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Pivot Duality of Universal Interpolation and Extrapolation Spaces

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Pivot Duality of Universal Interpolation and Extrapolation Spaces PIVOT DUALITY OF UNIVERSAL INTERPOLATION AND EXTRAPOLATION SPACES CHRISTIAN BARGETZ1 AND SVEN-AKE WEGNER2 Abstract. It is a widely used method, for instance in perturbation theory, to associate with a given C0-semigroup its so-called interpolation and extrapolation spaces. In the model case of the 2 shift semigroup acting on L (R), the resulting chain of spaces recovers the classical Sobolev scale. In 2014, the second named author defined the universal interpolation space as the projective limit of the interpolation spaces and the universal extrapolation space as the completion of the inductive limit of the extrapolation spaces, provided that the latter is Hausdorff. In this note we use the notion of the dual with respect to a pivot space in order to show that the aforementioned inductive limit is Hausdorff, already complete, and can be represented as the dual of the projective limit whenever a power of the generator of the initial semigroup is a self-adjoint operator. In the case of the classical Sobolev scale we show that the duality holds, and that the two universal spaces were already studied by Laurent Schwartz in the 1950s. Our results and examples complement the approach of Haase, who in 2006 gave a different definition of universal extrapolation spaces in the context of functional calculi. Haase avoids the inductive limit topology precisely for the reason that it a priori cannot be guaranteed that the latter is always Hausdorff. We show that this is indeed the case provided that we start with a semigroup defined on a reflexive Banach space. 1. The classical Sobolev scale We start by considering the following generic example. Let (T (t))t>0 denote the left shift semigroup 2 d d on the Hilbert space L (R) generated by the first derivative dx defined on the domain D( dx ) = f 2 d 2 f 2 L (R); dx f L (R) . Writing down the abstract interpolation and extrapolation spaces [5, Chapter II.5] gives the2 classicalg scale of Sobolev spaces 1 2 3 2 1 0 i− i− 3 i3 2 i2 1 i1 2 i0− 1 1 2 2 3 H (R) H (R) H (R) L (R) H− (R) − H− (R) − H− (R) · · · −! −! −! −! −! −! −! !··· where the maps are all continuous. Taking the projective limit of this chain of spaces, i.e., endowing the n n intersection n H (R) with the coarsest linear topology which makes the inclusions n H (R) \ 2N \ 2N ! Hk(R) for all k N continuous, yields the classical function space 2 n DL2 (R) = projn H (R) 2N n studied by Schwartz [10, 8, p. 199]. Taking the inductive limit, i.e., endowing the union n NH− (R) x k n [ 2 with the finest linear topology which makes the inclusions H (R) n NH− (R) for all k N continuous, yields a subspace of the space of distributions which turns![ out2 to be isomorphic to2 the strong dual of DL2 (R), i.e., n DL0 2 (R) = indn H− (R) ∼ 2N in a natural way. Also this space was investigated by Schwartz [10, 8, p. 200]. Indeed, we have the following commutative diagram x 1 2 3 arXiv:1604.00763v3 [math.FA] 16 Nov 2017 i− i− i− 2 0 1 1 2 2 3 L (R) H− (R) − H− (R) − H− (R) ··· id 2 Á Á Á (1) L (R) 1 2 3 2 1 2 3 L (R) 0 H (R)0 1 H (R)0 2 H (R)0 (i1)0 (i2)0 (i3)0 ··· 2010 Mathematics Subject Classification: Primary 47A06, Secondary 47D03, 46A13. Key words and phrases: Dual with respect to a pivot space, extrapolation space, interpolation space. 1 University of Innsbruck, Department of Mathematics, Technikerstraße 13, 6020 Innsbruck, Austria (current address) and The Technion|Israel Institute of Technology, Department of Mathematics, Haifa 320000, Israel, E-Mail: [email protected]. 2 Corresponding author: University of Wuppertal, School of Mathematics and Natural Sciences, Gaußstraße 20, 42119 Wuppertal, Germany (current address) and Nazarbayev University, School of Science and Technology, 53 Kabanbay Batyr Ave, 010000 Astana, Kazakhstan, E-Mail: [email protected], Phone: +49 (0) 202 / 439 - 2590. 1 where the maps Á¡n for n N are isomorphisms. Our first aim is to see that the corresponding inductive 2 N limits are isomorphic. We2 emphasize that this is not trivial just by having “step-wise”\step-wise" isomorphisms. n 2 2 n Indeed, we have for instance H− (R) = L (R) for each n N but L (R) = n NH− (R), which ⇠∼ 2 ⇠6∼ [ 2N shows that we have to be extremely careful when we “identify”\identify"2 isomorphic spaces.6 [ The suitable notion to address our first aim is that of equivalent inductive sequences. Each row in n+1 the diagram (1) is a so-called inductive sequence, i.e., a sequence (Xn,i; in )n N of Banach spaces n+1 2 Xn and linear and continuous maps in : Xn Xn+1 for n N.Twosuchinductivesequences. Two such inductive sequences n+1 n+1 ! 2 (Xn;,i in )n N and (Yn;,j jn )n N are said to be equivalent, if there are increasing sequences (k(n))n N 2N 2N 2N and (`(n))n N of natural numbers with n 6 `(n) 6 k(n) 6 `(n + 1) and linear and continuous maps 2N α↵n : Y`(n) Xk(n), βn : Xk(n) Y`(n+1) such that n `(n) ! k(n) n k(n) ! `(n+1) ik(n+1) X k(n) X ··· k(n) k(n+1) ··· ↵n βn ↵n+1 βn+1 Y`(n) `(n+1) Y`(n+1) `(n+2) Y`(n+2) ··· j`(n) j`(n+1) ··· commutes. AsBy a a matter matter of of fact, fact, two two equivalent equivalent inductive sequences have isomorphic inductive limits. We now see that the two inductive sequences in (1) are equivalent and we thus get that n n indn H−n(R) = indn Hn(R)0 indn2N H− (R) ⇠= indn2N H (R)0 2N ∼ 2N holds. Now we would like to conclude that the dual of a projective limit (=intersection) is equal to the inductive limitlimit(=union) (=union) of the duals of the spaces in the sequence. This is indeed true but requires an open mapping theorem for topological vector spaces. Note We advise that the those important readers, conclusion who are not of toothe familiar with the latter, to take the following a bit easy. The important conclusion is, that we finally following argument is that we finally get the isomorphism (2) below. We observe that DL2 (R) is a D 2 reflexiveget the isomorphism Fr´echet space (2) by below. JarchowWe [8, observe Proposition that 11.5.5L (R) and is a Corollary reflexive 11.4.3] Fr´echet and space thus by distinguished Jarchow [8, Proposition 11.5.5 and Corollary 11.4.3] and thus distinguished by [8, Remark after 13.4.5D on p. 280]. by [8, Remark after 13.4.5 on p. 280]. Then by [8, Corollary 13.4.4] it followsn that L0 2 (R(n)+1) is ul- Then by [8, Corollary 13.4.4] it follows that D 2 ( ) is ultrabornological. As H ( ) H ( ) trabornological. As H n( ) H (n+1)( ) isL0 denseR for all n , we have that− theR corresponding− R is dense for all n , we− haveR that− the correspondingR sequence isN reduced and [8, Proposition✓ 8.8.7] sequence is reducedN and [8, Proposition⊆ 8.8.7] implies that there2 is a natural isomorphism implies that there2 is a natural isomorphism n ' n H ' H D indn N n (R)0 (projn N n (R))b0 = L0 2 (R) (2) indn2 H (R)0 −! (projn2 H (R))b0 = DL0 2 (R) (2) 2N N n ! n2 n H H of linear spaces. As ' H (R)0 : n(R)0 (projn n(R))b0 is continuous, it follows that ' is continu- of linear spaces. As 'jHn : H ( )0 ! (proj 2N H ( ))0 is continuous, it follows that ' is continu- ous and hence an isomorphism(R)0 byR the open mappingn N theoremR b [8, Theorem 5.5.2] as its domain is|as ous and hence an isomorphism| by the! open mapping2 theorem [8, Theorem 5.5.2] as its domain is—as an LB-space|in particular webbed [8, Corollary 5.3.3] and its codomain is ultrabornological. an LB-space—in particular webbed [8, Corollary 5.3.3] and its codomain is ultrabornological. We point out out again, again, that that the the crucial crucial fact fact for for the the above above arguments arguments is not is not just just the the existence existence of isomor- of iso- morphisms¡ Án. In the current example this indeed might appear to be trivial by a well-known phisms n. In the current example this indeed mightn appear to be trivial by well-known characteri-n characterization of the Sobolev spaces, e.g.,n that H− (R) can be identified with then dual of H (R). zation of the Sobolev spaces, e.g., that H− (R) can be identified with the dual of H (R). However, However,exactly these exactly identifications these identifications have to be have compatible to be compatible with the mapswith the in the maps inductive in the inductive sequences sequences in order into order induce to an induce equivalence an equivalence of the latter. of the latter. Below, Below, in Definition in Definition 2, we 2, employ we employ a generalization a generalization of the of thenotion notion of the of the “dual \dual with with respect respect to to a pivot a pivot space”, space", see see Tucsnak, Tucsnak, Weiss Weiss [11, [11, Chapter Chapter 2.10], 2.10], in in order to construct families of isomorphisms that are compatible with the linking maps of the corresponding inductivespectra.
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