA,B1) ρ(t). If anj(TA,Bj)= 0, for j=0,1, then the choice t=an0(TA,B0)/an1(TA,B1) yields a). If anj(TA,Bj) = 0, for j=0orj= 1, then an0+n1−1(TA,B)=0 as well, because rank T<n j. The proof of b) is similar, using now the splitting T=TV0(t)+TV1(t). See [7] for some remarks on the connection between this result and the study of approximation numbers of embeddings from Besov spaces into spaces of Lipschitz type. We are now ready to establish embedding formulæ for L(a) p-ideals. Theorem 4.8. Let Abe a Banach space and let B=(B0,B 1)be a quasi-linearizable couple. Assume that 0<p 0,p 1<∞,0<θ<1, 1 p=1−θ p0+θ p1and 1 q= max{1,1 p}+1 p. Then the following holds: a) (L(a) p0(A, B0),L(a) p1(A, B1))θ,q →L (a) p(A, (B0,B 1)θ,1). b) Moreover, if Bis regular, (L(a) p0(B0,A),L(a) p1(B1,A))θ,q →L (a) p((B0,B 1)o θ,∞,A). Proof: Let T∈L (a) p0(A, B0)∩L (a) p1(A, B1). Using Lemma 4.7/a with B=(B0,B 1)θ,1=(B0,B 1)f,1;Jwhere f(t)=tθ, we have that a2n−1(TA,(B0,B1)θ,1)≤21+θ(1 + k)an(TA,B0)1−θan(TA,B1)θ. Whence σp(TA,(B0,B1)θ,1)≤21 p∞  n=1 a2n−1(TA,(B0,B1)θ,1)p1 p ≤2θp+p+1 p(1 + k)∞  n=1 an(TA,B0)(1−θ)pan(TA,B1)θp1 p ≤2θp+p+1 p(1 + k)σp0(TA,B0)1−θσp1(TA,B1)θ ≤2θp+p+1 p(1 + k)2−θmJ(2m,T) where m∈Zis arbitrary. 476 F. Cobos, T. Signes Take now T∈(L(a) p0(A, B0),L(a) p1(A, B1))θ,q and let T=∞ m=−∞ Tm be any J-representation of Twith (Tm)⊂L (a) p0(A, B0)∩L (a) p1(A, B1). Since the constant in the triangle inequality for σpis 21 pmax{21 p−1,1} and (2·21/p max{21/p−1,1})q= 2, we derive from our previous estimates that σp(TA,(B0,B1)θ,1)≤21 q∞  m=−∞ σp(Tm)q1 q ≤C∞  m=−∞ J(2m,T m) 2θm q1 q . This implies a). A similar argument, but using Lemma 4.7/b and (2.10), proves b). Remark 4.9.In fact, Lemma 4.7 is valid for any additive sfunction sin the sense of Pietsch [18] and [19], and Theorem 4.8 works for operator ideals generated by any additive s-function. 5. Injectivity and surjectivity Let [J,τ] be a quasi-normed operator ideal. An operator S∈L(A, B) belongs to the surjective hull Jsur if there are a Banach space Eand an operator T∈J(E,B) so that S(UA)⊆T(UE) where UAand UEstand for the closed unit balls of Aand E, respectively. The function τsur(S) = inf{τ(T):S(UA)⊆T(UE)} defines a quasi-norm in Jsur and [Jsur,τsur] turns out to be a quasinormed operator ideal (see [18]). It is clear that cτsur ≤cτ. An operator S∈L(A, B) is said to belong to the injective hull Jinj if there are a Banach space Fand an operator T∈J(A, F) such that SxB≤TxFfor all x∈A. The quasi-norm on Jinj is given by τinj(S) = inf{τ(T):SxB≤TxF,x∈A} and [Jinj,τinj] is a quasi-normed operator ideal (see [18]). Again cτinj ≤ cτ. If [J,τ]=[Jsur,τsur] (resp. [J,τ]=[Jinj,τinj]), then the quasinormed operator ideal Jis called surjective (resp. injective). Of course [L,·] satisfies these two conditions. Other examples are compact operators [K,·] and weakly compact operators [W,·]. A normed operator ideal which is injective but fails to be surjective is the ideal [Π,π] On A Result of Peetre 477 of all absolutely summing operators. Recall that T∈L(A, B) is absolutely summing if there is a constant c>0 such that for every finite set {aj}n j=1 ⊆A n  j=1 TajB≤csupn  j=1 |f(aj)|:f∈U A∗. We write π(T) for the least constant cfor which the above inequality holds. We refer to [18] and [4] for more details on these notions. Given two Banach spaces E,F, we denote by E⊕Fthe direct sum of Eand F, normed by (x, y)= max{xE,yF}. We shall use some ideas developed in [3] to establish the following theorems. Theorem 5.1. Let B=(B0,B 1)be a regular Banach couple, let Bbe an intermediate space with respect to Bwith B0∩B1dense in Band let ψ(t)be the ψ-function associated to Band B.IfAis any Banach space, [J,τ]is a quasi-normed operator ideal and qis defined by (2cτ)q=2, then (Jsur(B0,A),Jsur(B1,A))ψ∗,q;J→Jsur(B,A). Proof: Let T∈J sur(B0,A)∩Jsur(B1,A). Take any :>0, and let Ej(j=0,1) be Banach spaces and Rj∈J(Ej,A) so that T(UBj)⊆ Rj(UEj) and τ(Rj)≤(1 + :)τsur(TBj,A). Given any b∈U B, we can find bj∈Bjsuch that b=b0+b1and b0B0+tb1B1≤(1 + :)ψ(t). Hence UB⊆(1 + :)ψ(t)UB0+(1+:)ψ(t) tUB1. Consider the Banach space E0⊕E1and the operator R(x, y) = (1 + :)ψ(t)R0x+(1+:)ψ(t) tR1y. Then R∈J(E0⊕E1,A) and T(UB)⊆(1 + :)ψ(t)R0(UE0)+(1+:)ψ(t) tR1(UE1)⊆R(UE0⊕E1). Consequently, τsur(TB,A)≤τ(R)≤cτ(1 + :)2ψ(t)[τsur(TB0,A)+1 tτsur(TB1,A)] ≤2cτ(1 + :)2J(t−1,T) ψ∗(t−1). Combining this estimate with [1, Lemma 3.10.2], the result follows. 478 F. Cobos, T. Signes Since [J,τ]→[Jsur,τsur], we also have that (J(B0,A),J(B1,A))ψ∗,q;J→Jsur(B,A). This means that the assumption that Bis quasi-linearizable in Theorem 3.8 and Corollary 3.9/b can be eliminated, provided we take [Jsur,τsur] as the last operator space. Writing down Theorem 5.1 for B=(B0,B 1)o θ,∞and Jsurjective we conclude: Corollary 5.2. Let B=(B0,B 1)be a regular Banach couple and let A be a Banach space. If [J,τ]is a surjective quasi-normed operator ideal, (2cτ)q=2and 0<θ<1, then (J(B0,A),J(B1,A))θ,q →J((B0,B 1)o θ,∞,A). Next we show by means of an example that surjectivity is essential in the former result. We shall use the norm ideal Π which is not surjective. Example 5.3. Let Ω = c0⊕41⊕41and let A=(A0,A 1) be the Banach couple constructed by Garling and Montgomery-Smith in [9, Theorem 2]. Then Ajis isometric to 41; the projection P:Ω−→ c0given by P(x, y, z)=xbelongs to L(Ao θ,∞,c 0) and the embedding J:c0−→ Ω given by Jx =(x, 0,0) belongs to L(c0, Ao θ,∞). Moreover, it is easy to check that Ais regular. Let now R∈L(c0,4 2) be the operator defined by R(ζn)=(ζn/n). A direct computation shows that R/∈Π(c0,4 2). Put T=RP. Since Ajis isometric to 41, Grothendieck’s theorem implies that T∈Π(A0,4 2)∩Π(A1,4 2). However, Tdoes not belongs to Π(Ao θ,∞,4 2) because TJ =RPJ =R which is not absolutely summing. We consider now the injective hull. On A Result of Peetre 479 Theorem 5.4. Let B=(B0,B 1)be a Banach couple, let Bbe an intermediate space with respect to Band let ρ(t)be the ρ-function associated to Band B. Assume that Ais any Banach space, that [J,τ]is a quasinormed operator ideal and let qbe defined by (2cτ)q=2. Then (Jinj(A, B0),Jinj(A, B1))ρ,q;J→Jinj(A, B). Proof: Let T∈J inj(A, B0)∩Jinj(A, B1). Given any :>0 we can find Banach spaces Fj(j=0,1) and operators Rj∈J(A, Fj) with TxBj≤RjxFjfor all x∈A, and τ(Rj)≤(1 + :)τinj(TA,Bj). Given t>0, put Rx =(R0x, tR1x). Then R∈J(A, F0⊕F1). It follows from T(x)B≤J(t, Tx) ρ(t)≤1 ρ(t)max{R0xF0,tR1xF1}=1 ρ(t)RxF0⊕F1 that T∈Jinj(A, B) with τinj(TA,B)≤1 ρ(t)τ(R)≤cτ ρ(t)(τ(R0)+tτ(R1)) ≤2cτ(1 + :) ρ(t)J(t, T). Now we conclude the result by appealing to [1, Lemma 3.10.2]. Since [J,τ]→[Jinj,τinj], Theorem 5.4 shows that we can eliminate the assumption of quasi-linearizability in Theorem 3.7 and Corollary 3.9/a provided we take Jinj(A, B) as the last operator space. In particular, we have Corollary 5.5. Let B=(B0,B 1)be a Banach couple and let Abe a Banach space. If [J,τ]is an injective quasi-normed operator ideal, (2cτ)q=2and 0<θ<1, then (J(A, B0),J(A, B1))θ,q →J(A, (B0,B 1)θ,1). Working with the dual ideal [Πd,πd] of absolutely summing operators, it is not hard to derive from Example 5.3 that injectivity is essential in Corollary 5.5. Acknowledgement. We are grateful to Quanhua Xu for fruitful discussions and for drawing our attention to Lemma 4.4. We would also like to thank the referee for his comments. References [1] J. Bergh and J. L¨ ofstr¨ om,“Interpolation spaces. An introduction”, Grundlehren der Mathematischen Wissenschaften 223, Springer-Verlag, Berlin, 1976. 480 F. Cobos, T. Signes [2] F. Cobos, M. Cwikel and P. 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Triebel,“Interpolation theory, function spaces, differential operators”, North-Holland Mathematical Library 18, North-Holland Publishing Co., Amsterdam, 1978. Departamento de An´alisis Matem´atico Facultad de Matem´aticas Universidad Complutense de Madrid 28040 Madrid Spain E-mail address:[email protected] E-mail address:Tesera−[email protected] Primera versi´o rebuda el 26 d’octubre de 1999, darrera versi´o rebuda el 2 de maig de 2000.