Stability properties of the differential process generated by complex ...

STABILITY PROPERTIES OF THE DIFFERENTIAL PROCESS GENERATED BY COMPLEX INTERPOLATION

arXiv:1712.09647v1 [math.FA] 27 Dec 2017

´ M. F. CASTILLO, WILLIAN H. G. CORREA, ˆ VALENTIN FERENCZI, MANUEL GONZALEZ ´ JESUS Abstract. We obtain stability results for the bounded, trivial or singular character of the differential process of Rochberg and Weiss associated to complex interpolation of an analytic family of Banach spaces. Among other results, it is proved that there is global splitting stability for couples of K¨ othe spaces while there is no bounded or splitting stability for families of K¨othe spaces. This completes the results of Kalton who proved the existence of global bounded stability for couples of K¨ othe spaces. For general couples, we present a complete analysis of pairs whose differential process is linear at a point.

Contents 1. Introduction 2. Preliminaries 2.1. Elements from the theory of twisted sums 2.2. Elements from perturbation theory 2.3. Analytic families and twisted sums of Banach spaces 2.4. Derivations and centralizers 3. Preliminary results 3.1. Distances and isomorphisms 3.2. Bounded splitting and decompositions 4. Stability results for scales of K¨othe spaces 4.1. No bounded stability for families of K¨othe spaces 4.2. Splitting stability for pairs of K¨othe spaces 4.3. No splitting stability for families of K¨othe spaces 5. Stability results for arbitrary pairs 5.1. Optimal interpolation pairs 5.2. Strict interpolation pairs 6. Remarks about singular stability References

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2010 Mathematics Subject Classification. 46B70, 46E30, 46M18 . The research of the first author has been supported in part by Project IB16056 and Ayuda a Grupos GR15152 de la Junta de Extremadura; the research of the first and fourth authors has been supported in part by Project MTM2016-76958, Spain. The research of the second author has been supported in part by CNPq, grant 140413/2016-2, and CAPES, PDSE program 88881.134107/2016-0. The research of the third author has been supported by FAPESP, grants 2013/11390-4, 2015/17216-1 and 2016/25574-8. 1

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´ M. F. CASTILLO, WILLIAN H. G. CORREA, ˆ VALENTIN FERENCZI, MANUEL GONZALEZ ´ JESUS

1. Introduction The stability of the differential process associated to an interpolation scale has been a central topic in the theory since its inception. Assume that (Xt )t∈U is an interpolation family, where U is an open subset of C conformally equivalent to the open unit disc. For this introduction we shall choose as U the unit strip S = {z ∈ C : 0 ≤ Re z ≤ 1}. At the ground zero level (see below the explanation of this term) a stability problem can be described, paraphrasing [22, p. 657], as the study of properties P so that whenever the interpolation space Xθ (or an interpolated operator Tθ ) has P there is an open neighborhood of θ so that all spaces Xt (all interpolated operators Tt ) have P for all t in that neighborhood. We refer to [33] and the references given therein for results of this type in the space setting. In the operator setting, stability results for properties such as “to be an isomorphism”, “to be a Fredholm operator”, etc. have been studied in, e.g., [31] (complex method), [49] (real method) or [36] (orbits method). We describe a typical stability result next. The reader is invited to go to the background Section for all unexplained notation. Recall that the Kadets distance dK (A, B) between two Banach spaces A and B is the infimum of the gap g(i(A), j(B)) between the images i(A) and j(B) by all isometric embeddings i, j of the spaces A, B into a bigger superspace. Kalton and Ostrovskii [33, Thm. 4.5] show that a complex interpolation scale (Xt )t∈S satisfies dK (Xt , Xs ) ≤ 2h(t, s), where h is the pseudo-hyperbolic distance on S. A clever variation of Krugljak and Milman [36, Thm. 1] shows that a similar result holds valid for the orbit interpolation method. Then, after defining [33, p. 38] a property P to be open (resp. stable) if whenever a space X has P then there is ε > 0 so that every space Y with dK (X, Y ) < ε has P (resp. there is c > 0 so that if X has P and dK (X, Y ) < c then also Y has P), they state ([33, Propositions 5.4 and 5.5]): If P is a stable (resp. open) property and Xθ has P then Xt has P for all 0 < t < 1 (resp. all |t − θ| < ε for some ε > 0). The meaning of “ground zero level” in the opening paragraph is that an interpolation space Xθ can be considered as a 0-derived space of the scale (Xω )ω∈∂S . The 1-derived space, usually called the derived space, is the space dXθ that appears as the middle “twisted sum”space in the exact sequence (1)

0 −−−→ Xθ −−−→ dXθ −−−→ Xθ −−−→ 0

induced by the derivation map Ωθ associated to the interpolation method at θ. Iteration of the differentiation process produces the n-th derived spaces dn Xθ , induced by the n-th derivation map Ωθ,n (see [45]) which, according to [5], are connected by exact sequences 0 −−−→ dn Xθ −−−→ dn+m+1 Xθ −−−→ dm Xθ −−−→ 0 In this regard, we prove in Theorem 3.6 a generalized form of the result of Kalton and Ostrovskii mentioned earlier: that dK (dn Xθ , dn Xη ) ≤ 8(n + 1)h(θ, η).

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Probably the first stability results at level 1 are due to Cwikel, Jahberg, Milman and Rochberg [20], although referred to the minimal (θ, 1)-interpolation method for a pair (X0 , X1 ). They reinterpret the results of Zafran [49] to show that whenever Ωθ is bounded then X0 = X1 , up to a renorming; which, in particular means that all Ωt are bounded. A similar result is obtained by Kalton [30] in the context of complex interpolation for a pair of K¨othe spaces in the form: The induced derivation Ωθ is bounded if and only X0 = X1 , up to an equivalent renorming. See Theorem 2.5 for the precise and complete statement of Kalton’s theorem, which is much more powerful than what is cited here. Kalton’s global stability result leaves however two important questions unanswered in the domain of K¨othe spaces: what occurs for families (instead of pairs)? And what occurs regarding stability of splitting? Recall that an exact sequence like (1) is said to split (i.e., Xθ is complemented in dXθ ) if and only if Ωθ can be written as the sum of a bounded map plus a linear map. Thus, even if the study of when Ωθ is bounded is relevant from the point of view of K¨othe spaces, the right notion to study from the Banach space point of view is that of splitting. Thus, our first objective is to complete Kalton’s result in Theorem 4.7 by showing that given an interpolation pair (X0 , X1 ) of superreflexive K¨othe spaces Ωθ splits if and only if there is a weight function w so that X0 = X1 (w), up to an equivalent renorming. This solves the stability problem for splitting in the context of pairs of K¨othe spaces. A consequence, Proposition 4.8, is that the twisted Hilbert space induced by interpolating a superreflexive K¨othe space with its dual, a typical situation in our context (see [12]), does no split unless it is trivial. The case for families is however different. Let us remark that interpolation of families is relevant in many constructions; for example, in Pisier’s definition of ”θ-hilbertian spaces” [43], or in the construction of a uniformly convex HI space [26]. We will show in Section 4 through several examples that no bounded or splitting stability exists when one considers families of at least four K¨othe spaces. To put in context these results and counterexamples, let us recall what Rochberg [44] calls a flat analytic family on the unit disk D. Let k · k be a norm on Cn and let (Tz ) a family of invertible linear maps on Cn which vary analytically with z ∈ D. Define kxkz = kTz−1 xk. The family (Cn , k · kz ) is called a flat analytic family. A special case of flat family is mentioned in [21, p. 254]: Suppose W is a (possibly unbounded) positive linear operator on a Banach space X and that W admits enough of a functional calculus so that we can make good sense of the semigroup Ws , s ≥ 0. In this case the family of spaces Xs defined by kxkXs = kW s xkX , 0 ≤ s ≤ 1 will be a (complex) interpolation scale. The transport of these ideas to infinite dimension causes a few algebraic problems and we introduce a notion of ”coherence” to handle most of them. Our Proposition 4.1 shows a flat interpolation family of K¨othe sequence spaces kxkz = ke−D(z) xk2 generated by an analytic family D(z) of diagonal operators for which the derivation is always linear and constant. In Section 5 we will show that this is essentially the only option for pairs; precisely, Proposition 5.16 shows the follwing nice linear stability result: under some technical assumptions, if some Ωθ is linear then the associated family is the flat family induced by the operator Ωθ in the form kxkz = ke−zΩθ xk0 .

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´ M. F. CASTILLO, WILLIAN H. G. CORREA, ˆ VALENTIN FERENCZI, MANUEL GONZALEZ ´ JESUS

Another series of examples can be obtained as the “level one” interpretation of some results in the pioneering work [17, Theorem 5.1] of Coifman, Cwikel, Rochberg, Sagher and Weis. Indeed, a rather general result for families is obtained in Theorem 4.15 as follows: starting with a interpolation pair (X0 , X1 ) with derivation Φz on the strip, a measurable function α : T → [0, 1] and the family of spaces (X0 , X1 )α(ω) on the unit circle T, one gets an analytic family (X0 , X1 )α(z) with derivation Ωz = w ′(z)Φα(z) , where w(z) = α(z) + i˜ α(z) with α ˜ the harmonic conjugate of α. This theorem explains, to some extent, the lack of stability in the previous counterexamples and can be used to obtain other natural counterexamples. In particular, in combination with Kalton’s theorem, provides a positive stability theorem in the context of K¨othe spaces. The case of three spaces, for which there is both bounded and trivial stability, will be treated in [19]. We thank B. Maurey and G. Pisier for their hints and suggestions on this part of the work. However, the case of pairs remains elusive. Indeed, the following problem remains open: Problem Assume (X0 , X1 ) is an interpolation pair of Banach spaces such that Ωθ is bounded for some 0 < θ < 1. Does it follow that X0 = X1 up to equivalence of norms? One motivation for the study of scales without K¨othe space structure is the following wellknown question: Is the unit sphere of a uniformly convex space always uniformly homeomorphic to the unit sphere of a Hilbert space? In the K¨othe space case the answer is known to be positive, and this can be seen by interpolation methods. According to a result of Daher [23], if X0 and X1 are uniformly convex spaces, then the unit spheres of Xθ and Xν are uniformly homeomorphic for every 0 < θ, ν < 1, and the uniform homeomorphism is induced by the extremal functions of the interpolation process. As Daher observes, this fact together with an extrapolation theorem of Pisier for K¨othe spaces [42] (or with Kalton’s Theorem 2.5) imply that the unit sphere of a uniformly convex K¨othe space is uniformly homeomorphic to the unit sphere of the Hilbert space, a result previously obtained in [16] by other methods. This suggests that the study of general scales of interpolation may be relevant to this question: if there existed an extrapolation theorem for a general uniformly convex space X, relative to the existence of a scale where both X and ℓ2 are interior points, then Daher’s result would imply that X and ℓ2 have uniformly homeomorphic spheres. More specifically the study and properties of possible Ω’s on X are extremely relevant to these extrapolation questions. Another motivation is the definition by Pisier of so-called θ-Hilbertian spaces as certain interpolation spaces [43] and their relation to a question of V. Lafforgue; see Section 5 for more comments about this. In Section 5 we prove several stability results for pairs of general Banach spaces. As we have already mentioned, there is bounded and splitting stability for pairs of K¨othe spaces but apparently nothing is known about stability properties of general interpolation scales. A key role in our analysis to obtain new stability results is played by the study of properties of the extremal functions and the obtention of differential estimates for the norm in an interpolation scale. To the best of our knowledge, the first differential estimate for the norm is [20, Theorem 5.2], where the estimate d kakθ,1 ∼ kakθ,1 + kΩθ akθ,1 dθ

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is obtained for the minimal (θ, 1)-method applied to a pair with X0 continuously embedded in X1 . Our version of this estimate for the complex method on an arbitrary pair (X0 , X1 ) (see Lemma 5.6) is d kakt dt ± ≤ kΩθ akθ t=θ

from which we derive a number of stability results for general pairs. A typical result is Theorem 5.10, which in particular implies: If (X0 , X1 ) is an interpolation pair of superreflexive spaces with a common Schauder basis and there exists M such that kΩt : Xt → Xt k ≤ M for all t in some interval ]θ0 , θ1 [, then X0 = X1 , up to equivalent renorming. A part of our analysis deals with the particular situation in which extremals are unique (a case that occurs, for instance, when the spaces X0 and X1 are uniformly convex) and thus the derivation map Ωθ is unique and it makes sense to study “exact” stability of properties (instead of “up to a bounded” or “up to a bounded plus linear” perturbation). To adequately treat this case we need to work with the alternative description of the complex interpolation method provided by Daher [23], and for this reason the Preliminary results in Section 3 have been obtained in the general context of analytic families. We show that exact stability is related to isometric characterizations of X0 and X1 instead of isomorphic ones. We prove in Proposition 5.15 that, under some technical restrictions, Ωθ = 0 for some θ if and only if X0 = X1 isometrically. Also, Theorem 5.16 provides a complete and explicit characterization of pairs (X0 , X1 ) of spaces for which Ωθ is linear: X0 and X1 must be isometric by the (properly defined) map x 7→ e−Ωθ x. This implies that if additionally Ωθ is bounded, then X0 = X1 up to equivalence of norms. In turn, these results imply stability results for the families considered in Theorem 4.12. Namely, if Ωθ is linear for some θ = w(η) with w ′ (η) 6= 0 then Φ = Φα(η) is linear and thus X0 and X1 are isometric via e−Φ , Proposition 5.18. 2. Preliminaries 2.1. Elements from the theory of twisted sums. For a rather complete background on the theory of twisted sums and diagrams see [1, 14]. A twisted sum of two Banach spaces Y , Z is a quasi-Banach space X which has a closed subspace isomorphic to Y such that the quotient X/Y is isomorphic to Z. An exact sequence 0 −−−→ Y −−−→ X −−−→ Z −−−→ 0 of Banach spaces and linear continuous operators is a diagram in which the kernel of each arrow coincides with the image of the preceding one. Thus, the open mapping theorem yields that the middle space X in a short exact sequence 0 → Y → X → Z → 0 is a twisted sum of Y and Z. The simplest exact sequence is 0 → Y → Y ⊕ Z → Z → 0 with embedding y → (y, 0) and quotient map (y, z) → z. Two exact sequences 0 → Y → X1 → Z → 0 and 0 → Y → X2 → Z → 0 are said to be equivalent if there exists an operator T : X1 → X2 such that the following diagram

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´ M. F. CASTILLO, WILLIAN H. G. CORREA, ˆ VALENTIN FERENCZI, MANUEL GONZALEZ ´ JESUS

commutes:

0 −−−→ Y −−−→



X1 −−−→   yT

Z −−−→ 0



0 −−−→ Y −−−→ X2 −−−→ Z −−−→ 0. The classical 3-lemma [14, p. 3] shows that T must be an isomorphism. An exact sequence is said to be trivial if it is equivalent to 0 → Y → Y ⊕ Z → Z → 0. In this case we say that the exact sequence splits. Observe that the exact sequence splits if and only if the subspace Y of X is complemented. According to Kalton [28] and Kalton and Peck [34], twisted sums of quasi-Banach spaces can be identified with quasi-linear maps; i.e., homogeneous maps Ω : Z → Y that satisfy kΩ(z1 + z2 ) − Ωz1 − Ωz2 k ≤ C(kz1 k + kz2 k) for some constant C > 0. Actually, a quasilinear map Ω : Z → Y induces a quasi-norm on Y × Z given by k(y, z)kΩ = ky − Ωzk + kzk. If we denote by Y ⊕Ω Z the space Y × Z endowed with the above quasi-norm, we have an exact sequence 0 → Y → Y ⊕Ω Z → Z → 0 with embedding y → (y, 0) and quotient map (y, z) → z. Conversely, given an exact sequence 0 → Y → Y ⊕ Z → Z → 0 if B (resp. L) denotes a homogenous bounded (resp. linear) selection for the quotient map then B − L is a quasi-linear map. Two quasi-linear maps Ω, Λ : Z → Y are said to be equivalent, denoted Ω ≡ Λ, if the difference Ω − Λ can be written as B + L, where B : Z → Y is a homogeneous bounded map (not necessarily linear) and L : Z → Y is a linear map (not necessarily bounded). It turns out that two exact sequences are equivalent if and only if any two associated quasi-linear maps are equivalent. Therefore, a quasi-linear map Ω : Z → Y is trivial if and only it is equivalent to the 0 map, which occurs if and only if it can be written as Ω = B + L, with B : Z → Y a homogeneous bounded map and L : Z → Y a linear map. When Z is superreflexive, the twisted sum space Y ⊕Ω Z is (isomorphic to) a Banach space. Given an exact sequence 0 → Y → X → Z → 0 with associated quasi-linear map Ω and an operator α : Y → Y ′ , there is a commutative diagram 0 −−−→ Y −−−→ Y ⊕Ω Z −−−→ Z −−−→ 0  

 

′ αy (2) yα

0 −−−→ Y ′ −−−→ PO −−−→ Z −−−→ 0 whose lower sequence is called the push-out sequence and PO is called the push-out space. It can be represented as Y ′ ⊕αΩ Z with the natural inclusion and quotient maps, in which case α′ (y, x) = (αy, x). Again by the 3-lemma, if α is surjective then also α′ is surjective.

2.2. Elements from perturbation theory. Given a non-zero operator T : X → Y, the reduced minimum modulus of T is defined by γ(T ) := inf{kT xk : dist(x, ker T ) = 1}. It is well-known that the range of T is closed if and only if γ(T ) > 0 [35, Theorem IV.5.2]. Let M, N be closed subspaces of a Banach space Z, and let SM denote the unit sphere of M. The gap g(M, N) between M and N is defined by  g(M, N) = max sup dist(x, N), sup dist(y, M) , x∈SM

y∈SN

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and the minimum gap γ(M, N) between M and N is defined by γ(M, N) = inf

u∈M \N

dist(u, N) . dist(u, M ∩ N)

Note that M + N is a closed subspace if and only if γ(M, N) > 0 [35, Theorem IV.4.2]. Proposition 2.1. Let M and N be closed subspaces of Z such that M + N is closed, and let us denote R = (1/2) min{γ(M, N), γ(N, M)}. If M1 and N1 are closed subspaces of Z and g(M1 , M) + g(N1 , N) < R, then (1) M ∩ N = {0} implies M1 ∩ N1 = {0} and M1 + N1 is closed. (2) M + N = Z implies M1 + N1 = Z. In particular, if Z = M ⊕N and g(M1 , M) < R then Z = M1 ⊕N; i.e., the property of a subspace being complemented is open with respect to the gap. Proof. (1) Since M ∩ N = {0} and M + N is closed, γ(M, N) = inf u∈SM dist(u, N) > 0. Suppose that there exists u ∈ M1 ∩ N1 with kuk = 1. Since dist(u, M) ≤ g(M1 , M) and dist(u, N) ≤ g(N1 , N), our hypothesis implies 1 dist(u, M) + dist(u, N) < γ(M, N) 2 and dist(u, M ∩ N) = 1, contradicting [35, IV Lemma 4.4]. Hence M1 ∩ N1 = {0}. A similar argument shows that M1 + N1 is closed. Indeed, otherwise for every ε > 0 we could find u ∈ M1 and v ∈ N1 with kuk = kvk = 1 and ku − vk < ε. Therefore dist(u, M) ≤ g(M1 , M) and dist(u, N) ≤ g(N1 , N) + ε, and [35, IV Lemma 4.4] would imply 1 γ(M, N) ≤ g(M1 , M) + g(N1 , N) + ǫ 2 for every ǫ > 0, contradicting the hypothesis. (2) Let M ⊥ denote the annihilator of M in Z ∗ . Since M +N = Z if and only if M ⊥ ∩N ⊥ = {0} and M ⊥ + N ⊥ is closed, g(M, N) = g(M ⊥ , N ⊥ ) and γ(M, N) = γ(N ⊥ , M ⊥ ) [35, Chapter IV], it is a consequence of (1).  The Kadets distance dK (X, Y ) between two Banach spaces X and Y is the infimum of the gap g(i(X), j(Y )) taken over all the isometric embeddings of i, j of X, Y into a common superspace. This notion of gap g(M, N) is different, but equivalent to the one gˆ(M, N) used in [33]: g(M, N) ≤ gˆ(M, N) ≤ 2g(M, N) [35, Section IV.2.1]. Thus Theorem 4.1 in [33] implies the following result. Proposition 2.2. Let E and F be closed subspaces of Z. Then dK (Z/E, Z/F ) ≤ 4g(E, F ). 2.3. Analytic families and twisted sums of Banach spaces. We will consider analytic families within the framework described in [32, Section 10], which is based in the notion of admissible space of functions on an open subset U of the complex plane conformally equivalent to the open unit disc. Sometimes we will assume that U is the open unit disc D or the open unit strip S = {z ∈ C : 0 < ℜ(z) < 1} for convenience.

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Definition 2.3. Let U be an open subset of C conformally equivalent to D and let Y be a complex Banach space. An admissible space of analytic functions on U is a Banach space F ≡ (F(U, Y ), k · kF ) of analytic functions F : U → Y satisfying the following conditions: (a) For each z ∈ U, the evaluation map δz : F → Y is bounded. (b) If ϕ : U → D is a conformal equivalence and F : U → Y is a map, then F ∈ F if and only if ϕ · F ∈ F, and in this case kϕ · F kF = kF kF . Given an admissible family of functions F on U, for each z ∈ U we define Xz = {x ∈ Y : x = F (z)

for some F ∈ F}

with the norm kxk = inf{kF kF : x = F (z)} so that Xz is isometric to F/ ker δz with the quotient norm. The family (Xz )z∈U is called an analytic family of Banach spaces on U, and a function fxz ∈ F such that fxz (z) = x and kfxz kF = kxkz is called an extremal function (for x in z). There are many methods to generate analytic families. The simplest one is the complex interpolation method generated by an interpolation pair of Banach spaces (X0 , X1 ), which is a pair of Banach spaces, both of them contained as subspaces of a bigger vector space Σ = X0 + X1 . The pair will be called regular if, additionally, ∆ = X0 ∩ X1 is dense in both X0 and X1 . Both Σ and ∆ have their own Banach space structure and the canonical inclusions ∆ → Xi → Σ are contractions. The Calderon space C is formed by those bounded continuous functions F : S → X0 + X1 which are analytic on S and satisfy the boundary conditions F (k + ti) ∈ Xk for k = 0, 1, endowed with the norm kF kC = sup{kF (k + ti)kXk : t ∈ R, k = 0, 1} < ∞. Clearly C ≡ C(S, X0 + X1 ) is an admissible space of analytic functions on S. We can replace two spaces on the boundary of S by a suitably family of spaces distributed along the boundary of U. To simplify the description, we take U = D and denote by T the boundary of D. We will briefly describe the interpolation method for families we are going to use, which is a slight modification presented in [18] of the method developed in [17]. We assume that each space Xw (w ∈ T) is continuously linearly contained in a Banach space Σ, and that there is a subspace ∆ ⊂ ∩w∈T Xw such that for every x ∈ ∆ the function w 7→ kxkω (w ∈ Γ) R 2π is measurable and satisfies 0 log+ kxkeit dt < ∞, where log+ (0, y) = max{0, log y}. We also R 2π suppose that there is a measurable function k : [0, 2π) → [0, ∞) satisfying 0 log+ k(t)dt < ∞ and such that kxk± ≤ k(t)kxkeit for every x ∈ ∆ and every t ∈ [0, 2π). Then {Xw : w ∈ T} is called an interpolation family on T. P We denote by G0 the space of all analytic functions on D of the form g = nj=1 ψj xj , with ψj in the Smirnov class N + [25] and xj ∈ X, endowed with the norm kgk = ess supz∈T kg(z)kz < ∞. Moreover G is the completion of G0 . For each z0 ∈ D we define two interpolation spaces. The first one is X{z0 } , the completion of ∆ with respect to the norm kxk{z0 } = inf{kgk : g ∈ G0 , g(z0 ) = x}, and the second one is X[z0 ] = {f (z0 ) : f ∈ G} endowed with the natural quotient norm. If X{z0 } = X[z0 ] isometrically for every z0 ∈ D then G ≡ G(D, Σ) is an admissible space of functions on D. Given an admissible family F ≡ F(U, Y ) and z ∈ U, the evaluation map δz′ : F → Y of the derivative at z is bounded for all z ∈ U (see Lemma 3.1). We also need the following fact.

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Proposition 2.4. For each z ∈ U, the map δz′ is continuous and surjective from ker δz to Xz . Proof. Let ϕ : U → D be a conformal equivalence such that ϕ(z) = 0. Each g ∈ ker δz can be written as g = ϕ · f for some f ∈ F, and g ′ (z) = ϕ′ (z)f (z) ∈ Xz , thus δz′ (ker δz ) ⊂ Xz and the continuity into Xz follows from the closed graph theorem. Moreover, given x ∈ Xz and f ∈ F with f (z) = x, g = ϕ(z)−1 ϕ · f ∈ ker δz and ϕ′ (z) = x, hence δz′ (ker δz ) = Xz .  Thus, the following push-out diagram which exhibits the connection between analytic families and twisted sums of interpolation spaces: 0   y

(3)

0 −−−→

0 −−−→

ker δz ∩ ker δz′   y ker δz   δz′ y Xz   y 0

0   y

−−−→

ker ∆z   y

δ

z F −−− →  ∆ y z

Xz −−−→ 0



−−−→ POz −−−→ Xz −−−→ 0 qz jz   y 0

Here POz is the push-out space and can be described as the space {(f ′(z), f (z)) : f ∈ F } endowed with its natural quotient norm k(a, b)k = inf{kf kF : f ∈ F : f ′ (z) = a, f (z) = b}. The quotient operator ∆z comes clearly defined by ∆z (f ) = (f ′ (z), f (z)). 2.4. Derivations and centralizers. Thus we have a method to obtain twisted sums of the interpolation spaces Xz generated by an admissible family. Twisted sums can be always described by quasi-linear maps, but in this context is more useful to describe them using a linear perturbation of the quasi-linear map, called the derivation map (see [21]), which is a more canonical choice. The difference is that while the quasi-linear map is defined as a map F : Xz → Xz , the derivation is a map Ω : Xz → Σ with the additional property that for all x, y ∈ Xz its Cauchy differences satisfy Ω(x + y) − Ω(x) − Ω(y) ∈ Xz and there is a constant C so that kΩ(x + y) − Ω(x) − Ω(y)kXz ≤ C(kxk + kyk). A derivation Ω : Xz → Σ induces the twisted sum space, called the derived space, dXz = {(ω, x) : ω ∈ Σ, x ∈ Xz , ω − Ω(x) ∈ Xz } endowed with the quasi-norm k(ω, x)kz = kω − Ω(x)k + kxk, and we get an exact sequence 0 → Xz → dXz → Xz → 0 with natural inclusion and quotient map. This sequence splits if and only if Ω is trivial, which means that there is a linear map L : Xz → Σ such that Ω−L : Xz → Xz is bounded.

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The derivation associated to diagram (3) can be obtained as follows: Pick Bz : Xz → F a homogeneous bounded selection for δz and set Ωz = δz′ Bz ; thus Ωz (x) = Bz (x)′ (z) (x ∈ Xz ). It is not difficult to verify that dXz = POz . Note that, since Ωz depends on the choice of the selection Bz , different choices of selections lead to different derivations Ωz , but the difference between two of these derivations is always a bounded map, so both choices produce equivalent twisted sums. Furthermore, when there is a unique extremal fxz for every x, the canonical choice of Bz (x) = fxz induces a canonical associated derivation Ωz (x) = (fxz )′ (z). Kalton [29, 30] developed a deep theory connecting derived spaces and twisted sums in the specific case of K¨othe function spaces which we briefly describe now because it is essential to understand our work. Let X be a K¨othe function space contained in the space L0 (µ) of µmeasurable functions, where µ is a finite measure. We consider on X the Banach L∞ (µ)-modulus structure it naturally carries. An L∞ -centralizer on X is a quasi-linear map Ω : X → L0 (µ) for which there is a constant C such that, given f ∈ L∞ (µ) and x ∈ X, the difference Ω(f x) − f Ω(x) belongs to X and kΩ(f x)−f Ω(x)kX ≤ Ckf k∞ kxkX . We say that Ω is real if Ω(x) is real whenever x is real. The centralizer Ω is said to be trivial if there is a linear map L : X → L0 (µ) such that Ω − L : X → X is bounded. Of course, this means that the exact sequence induced by Ω splits. Derivations induced by complex interpolation scales of K¨othe function spaces are L∞ centralizers, real without loss of generality, and Kalton’s theorem establishes that essentially all real L∞ -centralizers arise in this way. Precisely: Theorem 2.5. [29, 30] (1) Given a complex interpolation pair (X0 , X1 ) of K¨othe function spaces and a point 0 < θ < 1, the derivation Ωθ is an L∞ -centralizer on the space Xθ . (2) For every real L∞ -centralizer Ω on a separable superreflexive K¨othe function space X there is a number ε > 0 and an interpolation pair (X0 , X1 ) of K¨othe function spaces so that X = Xθ for some 0 < θ < 1 and εΩ − Ωθ : Xθ → Xθ is a bounded map. (3) The induced centralizer Ωθ is bounded as a map Xθ → Xθ for some (hence for all) θ if and only if X0 = X1 , up to an equivalent renorming.

3. Preliminary results In this section F ≡ F(U, Y ) will be an admissible spaces of functions on an open subset U of the complex plane conformally equivalent to the open unit disc. Sometimes we are interested in the cases in which U is the open unit disc D or the open unit strip S. It is not difficult to translate some results for D into results for S, and conversely. Indeed, if ϕ : S → D is a conformal map and (Xω )ω∈D is an interpolation family on D, then Yz = Xϕ(z) provides an interpolation family (Yz )z∈S on S. The corresponding derivation maps are related as follows: ΩSz = ϕ′ (z)ΩDϕ(z) .

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Given s ∈ U we denote by ϕs : U → D a conformal equivalence taking s to 0. In the case U = S an example is given by (4)

ϕs (z) =

sin (π(z − s)/2) sin (π(z + s)/2)

(z ∈ S)

for which ϕ′s (s) = π/(2 sin πs). The conformal equivalence ϕs is unique up to a multiplicative constant: any other conformal equivalence ψs taking s to 0 can be written as ψs = f ◦ ϕs , where f (z) = eiθ z [2, 13.14 Lemma]. Given an admissible space F of functions on U and z ∈ U, we denote by δzn : F → Y the evaluation of the n-th derivative at z. We will need the following estimates: Lemma 3.1. Let F ≡ F(U, Y ), s ∈ U and n ∈ N. Then (1) kδsn : F → Y k ≤ (n!)/dist(s, ∂U)n . (2) kδs′ : ker δs → Xs k = γ(δs′ : ker δs → Xs ) = |ϕ′s (s)|. Proof. Given a positively oriented closed rectifiable curve Γ in U for which z belongs to the inside of Γ, the Cauchy integral formula [37, Appendix A3] establishes that, for each n ∈ N0 , Z n! f (ω) (n) f (z) = dw. 2πi Γ (w − z)n+1 We take a number r with 0 < r < dist(s, ∂U) and denote by Γ the boundary of the open disc D(s, r). By the Cauchy integral formula Z n! kf (ω)k n! (n) kf (s)k ≤ d|w| ≤ kf kF , 2π Γ r n+1 rn and since we can take r arbitrarily close to dist(s, ∂U), we get estimate (1). (2) Given g ∈ ker δs the function f (z) = ϕ′s (s) · ϕs (z)−1 g(z) is in F and satisfies f (s) = g ′(s) and kf k = |ϕ′s (s)|kgk. Therefore kδs′ gks = kf (s)ks ≤ |ϕ′s (s)|kgk, and we get kδs′ : ker δs → Xs k ≤ |ϕ′s (s)|. Also, given x ∈ BXs and ε > 0, we can take f ∈ F with kf k < (1 + ε) and f (s) = x. Then g(z) = ϕ′s (s)−1 ϕs (z) · f (z) defines g ∈ ker δs with kgk < (1 + ε)/|ϕ′s (s)| and g ′(s) = x. Hence δs′ (Bker δs ) ⊃ |ϕ′s (s)|(1 + ε)−1 BXs , and we get γ(δs′ : ker δs → Xs ) ≥ |ϕ′s (s)|. Since γ(T ) ≤ kT k for each T , the equality is proved.  Part (2) of Lemma 3.1 says that δs′ : ker δs → Xs is not only surjective, but a multiple of a quotient map: the induced injective map ker δs /(ker δs′ ∩ker δs ) → Xs is |ϕ′s (s)| times an isometry.

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3.1. Distances and isomorphisms. In this subsection F = F(U, Y ) is an admissible space of functions. First we show some connections between derivation of analytic functions and the derivation map Ωθ . Lemma 3.2. For each f ∈ F and s ∈ U, we have Ωs (f (s)) − f ′ (s) ∈ Xs with kΩs (f (s)) − f ′ (s)ks ≤ 2kδs′ : ker δs → Xs kkf k ≤ 2kf k/dist(s, ∂U). Proof. From Ωs (f (s)) − f ′ (s) = δs′ (Bs (f (s)) − f ) with Bs (f (s)) − f ∈ ker δs , we get the first part. For the rest, note that the operator δs′ : ker δs → Xs is bounded by Lemma 3.1.  Proposition 3.3. Let s, t ∈ U. (1) The spaces ker δs and F are isomorphic. Consequently, ker δs and ker δt are isomorphic. (2) For every n ∈ N, ∩0≤k≤n ker δsk and F are isomorphic. In particular, ker ∆s and ker ∆t are isomorphic. Proof. The operator ds : F → ker δs given by ds (f )(z) = f (z)ϕs (z) is clearly well-defined and injective, and it is surjective because T each g ∈ ker δsTcan be written as g = ϕs · f with f ∈ F. To prove (2), just note that ds : 0≤k≤n ker δsk → 0≤k≤n+1 ker δsk is also an isomorphism. 

Let s, t ∈ U. The map ϕs · f ∈ ker δs → ϕt · f ∈ ker δt is a bijective isomorphism, but we need a more precise description. Note that the map ϕs,t : U → D defined by ϕs,t(z) =

ϕs (z) − ϕs (t) 1 − ϕs (t)ϕs (z)

(z ∈ U)

is a conformal equivalence satisfying ϕs,t (t) = 0. Moreover, denoting α = ϕs (t) ∈ D, one has kϕs − ϕs,tk∞ = sup |ϕs (z) − ϕs,t (z)| z∈U λ − α = sup λ − 1 − αλ λ∈D ω − α = sup ω − 1 − αω ω∈T α − αω 2 = sup 1 − αω ω∈T αω − αω = sup ω−α ω∈T ≤ 2|α|,

since |αω − αω| ≤ |αω − αα| + |αα − αω| = 2|α||ω − α|.

Definition 3.4. ([33]) The pseudo-hyperbolic distance h(·, ·) on U is defined by h(s, t) = |ϕs (t)|. This yields: Proposition 3.5. g

T

0≤n≤N

ker δsn ,

T

0≤n≤N


ker δtn ) ≤ 2(N + 1)h(s, t).

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Proof. We proceed inductively on N. For N = 0, we take a norm-one ϕs · f ∈ ker δs . Since ϕs,t · f ∈ ker δt and kϕs · f − ϕs,t · f kF = kϕs − ϕs,t k∞ ≤ 2h(s, t), and we can proceed similarly for each norm-one ϕt · f ∈ ker δt , we get g (ker δs , ker δt ) ≤ 2h(s, t). Moreover if the estimate holds for N − 1 then it also holds for N because aN +1 − bN +1 = aN +1 − aN b + aN b − bN +1 = aN (a − b) + (aN − bN )b. T Since F/ 0≤n≤N ker δsn = dN Xs , Propositions 2.2 and 3.5 provide the following result:



Theorem 3.6. Given s, t ∈ U and N ∈ N ∪ {0}, we have dK (dN Xs , dN Xt ) ≤ 8(N + 1)h(s, t). And consequently, Corollary 3.7. Let P be a stable (resp. open) property. Assume that there is s ∈ U so that dn Xs has P. Then dn Xt has P for all t ∈ U (resp. for all t in an open disc centered in s). Note that the notion of gap g(M, N) we use is different from the one gˆ(M, N) used in [33], but this causes no problem because g(M, N) ≤ gˆ(M, N) ≤ 2g(M, N) [35, Section IV.2.1]. 3.2. Bounded splitting and decompositions. Kalton’s theorem establishes the importance of the following notions: Let us say that the derivation Ωz is bounded when it takes values in Xz and it is bounded as an homogeneous map from Xz to Xz , and that the exact sequence induced by the derivation Ωz boundedly splits if Ωz is bounded. Since the difference between two derivations associated to different bounded homogeneous selections Bz is bounded, the notion makes sense in this context, independently of the choice of the homogeneous bounded selection inducing the derivation. Next we give some characterizations of bounded splitting in terms of decompositions of an admissible space of functions F = F(U, Y ). Theorem 3.8. Let s ∈ U. The following assertions are equivalent: (1) δs : ker δs′ → Xs is surjective. (2) F = ker δs + ker δs′ . (3) There exists M > 0 such that each f ∈ F can be written as f = g + h with g ∈ ker δs , h ∈ ker δs′ and max{kgkF , khkF } ≤ Mkf kF . (4) δs′ (F) ⊂ Xs . (5) δs′ : F → Xs is a bounded operator. (6) Ωs (Xs ) ⊂ Xs . (7) Ωs : Xs → Xs is bounded. Proof. Clearly (1) ⇐ (2) ⇐ (3), (4) ⇐ (5) and (6) ⇐ (7). Moreover (4) ⇔ (6) and (5) ⇔ (7) follow from Lemma 3.2. We will prove (1) ⇒ (3) ⇒ (5) and (4) ⇒ (2). (1) ⇒ (3): Let f ∈ F with kf k = 1. Since δs : ker δs′ → Xs is surjective, it is open. So there exists r > 0 such that we can find h ∈ ker δs′ with khk ≤ rkf (s)ks and h(s) = f (s). Since kf (s)ks ≤ kf k, taking g = f − h ∈ ker δs we obtain (3) with M = r + 1.

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(3) ⇒ (5): Let f ∈ F. We can be write f = g+h with g ∈ ker δs , h ∈ ker δs′ and kgkF ≤ Mkf kF . Then kδs′ (f )ks = kδs′ (g)ks ≤ kδs′ : ker δs → Xs k · kgkF ≤ Mkδs′ : ker δs → Xs k · kf kF . (4) ⇒ (2): We know that the operator δs′ : ker δs → Xs is surjective. So taking a linear selection ℓ : Xs → ker δs for δs′ , for each f ∈ F, ℓ(f ′ (s)) ∈ ker δs and f − ℓ(f ′ (s)) ∈ ker δs′ .  Condition (2) in Theorem 3.8 provides a neat description of how the twisted sum space dΩs Xs splits when Ωs is bounded. Indeed, since dΩs Xs = F/(ker δs ∩ker δs′ ) and the subspace Xs embeds in dΩs Xs as ker δs /(ker δs ∩ ker δs′ ), condition (2) gives F ker δs + ker δs′ ker δs ker δs′ = = ⊕ . ker δs ∩ ker δs′ ker δs ∩ ker δs′ ker δs ∩ ker δs′ ker δs ∩ ker δs′ An immediate consequence is the following stability result. Proposition 3.9. Let s ∈ U. δ

s (1) If the sequence 0 → ker δs → F → Xs → 0 splits, then there is ε > 0 so that the exact δt sequence 0 → ker δt → F → Xt → 0 splits and Xt is isomorphic to Xs for |t − s| < ε. ∆ (2) If the sequence 0 → ker ∆s → F →s dXs → 0 splits, then there is ε > 0 so that the exact ∆ sequence 0 → ker ∆t → F →t dXt → 0 splits and dXt is isomorphic to dXs for |t − s| < ε.

Proof. If the exact sequence in (1) splits and |s − t| is small enough, by Proposition 2.1 the complement of ker δs is also a complement for ker δt . Analogously for (2), since ker ∆s = ker
δs ∩ ker δs′ , the kernels ker ∆s and ker ∆t are isomorphic for 0 < s, t < 1, and g ker ∆s , ker ∆t → 0 as s → t. 

We do not know whether the spaces ker δs′ and ker δt′ are isomorphic for s 6= t, or if ker δs′ is isomorphic to F. The absence of bounded stability for families that will be shown in Section 4 implies that, at least in that case, a similar estimate to Proposition 3.5 for ker δs′ cannot hold: Claim. Given an analytic family F it is not true that lims→t g(ker δs′ , ker δt′ ) = 0. Otherwise, g(ker δt , ker δs )+g(ker δs′ , ker δt′ ) → 0 as |s−t| → 0. This would imply that whenever Ωt is bounded, the decomposition F = ker δt + ker δt′ plus Theorem 3.8 (2) would imply that also F = ker δs + ker δs′ for s sufficiently close to t, and thus Ωs would be bounded as well. But the examples in the forthcoming Section 4 show that no bounded stability exists for families. ¨ the spaces 4. Stability results for scales of Ko 4.1. No bounded stability for families of K¨ othe spaces. Kalton’s theorem establishes an optimal result for bounded stability when working with pairs of superreflexive K¨othe spaces: if Ωθ is bounded from Xθ to Xθ then the scale is trivial, i.e. X0 = X1 , up to an equivalence of norms. However, our first result in this context is that Kalton’s result is no longer true for families of K¨othe spaces. Let A denote the disc algebra, which is the space of all continuous functions on D which are analytic on D. A sequence (ωn ) in A induces a family of diagonal operators

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D(z) : c00 → c00 , (z ∈ D), given by D(z)(xn ) = (wn (z)xn ), so that for each s ∈ T we can define a Banach space Xs as the completion of c00 with respect to the norm kxks = ke−D(s) xk2 . For x ∈ c00 , we denote kxkΣ = inf{kx1 kz1 + · · · + kxn kzn }, where the infimum is taken over all n ∈ N, z1 , . . . , zn ∈ T, and x1 , . . . , xn ∈ c00 such that x = x1 + · · · + xn . Claim. k.kΣ is a norm on c00 . Proof. The only difficulty is to show that kxkΣ = 0 implies x = 0. Suppose that x = (xj ) ∈ c00 and xk 6= 0. Notice that e−D(z) is the multiplication operator associated to the sequence (e−wn (z) ). −w If |wk (z)| ≤ M, then e 1 (z) = e−Re(w1 (z)) ≥ e−M . Therefore X X X X xk , ≥ e−M |xk | e−wk (z) xk ≥ e−M kxj kzj = ke−D(zj ) xj k2 ≥ j j and we conclude kxkΣ > 0.



Let Σ be the completion of c00 with respect to k.kΣ . Then for each s ∈ T we have Xs ⊂ Σ with inclusion having norm at most 1. We will show in Proposition 4.1 that (Xs )s∈T is an interpolation family with containing space Σ and intersection space ∆ = c00 . First note that the projection Pn onto the first n coordinates is a norm one operator on Xs for each z ∈ T, and on Σ. Lemma 4.1. For every z0 in D one has: (1) X{z0 } = X[z0 ] . Thus we can denote Xz0 = X{z0 } = X[z0 ] . (2) The space Xz0 is the completion of c00 with respect to the norm kxkz0 = ke−D(z0 ) xk2 . (3) Ωz0 = D ′ (z0 ). Proof. (1) Let x ∈ c00 . Clearly kxk[z0 ] ≤ kxk{z0 } . Let f ∈ G (see subsection 2.3) Pn such that f (z0 ) = x. Take n such that Pn (x) = x and define g(z) = Pn (f (z)). Then g(z) = j=1 ψj (z)ej , where (ej ) is the canonical basis of ℓ2 . Since ψj (z)ej = (Pj − Pj−1 )f (z) and f is analytic when viewed as an Σ-valued function, we get that ψj is analytic. If z ∈ D, then |ψj (z)| kej kU = k(Pj − Pj−1 )f (z)kU ≤ 2kf (z)kU ≤ 2kf kG . Hence ψj ∈ H ∞ ⊂ N + , where H ∞ is the space of bounded analytic functions on D. Also, for almost every z ∈ T we have: kg(z)kXz = kPn (f (z))kXz ≤ kf (z)kXz ≤ kf kG . Thus g ∈ G and kgkG ≤ kf kG . Since g(z0 ) = Pn (f (z0 )) = x, we get kxk{z0 } = kxk[z0 ] . To prove (2), let x ∈ c00 and let g(z) = eD(z)−D(z0 ) x ∈ G. Then g(z0 ) = x, and for z ∈ T, kg(z)kXz = ke−D(zz ) xk2 . Thus kxkz0 ≤ ke−D(z0 ) xk2 . Take f ∈ G such that f (z) = x. Given a non-zero y ∈ c00 , define Then h ∈ H ∞ and khkH ∞

h(z) = he−D(z) f (z), yi. ≤ kf kG kyk2. Hence |h(z0 )| = he−D(z0 ) x, yi ≤ kf kG kyk2 .

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Since kf kG can be taken arbitrarily close to kxkz0 and y is arbitrary, ke−D(z0 ) xk2 ≤ kxkz0 . Thus we have shown that the following function g is an extremal function for x at z0 : g(z) = eD(z)−D(z0 ) x = (ew1 (z)−w1 (z0 ) x1 , ew2 (z)−w2 (z0 ) x2 , . . .).
Then g ′ (z) = wn′ (z)ewn (z)−wn (z0 ) xn , hence Ωz0 x = g ′ (z0 ) = (wn′ (z0 )xn ) = D ′ (z0 )x.



Proposition 4.2. Let D(xn ) = (wn xn ) be an unbounded diagonal operator on ℓ2 . (1) The choice D(z) = zD yields an interpolation family such that Ωz = D for every z ∈ D. (2) The choice D(z) = z 2 D yields and interpolation family such that Ωz = 2zD for every z ∈ D. Therefore Ω0 = 0 while Ωz is unbounded for every z 6= 0. These examples are instances of the following more general situation:

Theorem 4.3. Let w ∈ A and let T : X → X be an operator. Set the flat family kxks = kew(s)T xkX (s ∈ T). Then for each z ∈ D, kxkz = kew(z)T xkX and the associated derivation is Ωz (x) = w ′ (z)T (x). 4.2. Splitting stability for pairs of K¨ othe spaces. Kalton’s work justifies the importance of bounded derivations and bounded splitting. However the natural notions for exact sequences are that of splitting sequence and trivial derivation. Thus, for an interpolation scale of K¨othe spaces, it is a reasonable guess that “nontrivial scales” correspond to “nontrivial centralizers”. The difficulty is that the non-triviality notion for exact sequences, hence for quasi-linear maps, centralizers or derivations, involves wildly uncontrolled linear maps. Recall that F is trivial if and only if F = B + L, with B homogeneous bounded and L linear. Thus, while Kalton shows [30] that the centralizer Ωθ associated to the scale (X0 , X1 )θ of K¨othe function spaces is bounded if and only if X0 = X1 (up to equivalence of norms), the following question remained open: Question 4.4. Does the triviality of Ωθ imply that X0 and X1 are equal, or at least isomorphic? In fact, all the examples in Theorem 4.2 induce trivial derivations on the whole domain. So they are counterexamples to bounded stability, but not to splitting stability. We shall now prove stability of splitting for interpolation of a pair of K¨othe spaces. The following sentence in [9, p. 364] clearly suggests that a positive answer was known to Nigel, at least for K¨othe sequence spaces: If (Z0 , Z1) are two super-reflexive sequence spaces and Zθ = [Z0 , Z1 ]θ for 0 < θ < 1 is the usual interpolation space by the Calderon method, one can define a derivative dXθ which is a twisted sum Xθ ⊕Ω Xθ which splits if and only if Z1 = wZ0 for some weight sequence w = (w(n)) where w(n) > 0 for all n. These remarks follow easily from the methods of [30]. Given a K¨othe function space X (on a Polish space), a weight w is a scalar positive measurable. When X is a sequence space we denote ϕ(X) the space of finitely supported elements of X. When X is a (non-atomic) function space, ϕ(X) denotes the space of functions of compact support of X. The space ϕ(X) is dense in X. We denote X(w) the space of all measurable scalar functions f such that wf ∈ X, endowed with the norm kxkw = kwxkX . Since wx ∈ X for every x ∈ ϕ(X), the space ϕ(X) is dense in X(w) and X(w) is the completion of ϕ(X) with respect to k · kw .

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From the approach in [12] we get the following general version of a well-known result for scales of Lp -spaces [3, Theorem 5.4.1] (see also [22, p. 655]): Proposition 4.5. Let X be a K¨othe function space with the Radon-Nikodym property, and let w0 , w1 be two weights. Then for any 0 < θ < 1, (X(w0 ), X(w1))θ = X(w01−θ w1θ ) with associated linear centralizer Ωθ (x) = log(w0 /w1 )x for x ∈ ϕ(X). Proof. By [32, Theorem 4.6], the space (X(w0 ), X(w1 ))θ is isometric to the space X(w0 )1−θ X(w1 )θ endowed with the norm θ 1−θ θ kxkθ = inf{kak1−θ b} w0 kbkw1 : a ∈ X(w0 ), b ∈ X(w1 ), x = a θ 1−θ θ = inf{kw0 ak1−θ b }. X kw1 bkX : a ∈ X(w0 ), b ∈ X(w1 ), x = a

Standard lattice estimates such as [39, Proposition 1.d.2] imply that kxkθ ≥ inf{kw01−θ a1−θ w1θ bθ kX :

x = a1−θ bθ } = kxkX(w1−θ wθ ) , 0

1

w01−θ w1θ x.

and the reverse inequality can be obtained by using w0 a = w1 b = We obtain Ωθ on ϕ(X). Observe that a bounded homogeneous optimal selector for the evaluation map δθ on ϕ(X) is defined by Bθ (x) = (w1 /w0 )θ−z x: indeed, Bθ (x)(θ) = x while kBθ xk = kxkXθ as it follows from kBθ (x)(0 + it)kw0 = kBθ (x)(1 + it)kw1 = kw01−θ w1θ xkX = kxkXθ .  It is a well-known fact that complex interpolation between two Hilbert spaces yields Hilbert spaces [32], although real interpolation between two Hilbert spaces can generate non-Hilbert spaces. Let us show that the induced derivation is trivial. We have Corollary 4.6. Let (H0 , H1 ) be an interpolation pair of Hilbert spaces. Then for every 0 < θ < 1 the derivation Ωθ is trivial. Proof. It is a consequence of Proposition 4.5 and the following fact [24, Lemma 2.2]: Given an interpolation pair of Hilbert spaces (H0 , H1 ), for every ε > 0 there is a set I, a positive weight w : I → R+ and an operator T : (H0 , H1 ) → (ℓ2 (I), ℓ2 (I, w)) such that both T0 : H0 → ℓ2 (I) and T1 : H1 → ℓ2 (I, w) are (1 + ε)-isometries.  We show now how to pass this information to any scale of superreflexive K¨othe function spaces. In this form, we solve both the stability problem for the property of splitting in the case of K¨othe spaces, by a scale stability result, and complete Theorem 1: Theorem 4.7. Let (X0 , X1 ) be an interpolation pair of superreflexive K¨othe spaces and let 0 < θ < 1. Then Ωθ is trivial if and only if there is a weight function w so that X1 = X0 (w) up to equivalence of norms. Proof. Assume L0 (µ) is the base space for the K¨othe structure of X0 , X1 . The proof goes in two steps. Step 1. If Ωθ is trivial then there is a function f ∈ L0 (µ) so that Ωθ (x) − f x is bounded.

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Indeed, since Ωθ is a centralizer, there is a constant c > 0 such that for every a ∈ L∞ (µ) and every x ∈ X we have kΩθ (ax) − aΩθ (x)k ≤ ckakkxk, and since it is trivial, there is a linear map L so that Ωθ − L takes values in Xθ and is bounded there. The techniques in [12] (Lemmas 3.10 and 3.13) show that after some averaging it is possible to get a linear map Λ such that Ωθ − Λ takes values in Xθ and is bounded there and, moreover, Λ(ux) = uΛx for every unit of X. P Since characteristic functions can be written as the mean of two units one gets that if if s = N λi 1Ai is a simple function then Λ(sx) = sΛ(x). Now, simple functions are dense in L∞ , so given a ∈ L∞ pick a simple s so that kask ≤ ε. Since Λ(ax) = Λ((a − s)x) + Λ(sx) and aΛ(x) = (a − s)Λ(x) + sΛ(x), it follows that kΛ(ax) − aΛ(x)k = kΛ((a − s)x) − (a − s)Λ(x)k ≤ cka − skkxk ≤ cεkxk which shows that Λ actually verifies Λ(ax) = aΛ(x) for every a ∈ L∞ . It is then a standard fact that Λ must have the form Λ(x) = f x for some function f . Step 2. The spaces X0 , X1 are weighted versions to each other. Pick w0 = eθf and w1 = e(θ−1)f . By the previous proposition, (Xθ (w0 ), Xθ (w1 ))θ = Xθ (w01−θ w1θ ) = Xθ with associated centralizer Ω(x) = log(w0 /w1 )x = f x = Λ(x). Thus Ωθ − Ω is bounded and, by Kalton’s uniqueness theorem (part (3) of Theorem 2.5), we get X0 = Xθ (w0 ) and X1 = Xθ (w1 ).  Theorem 4.7 implies that the map Ωθ , when trivial, is a bounded perturbation of a diagonal map. This is a consequence of the symmetry properties induced by the K¨othe space structure. Now we can complete Corollary 4.6 with the following result stating that twisted Hilbert spaces induced by interpolation of K¨othe spaces are trivial only in the obvious cases. Proposition 4.8. A twisted Hilbert space induced by interpolation at θ = 1/2 between a superreflexive K¨othe space X and its dual is trivial if and only if X = L2 (w) for some weight function w. Proof. If the twisted space is trivial then since X1/2 = L2 (see, e.g., [12]), and since spaces on the whole scale are weighted versions of each other, X and X ∗ are equal to L2 (w) and L2 (w −1) respectively, for some weight.  4.3. No splitting stability for families of K¨ othe spaces. We shall now give some examples which show that there is no splitting stability in general for families of K¨othe spaces. In our first example we explicitly define the initial configuration of ℓp spaces. Lemma 4.9. Let p : ∂D → [1, ∞) be a measurable function, let α : D → C be an analytic function 1 1 = p(z) on ∂D, and consider the interpolation family (ℓp(z) )z∈∂D . Given on D satisfying Re α(z)

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z0 ∈ D with α(z0 ) ∈ R, the interpolation space at z0 is ℓp(z0 ) and the induced derivation is ! α′ (z0 ) |xn | Ωz0 ((xn )) = − xn log . α(z0 ) kxkℓp(z0 ) Proof. We first check that  xn f (z) = |xn | |xn | is an extremal function for x ∈ c00 with kxkp(z0 ) = 1. The function f is analytic and f (z0 ) = x, and f ∈ G because each coordinate is bounded (as we can see in the following calculation), and for every z ∈ D we have X α(z0 ) p(z) kf (z)kp(z) = |xn |Re( α(z) )p(z) X 1 = |xn |α(z0 )Re( α(z) )p(z) X = |xn |α(z0 ) X = |xn |p(z0 ) = 1. 

α(z0 ) α(z)

In particular, it is an extremal function. Moreover for non-zero x = (xn ) ∈ ℓp(z0 ) , ! x Ωz0 (x) = xΩz0 = f ′ (z0 ) kxkℓp(z0 )   α(z0 ) α(z ) xn 0 ′ α(z0 ) |xn | α (z0 ) log |xn | = − |xn | α(z0 )2 ! α′ (z0 ) |xn | = − xn log , α(z0 ) kxkℓp(z0 ) and the proof is complete.



Definition 4.10. A quasi-linear map is said to be singular when it admits no trivial restriction to an infinite dimensional subspace It is not hard to check that a quasi-linear map Ωz singular when the quotient map of the associated exact sequence is a strictly singular operator [13].
1 Proposition 4.11. A configuration (ℓp(z) )z∈∂D with p(z) = Re z 21+2 yields Ω0 = 0 and Ωz is singular for any z ∈ D, z 6= 0. 2

2

+2) +2) 1 = Re(z ∈ [ 13 , 1] it turns out that p(z) ∈ [1, 3]. We thus Proof. Since p(z) = Re( z 21+2 ) = Re(z |z 2 +2|2 |z 2 +2|2 set α(z) = z 2 + 2 on D. In that case we get α(z) ∈ R if and only if z = t or z = it, t ∈ R. By the previous lemma, Ω0 = 0, and for z = t and z = it, t 6= 0, Ωz is a nonzero multiple of the KaltonPeck map on ℓp(z0 ) , and therefore it is singular. Moreover, the choice αz0 (z) = z 2 + 2 − iIm(α(z0 )) yields that Ωz0 is a nonzero multiple of the Kalton-Peck map for any z0 ∈ D, z0 6= 0. 

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´ M. F. CASTILLO, WILLIAN H. G. CORREA, ˆ VALENTIN FERENCZI, MANUEL GONZALEZ ´ JESUS

A more general and formally different way of acting is the following one. Recall that a measurable function α on ∂D which is integrable with respect to the harmonic measure on ∂D can be extended to an harmonic function on D. Precisely, if ω = eit , z = reiθ and Pz (ω) = then α(z) = where

R

∂D

1 − r2 1 + r 2 − 2r cos(θ − t)

α(ω)dPz (ω). The harmonic conjugate of α is defined as α(z) ˜ =

R

∂D

α(ω)dP˜z (ω),

2r sin(θ − t) . 1 + r 2 − 2r cos(θ − t) We use now the following reiteration theorem of [17] to get a initial suitable configuration. P˜z (ω) =

Proposition 4.12. ([17, Theorem 5.1]) Let α : ∂D → [0, 1] be a measurable function, let (X0 , X1 ) be an interpolation pair of Banach spaces, and let Xω = (X0 , X1 )α(ω) (ω ∈ ∂D). If inf ∂D α(ω) and sup∂D α(ω) are attained then (Xω )ω∈∂D is an interpolation family and X{z} = X[z] = (X0 , X1 )α(z) R for each z ∈ D, with equality of norms, where α(z) = ∂D α(ω)dPz (ω). The level one result is now:

Theorem 4.13. Let α and (X0 , X1 )α(ω) (ω ∈ ∂D) be as in Theorem 4.12, and let Ωs denote a derivation corresponding to (X0 , X1 )s for 0 < s < 1. Then a derivation corresponding to the family (X0 , X1 )α(z) at z ∈ D is Φz = w ′(z)Ωα(z) , where w = α + i˜ α. Proof. Fix z ∈ D and x ∈ X0 ∩ X1 , and take f in the Calderon space C(X0 , X1 ) such that f (α(z)) = x and kf k = kxkα(z) . By the proof of Theorem 4.12 we may assume that f is a linear combination of continuous functions with image in X0 ∩ X1 . Included in the proof of [17, Theorem 5.1] is the fact that the function g = f ◦ w is an extremal for the family (X0 , X1 )α(z) since g(z) = x, and kgk ≤ kf k. Therefore Φz (x) = (f ◦ w)′(z) = w ′ (z)Ωw(z) (x). Finally Ωw(z) may be chosen as Ωα(z) by vertical symmetry (in case of uniqueness of the extremal functions they are actually equal, see Lemma 5.14).  The previous result means that under the hypothesis of Theorem 4.12 the derivation for the family is always a multiple of the derivation for the initial pair. Corollary 4.14. Let (X0 , X1 ) be an interpolation pair, with associated derivations Ωs , s ∈]0, 1[. Let B0 = {eiθ : θ ∈ [0, π2 ]∪[π, 3π ]} and α = χB0 . Consider the interpolation family {Xw : w ∈ ∂D} 2 with Xw = (X0 , X1 )α(w) for w ∈ T, and for z ∈ D, Xz the interpolated space and Φz the associated derivation. Then Xz = (X0 , X1 ) 1 for every z = t, z = it, t ∈ (−1, 1), Φ0 = 0 and for a non-zero 2 z ∈ D, Φz is a multiple of the derivation Ωs for the pair in s = α(z). This example was suggested to us by B. Maurey. A special case of Proposition 4.11 can be obtained with just two spaces distributed on four arches as above: just consider X0 = ℓ∞ and X1 = ℓ1 . We get a family where, for every non-zero z ∈ D, Φz is a multiple of the Kalton-Peck

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21

map on Xz , Φ0 is 0, and Xz = ℓ2 for every z = t, z = it, t ∈ (−1, 1). So while we have bounded splitting in 0 we have non trivial (even singular) Φz for z arbitrarily close to 0. We obtain in this way a stability result that also justifies the exceptions that have appeared in the previous examples. Theorem 4.15. Let (X0 , X1 ) be an interpolation pair of K¨othe function spaces. With the notation of Theorem 4.12, and w = α + i˜ α as in Proposition 4.13, we have: (1) If the derivation Ωz0 is bounded for some z0 ∈ D such that 0 < α(z0 ) < 1 and w ′ (z0 ) 6= 0, then X0 = X1 with equivalence of norms, and the derivation Ωz is bounded for all z ∈ D. (2) If the derivation Ωz0 is trivial for some z0 ∈ D such that 0 < α(z0 ) < 1 and w ′(z0 ) 6= 0, then X1 is a weighted version of X0 with equivalence of norms, and the derivation Ωz is trivial for all z ∈ D. 5. Stability results for arbitrary pairs In the case of interpolation spaces obtained from a pair of arbitrary spaces we will prove some uniqueness theorems of an isometric nature, instead of up to equivalent renorming like in the case of K¨othe spaces. To do this we need to make a close inspection of the properties of the extremal functions and of the function z → k · kz . In this section we consider the following alternative description of the complex interpolation method applied to an interpolation pair (X0 , X1 ) as given in [23]: Let F ∞ (X) ≡ F ∞ (S, X0 +X1 ) denote the space of all functions F : S → X0 +X1 analytic on the unit strip S such that F (j + it) ∈ Xj for j = 0, 1 and t ∈ R, the maps fj : t ∈ R → F (j + it) ∈ Xj (j = 0, 1) are Bochner measurable, kF kF∞ (X) = maxj=0,1 kfj kL∞ (R,Xj ) < ∞, and Z Z F (z) = F (it)Q0 (z, t) dt + F (1 + it)Q1 (z, t) dt for each z ∈ S, R

where

Qi (a + ib, t) =

R

e−π(t−b sin(πa) sin2 (πa) + (cos(πa) − eijπ−π(t−b) )

2,

j = 0, 1.

It is not difficult to check that the space F ∞ (X) endowed with k · kF∞ (X) is an admissible space of analytic functions on S (Definition 2.3). Moreover, for 0 < θ < 1, the associated spaces Xθ coincide (with equality of norms) with the spaces obtained using the usual description of the complex interpolation method [23, p. 288]. Remark 1. Let 0 < θ < 1 and t ∈ R. The invariance under vertical translations of the strip S implies that given f in the Calderon space C such that f (θ) = x, the function g(z) = f (z − it) is in C and satisfies kf kC = kgkC and g(θ + it) = x; and the same is true for the space F ∞ (X). In particular Xθ = Xθ+it isometrically, and it is enough to study the scale (Xθ )0 θ by the assumptions of 6.2.

´ M. F. CASTILLO, WILLIAN H. G. CORREA, ˆ VALENTIN FERENCZI, MANUEL GONZALEZ ´ JESUS

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We prove (2), and the proof of (3) is similar: observe that if it happens otherwise we may find a normalized block-sequence (wi )i in Xθ which is c-equivalent to a block-sequence (vi )i in Xt . For each n ∈ N, we may find a normalized disjoint sequence (yi )i=1,...,n of (wi )i such that ky1 + · · · + yn k ≥ C −1 MXθ (n). The sequence (yi ) is c-equivalent to a semi-normalized disjoint sequence (zi ) in Xt . By assumption kz1 + . . . + zn k ≤ cMX0 (n)1−t MX1 (n)t . Therefore C −1 MX0 (n)1−θ MX1 (n)θ ≤ CcMX0 (n)1−t MX1 (n)t and  θ−t MX1 (n) ≤ cC 2 , MX0 (n) which yields a contradiction.  All this suggests the following problem: Problem 6.4. Assume that Xθ is totally incomparable with Xt for t 6= θ in a neighborhood of θ. Is Ωθ singular? A partial result in this direction is Proposition 6.5. If there is ε > 0 so that the spaces Xs and Xθ are pairwise totally incomparable for |s − θ| < ε then the exact sequences δ

θ 0 −−−→ ker δθ −−−→ F −−− → Xθ −−−→ 0

and ∆

0 −−−→ ker ∆θ −−−→ F −−−θ→ dXθ −−−→ 0 are singular. Proof. Suppose that M is an infinite dimensional closed subspace of C and that the restriction δθ |M is an isomorphism. Then the sum M + ker δθ is direct and closed. By Propositions 2.1 and 3.5, when |θ − t| is small enough the sum M + ker δt is also direct and closed. Hence both Xθ and Xt contain subspaces isomorphic to M. As for the second part, observe that the spaces dXs are pairwise totally incomparable when the spaces Xs are so by a 3-space argument.  References [1] A. Avil´es, F. Cabello S´ anchez, J.M.F. Castillo, M. Gonz´alez, Y. Moreno, Separably injective Banach spaces. Lecture Notes in Mathematics, 2132. Springer, 2016. [2] J. Bak, D. J. Newman, Complex analysis. Springer, 1982. [3] J. Bergh, J. L¨ ofstr¨ om, Interpolation spaces. An introduction. Springer, 1976. [4] F. Cabello, J.M.F. Castillo, S. Goldstein, Jes´ us Su´ arez, Twisting noncommutative Lp -spaces, Adv. in Math. 294 (2016) 454–488. [5] F. Cabello, J.M.F. Castillo, N. J. Kalton, Complex interpolation and twisted twisted Hilbert spaces, Pacific J. Math. 276 (2015) 287–307. [6] W. Cao, Y. Sagher, Stability of Fredholm properties on interpolation scales, Ark. Mat. 28, (1990) 249–258. [7] W. Cao, Y. Sagher, Stability in interpolation families of Banach spaces, Proc. Amer. Math. Soc. 112 (1991) 91-100.

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[32] N.J. Kalton, S. Montgomery-Smith, Interpolation of Banach spaces, Chapter 36 in Handbook of the Geometry of Banach spaces vol. 2, (W.B. Johnson and J. Lindenstrauss eds.), pp. 1131–1175, North-Holland 2003. [33] N.J. Kalton, M. Ostrovskii, Distances between Banach spaces, Forum Math. 11 (1999) 17-48. [34] N.J. Kalton, N.T. Peck, Twisted sums of sequence spaces and the three space problem, Trans. Amer. Math. Soc. 255 (1979) 1–30. [35] T. Kato, Perturbation theory for linear operators, Corrected printing of the 2nd. ed. Springer, 1980. [36] N. Kruglijak, M. Milman, A distance between orbits that controls commutator estimates and invertibility of operators, Adv. Math. 182 (2004) 78-123. [37] K.B. Laursen, M.M. Neumann, An introduction to local spectral theory, London Mathematical Society Monographs. New Series, 20. The Clarendon Press, Oxford University Press, New York, 2000. [38] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces I. Sequence spaces, Springer, 1977. [39] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces II. Function spaces, Springer, 1979. [40] Odell Schlumprecht, The distortion problem, Acta Math. 173 (1994) 259-281. [41] V.I. Ovchinnikov, The method of orbits in interpolation theory, Math. Reports vol. 1, part 2, pp. 349-516. Harwood Acad. Publ., 1984. [42] G. Pisier, Some applications of the complex interpolation method to Banach lattices, J. Anal. Math. 35 (1979), 264–281 [43] G. Pisier, Complex Interpolation Between Hilbert, Banach and Operator Spaces, Mem. Amer. Math. Soc. 978, 2010. [44] R. Rochberg, Function theoretic results for complex interpolation families of Banach spaces, Trans. Amer. Math. Soc. 284 (1984) 745-758. [45] R. Rochberg, Higher order estimates in complex interpolation theory, Pacific J. Math. 174 (1996) 247–267. [46] R. Rochberg, G. Weiss, Derivatives of Analytic Families of Banach Spaces, Annals of Math. 118 (1983) 315–347. [47] I.Ya. Schneiberg, Spectral properties of linear operators in interpolation families of Banach spaces, Mat. Issled 9 (1974) 214-229. [48] J. Stafney, Analytic interpolation of certain multiplier spaces, Pacific J. Math 32 (1970) 241-248. [49] M. Zafran, Spectral theory and interpolation of operators, J. Funct. Anal. 36 (1980) 185-204. ´ticas, Universidad de Extremadura, Avenida de Elvas s/n, 06011 Departamento de Matema Badajoz, Spain. E-mail address: [email protected] ´tica, Instituto de Matema ´tica e Estat´ıstica, Universidade de Sa ˜o Departamento de Matema ˜ ˜ Paulo, rua do Matao 1010, 05508-090 Sao Paulo SP, Brazil E-mail address: [email protected] ´tica, Instituto de Matema ´tica e Estat´ıstica, Universidade de Sa ˜o Departamento de Matema ˜o 1010, 05508-090 Sa ˜o Paulo SP, Brazil, and Paulo, rua do Mata Equipe d’Analyse Fonctionnelle, Institut de Math´ ematiques de Jussieu, Universit´ e Pierre et Marie Curie - Paris 6, Case 247, 4 place Jussieu, 75252 Paris Cedex 05, France. E-mail address: [email protected] ´ticas, Universidad de Cantabria, Avenida de los Castros s/n, 39071 Departamento de Matema Santander, Spain. E-mail address: [email protected]