$H^\infty$ functional calculus and square functions on noncommutative $L^p$-spaces

H^{\infty}

functional calculus and square functions on noncommutative

L^{p}

-spaces

Junge, Marius ; Le Merdy, Christian ; Xu, Quanhua

Astérisque, no. 305 (2006) , 144 p.

Zbl | 1 citation dans Numdam

@book{AST_2006__305__R1_0,
     author = {Junge, Marius and Le Merdy, Christian and Xu, Quanhua},
     title = {$H^\infty$ functional calculus and square functions on noncommutative $L^p$-spaces},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {305},
     year = {2006},
     zbl = {1106.47002},
     language = {en},
     url = {http://www.numdam.org/item/AST_2006__305__R1_0/}
}

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%A Le Merdy, Christian
%A Xu, Quanhua
%T $H^\infty$ functional calculus and square functions on noncommutative $L^p$-spaces
%S Astérisque
%D 2006
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Junge, Marius; Le Merdy, Christian; Xu, Quanhua. $H^\infty$ functional calculus and square functions on noncommutative $L^p$-spaces. Astérisque, no. 305 (2006), 144 p. http://numdam.org/item/AST_2006__305__R1_0/

Bibliographie
Cité par

[1] M. Akcoglu - A pointwise ergodic theorem in $L_{p}$ -spaces, Canad. J. Math. 27 (1975), p. 1075-1082. | Zbl | DOI

[2] M. Akcoglu & L. Sucheston - Dilations of positive contractions on $L_{p}$ spaces, Canad. Math. Bull. 20 (1977), p. 285-292. | Zbl | DOI

[3] D. Albrecht, X. T. Duong & A. Mcintosh - Operator theory and harmonie analysis, in Instructional Workshop on Analysis and Geometry, Part III (Canberra, 1995), Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 34, Austral. Nat. Univ., 1996, p. 77-136.

[4] C. Anantharaman-Delaroche - On ergodic theorems for free group actions on noncommutative spaces, Probab. Theory Related Fields 135 (2006), p. 520-546. | Zbl | DOI

[5] P. Auscher, A. Mcintosh & A. Nahmod - Holomorphic functional calculi of operators, quadratic estimates and interpolation, Indiana Univ. Math. J. 46 (1997), p. 375-403. | Zbl | DOI

[6] J. Bergh & J. Löfström - Interpolation spaces, Springer-Verlag, 1970.

[7] E. Berkson & T. A. Gillespie - Spectral decompositions and harmonic analysis on UMD Banach spaces, Studia Math. 112 (1994), p. 13-49. | EuDML | Zbl

[8] E. Berkson, T. A. Gillespie & P. S. Muhly - Abstract spectral décompositions guaranteed by the Hilbert transform, Proc. London Math. Soc. 53 (1986), p. 489-517. | Zbl | DOI

[9] P. Biane - Free hypercontractivity, Commun. Math. Phys. 184 (1997), p. 457-474. | Zbl | DOI

[10] E. Blanchard & K. Dykema - Embeddings of reduced free products of operator algebras, Pacific J. Math. 199 (2001), p. 1-19. | Zbl | DOI

[11] M. Bożejko - Ultracontractivity and strong Sobolev inequality for $q$ -Ornstein-Uhlenbeck semigroup, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2 (1999), p. 203-220. | Zbl | DOI

[12] M. Bożejko & G. Fendler - A note on certain partial sum operatos, 2005, Preprint.

[13] M. Bożejko, B. Kümmerer & R. Speicher - $q$ -Gaussian processes : noncommutative and classical aspects, Commun. Math. Phys. 185 (1997), p. 129-154. | Zbl | DOI

[14] M. Bożejko & R. Speicher - An example of a generalized brownian motion, Commun. Math. Phys. 137 (1991), p. 519-531. | Zbl | DOI

[15] M. Bożejko & R. Speicher, Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces, Math. Ann. 300 (1994), p. 97-120. | EuDML | Zbl | DOI

[16] O. Bratelli & D. W. Robinson - Operator algebras and quantum statistical mechanics II, Springer-Verlag, 1981.

[17] D. Burkholder - Martingales and singular integrals in Banach spaces, in Handbook of the Geometry of Banach Spaces, Vol. I (W. B. Johnson & J. Lindenstrauss, eds.), Elsevier, 2001, p. 233-269. | Zbl | DOI

[18] M.-D. Choi - A schwarz inequality for positive linear maps on $C^{*}$ -algebras, Illinois J. Math. 18 (1974), p. 565-588.

[19] P. Clément, B. De Pagter, F. Sukochev & H. Witvliet - Schauder decompositions and multiplier theorems, Studia Math. 138 (2000), p. 135-163. | EuDML | Zbl

[20] M. Cowling Harmonic analysis on semigroups, Ann. of Math. 117 (1983), p. 267-283. | Zbl | DOI

[21] M. Cowling, I. Doust, A. Mcintosh & A. Yagi - Banach space operators with a bounded $H^{\infty}$ functional calculus, J. Austral. Math. Soc. Ser. A 60 (1996), p. 51-89. | Zbl | DOI

[22] E. B. Davies One-parameter semigroups, London Math. Soc. Monographs, vol. 15, Academic Press Inc., 1980. | Zbl

[23] J. Diestel & J. J. Uhl - Vector measures, vol. 15, Amer. Math. Soc, 1977, préface de J. Pettis, Mathematical Surveys no. 15. | Zbl

[24] X. T. Duong - $H^{\infty}$ functional calculus of second order elliptic partial differential operators on $L_{p}$ -spaces, Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 24, Austral. Nat. Univ., 1989, p. 91-102.

[25] T. Fack - Type and cotype inequalities for non-commutative $L^{p}$ -spaces, J. Operator Theory 7 (1987), p. 255-279. | Zbl

[26] T. Fack & H. Kosaki - Generalized $s$ -numbers of $τ$ -measurable operators, Pacific J. Math. 123 (1986), p. 269-300. | Zbl | DOI

[27] G. Fendler - Dilations of one parameter semigroups of positive contractions on $L_{p}$ -spaces, Canad. J. Math. 49 (1997), p. 736-748. | Zbl | DOI

[28] J. B. Garnett - Bounded analytic functions, Pure Appl. Math., Academic Press Inc., 1981. | Zbl

[29] J. A. Goldstein - Semigroups of linear operators and applications, Oxford University Press, 1985. | Zbl

[30] U. Haagerup - An example of a non nuclear $C^{*}$ -algebra, which has the metric approximation property, Invent. Math. 50 (1979), p. 279-293. | EuDML | Zbl | DOI

[31] A. Harcharras - Fourier analysis, Schur multipliers on $S^{p}$ and noncommutative $(p)$ -sets, Studia Math. 137 (1999), p. 203-260. | EuDML | Zbl | DOI

[32] M. Hieber & J. Prüss - Functional calculi for linear operators in vector valued $L^{p}$ -spaces via the transference principle, Adv. Differential Equations 3 (1998), p. 847-872. | Zbl

[33] M. Junge - Doob's inequality for non-commutative martingales, J. Reine Angew. Math. 549 (2002), p. 149-190. | Zbl

[34] M. Junge, C. Le Merdy & Q. Xu - Calcul fonctionnel et fonctions carrées dans les espaces $L^{p}$ non commutatifs, C. R. Acad. Sci. Paris Sér. I Math. 337 (2003), p. 93-98. | Zbl | DOI

[35] M. Junge & Q. Xu - Haagerup's reduction on noncommutative $L^{p}$ -spaces and applications, in préparation.

[36] M. Junge & Q. Xu, Noncommutative maximal ergodic theorems, J. Amer. Math. Soc. to appear. | Zbl

[37] M. Junge & Q. Xu, Théorèmes ergodiques maximaux dans les espaces $L_{p}$ non commutatifs, C. R. Acad. Sci. Paris, Sér. I Math. 334 (2002), p. 773-778. | Zbl | DOI

[38] M. Junge & Q. Xu, Noncommutative Burkholder/Rosenthal inequalities, Ann. Probab. 31 (2003), p. 948-995. | Zbl | DOI

[39] R. Kadison - A generalized Schwarz inequality and algebraic invariants for operator algebras, Ann. of Math. 56 (1952), p. 494-503. | MR | Zbl | DOI

[40] R. Kadison & J. R. Ringrose - Fundamentals of the theory of operator algebras, I and, Graduate Studies in Math., Amer. Math. Soc., 1997, 15 and | MR | Zbl | DOI

R. Kadison & J. R. Ringrose - Fundamentals of the theory of operator algebras, II, Graduate Studies in Math., Amer. Math. Soc., 1997, 16. | Zbl | DOI

[41] N. Kalton & L. Weis - The $H^{\infty}$ calculus and sums of closed operators, Math. Ann. 321 (2001), p. 319-345. | MR | Zbl | DOI

[42] H. Kosaki - Applications of the complex interpolation method to a von Neumann algebra : noncommutative $L^{p}$ -spaces, J. Funct. Anal. 56 (1984), p. 29-78. | MR | Zbl | DOI

[43] P. C. Kunstmann & L. Weis - Maximal $L_{p}$ -regularity for parabolic équations, Fourier multiplier theorems, and $H^{\infty}$ -functional calculus, in Functional analytic methods for evolution equations (M. Iannelli, R. Nagel & S. Piazzera, eds.), Lect. Notes in Math., vol. 1855, Springer-Verlag, 2004, p. 65-311. | MR | Zbl | DOI

[44] F. Lancien, G. Lancien & C. Le Merdy - A joint functional calculus for sectorial operators with commuting resolvents, Proc. London Math. Soc. 77 (1998), p. 387-414. | MR | Zbl | DOI

[45] C. Le Merdy - The similarity problem for bounded analytic semigroups on Hilbert space, Semigroup Forum 56 (1998), p. 205-224. | MR | Zbl | DOI

[46] C. Le Merdy, $H^{\infty}$ functional calculus and applications to maximal regularity, Pub. Math. UFR Sci. Tech. Besançon 16 (1999), p. 41-77. | MR | Zbl

[47] C. Le Merdy, Two results about $H^{\infty}$ functional calculus on analytic UMD banach spaces, J. Austr. Math. Soc. 74 (2003), p. 351-378. | MR | Zbl | DOI

[48] C. Le Merdy, The Weiss conjecture for bounded analytic semigroups, J. London Math. Soc. 67 (2003), p. 715-738. | MR | Zbl | DOI

[49] C. Le Merdy, On square functions associated to sectorial operators, Bull. Soc. Math. France 132 (2004), p. 137-156. | MR | EuDML | Zbl | Numdam | DOI

[50] J. Lindenstrauss & L. Tzafriri - Classical Banach spaces II, Springer-Verlag, 1979. | MR | DOI

[51] F. Lust-Piquard - Inégalités de Khintchine dans $C_{p} (1 < p < \infty)$ , C. R. Acad. Sci. Paris 303 (1986), p. 289-292. | Zbl | MR

[52] F. Lust-Piquard & G. Pisier - Non-commutative Khintchine and Paley inequalities, Arkiv för Math. 29 (1991), p. 241-260. | MR | Zbl | DOI

[53] A. Mcintosh - Operators which have an $H^{\infty}$ functional calculus, in Miniconference on operator theory and partial differential equations, Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 14, Austral. Nat. Univ., 1986, p. 210-231. | MR

[54] A. Mcintosh & A. Yagi - Operators of type $ω$ without a bounded $H^{\infty}$ functional calculus, in Miniconference on operators in analysis, Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 24, Austral. Nat. Univ., 1989, p. 159-172. | MR

[55] T. Mei - Operator valued Hardy spaces, Mem. Amer. Math. Soc, to appear. | Zbl | MR

[56] R. Nagel - One parameter semigroups of positive operators, Lect. Notes in Math., vol. 1184, Springer-Verlag, 1986. | MR

[57] K. R. Parthasarathy - An Introduction to Quantum Stochastic Calculus, Monographs in Math., vol. 85, Birkhäuser Verlag, 1992. | MR | Zbl

[58] V. I. Paulsen - Completely bounded maps and operator algebras, Cambridge Studies in Adv. Math., vol. 78, Cambridge University Press, 2002. | MR | Zbl

[59] G. Pisier - Holomorphic semigroups and the geometry of Banach spaces, Ann. of Math. 115 (1982), p. 375-392. | MR | Zbl | DOI

[60] G. Pisier, Factorization of linear operators and geometry of Banach spaces, CBMS Regional Conf. Series in Math., vol. 60, Amer. Math. Soc, 1986. | MR | Zbl | DOI

[61] G. Pisier, Complex interpolation and regular operators between Banach lattices, Arch. Math. 62 (1994), p. 261-269. | MR | Zbl | DOI

[62] G. Pisier, Non-commutative vector valued $L_{p}$ -spaces and completely $p$ -summing maps, Astérisque, vol. 247, Soc. Math. France, 1998. | MR | Zbl | Numdam

[63] G. Pisier, Introduction to operator space theory, London Math. Soc. Lect. Note Series, vol. 294, Cambridge University press, 2003. | MR | Zbl

[64] G. Pisier & Q. Xu - Non-commutative martingale inequalities, Commun. Math. Phys. 189 (1997), p. 667-698. | MR | Zbl | DOI

[65] G. Pisier & Q. Xu, Non-commutative $L^{p}$ -spaces, in Handbook of the geometry of Banach spaces, Vol. II (W. Johnson & J. Lindenstrauss, eds.), Elsevier, 2003, p. 1459-1517. | MR | Zbl | DOI

[66] R. J. Plymen & P. L. Robinson - Spinors in Hilbert space, Cambridge Tracts in Math., vol. 114, Cambridge University Press, 1994. | MR | Zbl

[67] J. Prüss - Evolutionary integral equations and applications, Monographs in Math., vol. 87, Birkhäuser Verlag, 1993. | MR | DOI

[68] N. Randrianantoanina - Non-commutative martingale transforms, J. Funct. Anal. 194 (2002), p. 181-212. | MR | Zbl

[69] S. Sakai - $C^{*}$ -algebras and $W^{*}$ -algebras, Classics in Mathematics, Springer-Verlag, 1998. | MR | DOI

[70] E. M. Stein - Topics in harmonic analysis related to the Littlewood-Paley theory., Ann. Math. Studies, Princeton University Press, 1970. | MR | Zbl

[71] B. Sz-Nagy & C. Foias - Harmonic analysis of operators on Hilbert space, Akademiai Kiadó, 1970. | MR | Zbl

[72] M. Takesaki Conditional expectations in von Neumann algebras, J. Functional Analysis 9 (1972), p. 306-321. | MR | Zbl | DOI

[73] M. Terp - $L^{p}$ -spaces associated with von Neumann algeras, 1981, Notes, Math. Instititue, Copenhagen University.

[74] M. Uijterdijk - Functional calculi for closed sectorial operators, Ph.D. Thesis, Delft, 1988.

[75] D. Voiculescu - Symmetries of some reduced free product $C^{*}$ -algebras, in Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983), Lect. Notes in Math., vol. 1132, Springer-Verlag, 1985, p. 556-588. | MR | DOI

[76] D. Voiculescu, K. Dykema & A. Nica - Free random variables, CRM Monograph Series, vol. 1, Amer. Math. Soc, 1992. | MR | Zbl | DOI

[77] J. Voigt - Abstract Stein interpolation, Math. Nachr. 157 (1992), p. 197-199. | MR | Zbl | DOI

[78] L. Weis - A new approach to maximal $L_{p}$ -regularity, in Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998) (G. Lumer & L. Weis, eds.), Lect. Notes in Pure Appl. Math., vol. 215, Marcel Dekker, 2001, p. 195-214. | MR | Zbl

[79] L. Weis, Operator-valued Fourier multiplier theorems and maximal $L_{p}$ -regularity, Math. Ann. 319 (2001), p. 735-758. | MR | Zbl | DOI

[80] Q. Xu - Recent developments on non-commutative martingale inequalities, in Functional space theory and its applications, Proc. of International Conference and 13th academic symposium in China, Research information Ltd, 2003, p. 283-313.