×

Complex interpolation between Hilbert, Banach and operator spaces. (English) Zbl 1213.46002

Mem. Am. Math. Soc. 978, v, 78 p. (2010).
Motivated by a question of Vincent Lafforgue, the author studies the Banach spaces \( X\) satisfying the following property: there is a function \( \varepsilon\to \Delta_X(\varepsilon)\) tending to zero with \( \varepsilon>0\) such that every operator \( T: L_2\to L_2\) with \( \| T\|\leq \varepsilon\) that is simultaneously contractive (i.e., of norm \( \leq 1\)) on \( L_1\) and on \( L_\infty\) must be of norm \( \leq \Delta_X(\varepsilon)\) on \( L_2(X)\). He shows that \( \Delta_X(\varepsilon) \in O(\varepsilon^\alpha)\) for some \( \alpha>0\) if and only if \( X\) is isomorphic to a quotient of a subspace of a \( \theta\)-Hilbertian space for some \( \theta>0\), where \( \theta\)-Hilbertian is meant in a slightly more general sense than it was defined initially in one of the author’s earlier papers [J.Anal.Math.35, 264–281 (1979; Zbl 0427.46048)].
Although this new notion of \(\theta\)-Hilbertian spaces seems to be quite involved (complex interpolation of families of finite-dimensional Banach spaces rather than complex interpolation of two spaces, as well as ultraproducts of the resulting finite-dimensional spaces), it offers new ways for the (admittedly rather abstract) description of certain complex interpolation spaces.
E.g., let \( B_{{r}}(L_2(\mu))\) be the space of all regular operators and \(B(L_2(\mu))\) the space of all bounded operators on \(L_2(\mu)\), respectively. The author is able to describe the complex interpolation space \((B_{{r}}(L_2(\mu)), B(L_2(\mu)))^\theta \) in the following way: an operator \( T: L_2(\mu)\to L_2(\mu)\) belongs to this space if and only if \( T\otimes id_X\) is bounded on \( L_2(X)\) for any \( \theta\)-Hilbertian space \( X\). More generally, as a generalization of the corresponding result for \(p_0=1\) and \(p_1 = \infty\) from the author’s paper [Arch.Math.62, 261–269 (1994; Zbl 0991.46007)], a description of the spaces \({ (B(\ell_{p_0}), B(\ell_{p_1}))^\theta }\) or \((B(L_{p_0}), B(L_{p_1}))^\theta \) for any pair \( 1\leq p_0,p_1\leq \infty\) and \( 0<\theta<1\) is given. In the same vein, given a locally compact Abelian group \( G\), let \( M(G)\) (resp., \( PM(G)\)) be the space of complex measures (resp., pseudo-measures) on \( G\) equipped with the usual norm \( \|\mu\| _{M(G)} = |\mu|(G)\) (resp., \( \|\mu\| _{PM(G)} = \sup\{|\hat\mu(\gamma)| \big| \gamma\in\widehat G\})\). The author describes similarly the interpolation space \( (M(G), PM(G))^\theta\).
Various extensions and variants of these results are given, e.g., to Schur multipliers on \( B(\ell_2)\) and to operator spaces.

MSC:

46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
46B70 Interpolation between normed linear spaces
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46M05 Tensor products in functional analysis
47A80 Tensor products of linear operators
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Jöran Bergh, On the relation between the two complex methods of interpolation, Indiana Univ. Math. J. 28 (1979), no. 5, 775-778. · Zbl 0394.41004 · doi:10.1512/iumj.1979.28.28054
[2] Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. · Zbl 0344.46071
[3] J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 21 (1983), no. 2, 163-168. · Zbl 0533.46008 · doi:10.1007/BF02384306
[4] J. Bourgain, On trigonometric series in super reflexive spaces, J. London Math. Soc. (2) 24 (1981), no. 1, 165-174. · Zbl 0475.46008 · doi:10.1112/jlms/s2-24.1.165
[5] J. Bourgain, A Hausdorff-Young inequality for \(B\)-convex Banach spaces, Pacific J. Math. 101 (1982), no. 2, 255-262. · Zbl 0498.46014
[6] J. Bourgain, Vector-valued Hausdorff-Young inequalities and applications, Geometric aspects of functional analysis (1986/87), Lecture Notes in Math., vol. 1317, Springer, Berlin, 1988, pp. 239-249. · doi:10.1007/BFb0081745
[7] Artur Buchholz, Norm of convolution by operator-valued functions on free groups, Proc. Amer. Math. Soc. 127 (1999), no. 6, 1671-1682. · Zbl 0916.43002 · doi:10.1090/S0002-9939-99-04660-2
[8] A. V. Buhvalov, Geometric properties of Banach spaces of measurable vector-valued functions, Dokl. Akad. Nauk SSSR 239 (1978), no. 6, 1279-1282 (Russian).
[9] A. V. Buhvalov, Continuity of operators in the spaces of measurable vector-valued functions with applications to the investigation of Sobolev spaces and analytic functions in t, Dokl. Akad. Nauk SSSR 246 (1979), no. 3, 524-528 (Russian).
[10] D. L. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 270-286.
[11] D. L. Burkholder, R. F. Gundy, and M. L. Silverstein, A maximal function characterization of the class \(H^{p}\), Trans. Amer. Math. Soc. 157 (1971), 137-153. · Zbl 0223.30048 · doi:10.1090/S0002-9947-1971-0274767-6
[12] A.- P. Calderón, Intermediate spaces and interpolation, Studia Math. (Ser. Specjalna) Zeszyt 1 (1963), 31-34. · Zbl 0124.31803
[13] R. Coifman, M. Cwikel, R. Rochberg, Y. Sagher, and G. Weiss, Complex interpolation for families of Banach spaces, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 269-282. · Zbl 0422.46032
[14] R. R. Coifman, R. Rochberg, G. Weiss, M. Cwikel, and Y. Sagher, The complex method for interpolation of operators acting on families of Banach spaces, Euclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979) Lecture Notes in Math., vol. 779, Springer, Berlin, 1980, pp. 123-153. · Zbl 0422.46032
[15] R. R. Coifman, M. Cwikel, R. Rochberg, Y. Sagher, and G. Weiss, A theory of complex interpolation for families of Banach spaces, Adv. in Math. 43 (1982), no. 3, 203-229. · Zbl 0501.46065 · doi:10.1016/0001-8708(82)90034-2
[16] R. R. Coifman and S. Semmes, Interpolation of Banach spaces, Perron processes, and Yang-Mills, Amer. J. Math. 115 (1993), no. 2, 243-278. · Zbl 0789.46021 · doi:10.2307/2374859
[17] Peter L. Duren, Theory of \(H^{p}\) spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. · Zbl 0215.20203
[18] Edward G. Effros and Zhong-Jin Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000. · Zbl 0969.46002
[19] Per Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), 1972, pp. 281-288 (1973).
[20] Tadeusz Figiel, Singular integral operators: a martingale approach, Geometry of Banach spaces (Strobl, 1989) London Math. Soc. Lecture Note Ser., vol. 158, Cambridge Univ. Press, Cambridge, 1990, pp. 95-110. · Zbl 0746.47026
[21] D. J. H. Garling and S. J. Montgomery-Smith, Complemented subspaces of spaces obtained by interpolation, J. London Math. Soc. (2) 44 (1991), no. 3, 503-513. · Zbl 0770.46030 · doi:10.1112/jlms/s2-44.3.503
[22] John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. · Zbl 0469.30024
[23] A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Resenhas 2 (1996), no. 4, 401-480 (French). Reprint of Bol. Soc. Mat. São Paulo 8 (1953), 1-79 [ MR0094682 (20 #1194)]. · Zbl 1019.46038
[24] M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1-295. · Zbl 0841.20039
[25] Uffe Haagerup and Gilles Pisier, Bounded linear operators between \(C^*\)-algebras, Duke Math. J. 71 (1993), no. 3, 889-925. · Zbl 0803.46064 · doi:10.1215/S0012-7094-93-07134-7
[26] Asma Harcharras, Fourier analysis, Schur multipliers on \(S^p\) and non-commutative \(\Lambda (p)\)-sets, Studia Math. 137 (1999), no. 3, 203-260. · Zbl 0948.43002
[27] Henry Helson, Lectures on invariant subspaces, Academic Press, New York-London, 1964. · Zbl 0119.11303
[28] Eugenio Hernández, Intermediate spaces and the complex method of interpolation for families of Banach spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13 (1986), no. 2, 245-266. · Zbl 0617.46078
[29] Roberto Hernandez, Espaces \(L^{p}\), factorisation et produits tensoriels dans les espaces de Banach, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 9, 385-388 (French, with English summary). · Zbl 0552.46039
[30] R. Hernandez, Espaces \(L^p\), factorisations et produits tensoriels, Seminar on the geometry of Banach spaces, Vol. I, II (Paris, 1983), Publ. Math. Univ. Paris VII, vol. 18, Univ. Paris VII, Paris, 1984, pp. 63-79 (French).
[31] Carl Herz, The theory of \(p\)-spaces with an application to convolution operators., Trans. Amer. Math. Soc. 154 (1971), 69-82. · Zbl 0216.15606 · doi:10.1090/S0002-9947-1971-0272952-0
[32] Robert C. James, Some self-dual properties of normed linear spaces, Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967) Princeton Univ. Press, Princeton, N.J., 1972, pp. 159-175. Ann. of Math. Studies, No. 69.
[33] John E. Jayne and C. Ambrose Rogers, Selectors, Princeton University Press, Princeton, NJ, 2002. · Zbl 1002.54001
[34] W. B. Johnson and L. Jones, Every \(L_{p}\) operator is an \(L_{2}\) operator, Proc. Amer. Math. Soc. 72 (1978), no. 2, 309-312. · Zbl 0391.46026 · doi:10.1090/S0002-9939-1978-0507330-1
[35] Junge, M. Factorization theory for spaces of operators. Habilitationsschrift, Kiel University, 1996. · Zbl 0851.47022
[36] Nigel Kalton and Stephen Montgomery-Smith, Interpolation of Banach spaces, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1131-1175. · Zbl 1041.46012 · doi:10.1016/S1874-5849(03)80033-5
[37] Gennadi Kasparov and Guoliang Yu, The coarse geometric Novikov conjecture and uniform convexity, Adv. Math. 206 (2006), no. 1, 1-56. · Zbl 1102.19003 · doi:10.1016/j.aim.2005.08.004
[38] M. Koskela, A characterization of non-negative matrix operators on \(l^{p}\) to \(l^{q}\) with \(\infty >p\geq q>1\), Pacific J. Math. 75 (1978), no. 1, 165-169. · Zbl 0374.47014
[39] Omran Kouba, On the interpolation of injective or projective tensor products of Banach spaces, J. Funct. Anal. 96 (1991), no. 1, 38-61. · Zbl 0753.46037 · doi:10.1016/0022-1236(91)90072-D
[40] S. Kwapień, Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients, Studia Math. 44 (1972), 583-595. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, VI. · Zbl 0256.46024
[41] Stanislaw Kwapień, On operators factorizable through \(L_{p}\) space, Actes du Colloque d’Analyse Fonctionnelle de Bordeaux (Univ. de Bordeaux, 1971) Soc. Math. France, Paris, 1972, pp. 215-225. Bull. Soc. Math. France, Mém. No. 31-32.
[42] S. G. Kreĭn, Yu. Ī. Petunīn, and E. M. Semënov, Interpolation of linear operators, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, R.I., 1982. Translated from the Russian by J. Szűcs.
[43] Vincent Lafforgue, Un renforcement de la propriété (T), Duke Math. J. 143 (2008), no. 3, 559-602 (French, with English and French summaries). · Zbl 1158.46049 · doi:10.1215/00127094-2008-029
[44] Damien Lamberton, Spectres d’opérateurs et géométrie des espaces de Banach, Dissertationes Math. (Rozprawy Mat.) 242 (1985), 58 (French). · Zbl 0568.47003
[45] Alexander Lubotzky, Discrete groups, expanding graphs and invariant measures, Progress in Mathematics, vol. 125, Birkhäuser Verlag, Basel, 1994. With an appendix by Jonathan D. Rogawski. · Zbl 0826.22012
[46] José L. Marcolino Nhani, La structure des sous-espaces de treillis, Dissertationes Math. (Rozprawy Mat.) 397 (2001), 50 (French, with English summary). · Zbl 0990.46007
[47] Jiří Matoušek, Lectures on discrete geometry, Graduate Texts in Mathematics, vol. 212, Springer-Verlag, New York, 2002. · Zbl 0999.52006
[48] Jiří Matoušek, On embedding expanders into \(l_p\) spaces, Israel J. Math. 102 (1997), 189-197. · Zbl 0947.46007 · doi:10.1007/BF02773799
[49] Bernard Maurey, Type, cotype and \(K\)-convexity, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1299-1332. · Zbl 1074.46006 · doi:10.1016/S1874-5849(03)80037-2
[50] Peter Meyer-Nieberg, Banach lattices, Universitext, Springer-Verlag, Berlin, 1991. · Zbl 0743.46015
[51] M. I. Ostrovskii, Coarse embeddability into Banach spaces, Topology Proc. 33 (2009), 163-183. · Zbl 1179.54042
[52] Narutaka Ozawa, A note on non-amenability of \({\scr B}(l_p)\) for \(p=1,2\), Internat. J. Math. 15 (2004), no. 6, 557-565. · Zbl 1056.46046 · doi:10.1142/S0129167X04002430
[53] Assaf Naor, Yuval Peres, Oded Schramm, and Scott Sheffield, Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces, Duke Math. J. 134 (2006), no. 1, 165-197. · Zbl 1108.46012 · doi:10.1215/S0012-7094-06-13415-4
[54] Javier Parcet and Gilles Pisier, Non-commutative Khintchine type inequalities associated with free groups, Indiana Univ. Math. J. 54 (2005), no. 2, 531-556. · Zbl 1091.46035 · doi:10.1512/iumj.2005.54.2612
[55] Stefanie Petermichl, Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 6, 455-460 (English, with English and French summaries). · Zbl 0991.42003 · doi:10.1016/S0764-4442(00)00162-2
[56] Albrecht Pietsch, Operator ideals, North-Holland Mathematical Library, vol. 20, North-Holland Publishing Co., Amsterdam-New York, 1980. Translated from German by the author. · Zbl 0434.47030
[57] Gilles Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), no. 3-4, 326-350. · Zbl 0344.46030
[58] G. Pisier, Some applications of the complex interpolation method to Banach lattices, J. Analyse Math. 35 (1979), 264-281. · Zbl 0427.46048 · doi:10.1007/BF02791068
[59] Gilles Pisier, Factorization of linear operators and geometry of Banach spaces, CBMS Regional Conference Series in Mathematics, vol. 60, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. · Zbl 0588.46010
[60] Gilles Pisier, Completely bounded maps between sets of Banach space operators, Indiana Univ. Math. J. 39 (1990), no. 1, 249-277. · Zbl 0747.46015 · doi:10.1512/iumj.1990.39.39014
[61] Gilles Pisier, Complex interpolation and regular operators between Banach lattices, Arch. Math. (Basel) 62 (1994), no. 3, 261-269. · Zbl 0991.46007 · doi:10.1007/BF01261367
[62] Gilles Pisier, Regular operators between non-commutative \(L_p\)-spaces, Bull. Sci. Math. 119 (1995), no. 2, 95-118. · Zbl 0826.46056
[63] Gilles Pisier, The operator Hilbert space \({\mathrm OH}\), complex interpolation and tensor norms, Mem. Amer. Math. Soc. 122 (1996), no. 585, viii+103. · Zbl 0932.46046 · doi:10.1090/memo/0585
[64] Gilles Pisier, Non-commutative vector valued \(L_p\)-spaces and completely \(p\)-summing maps, Astérisque 247 (1998), vi+131 (English, with English and French summaries). · Zbl 0937.46056
[65] Gilles Pisier, Similarity problems and completely bounded maps, Second, expanded edition, Lecture Notes in Mathematics, vol. 1618, Springer-Verlag, Berlin, 2001. Includes the solution to “The Halmos problem”. · Zbl 0971.47016
[66] Gilles Pisier, Introduction to operator space theory, London Mathematical Society Lecture Note Series, vol. 294, Cambridge University Press, Cambridge, 2003. · Zbl 1093.46001
[67] Richard Rochberg, Function theoretic results for complex interpolation families of Banach spaces, Trans. Amer. Math. Soc. 284 (1984), no. 2, 745-758. · Zbl 0592.46064 · doi:10.1090/S0002-9947-1984-0743742-6
[68] Richard Rochberg, Interpolation of Banach spaces and negatively curved vector bundles, Pacific J. Math. 110 (1984), no. 2, 355-376. · Zbl 0549.46040
[69] Richard Rochberg, The work of Coifman and Semmes on complex interpolation, several complex variables, and PDEs, Function spaces and applications (Lund, 1986) Lecture Notes in Math., vol. 1302, Springer, Berlin, 1988, pp. 74-90. · doi:10.1007/BFb0078864
[70] John Roe, Lectures on coarse geometry, University Lecture Series, vol. 31, American Mathematical Society, Providence, RI, 2003. · Zbl 1042.53027
[71] Marvin Rosenblum and James Rovnyak, Topics in Hardy classes and univalent functions, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1994. · Zbl 0816.30001
[72] Helmut H. Schaefer, Banach lattices and positive operators, Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 215. · Zbl 0296.47023
[73] Stephen Semmes, Interpolation of Banach spaces, differential geometry and differential equations, Rev. Mat. Iberoamericana 4 (1988), no. 1, 155-176. · Zbl 0716.46056 · doi:10.4171/RMI/67
[74] Zbigniew Slodkowski, Polynomial hulls with convex sections and interpolating spaces, Proc. Amer. Math. Soc. 96 (1986), no. 2, 255-260. · Zbl 0588.32017 · doi:10.1090/S0002-9939-1986-0818455-X
[75] Zbigniew Slodkowski, Complex interpolation of normed and quasinormed spaces in several dimensions. I, Trans. Amer. Math. Soc. 308 (1988), no. 2, 685-711. · Zbl 0657.46057 · doi:10.1090/S0002-9947-1988-0951623-1
[76] S. M. Srivastava, A course on Borel sets, Graduate Texts in Mathematics, vol. 180, Springer-Verlag, New York, 1998. · Zbl 0903.28001
[77] Bernard Virot, Extensions vectorielles d’opérateurs linéaires bornés sur \(L^{p}\), C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), no. 8, 413-415 (French, with English summary). · Zbl 0473.46023
[78] Lutz Weis, Integral operators and changes of density, Indiana Univ. Math. J. 31 (1982), no. 1, 83-96. · Zbl 0492.47017 · doi:10.1512/iumj.1982.31.31010
[79] Quanhua Xu, Interpolation of Schur multiplier spaces, Math. Z. 235 (2000), no. 4, 707-715. · Zbl 0984.47018 · doi:10.1007/PL00004817
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.