Noncommutative Boyd interpolation theorems
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Abstract:
We present a new, elementary proof of Boyd’s interpolation theorem. Our approach naturally yields a noncommutative version of this result and even allows for the interpolation of certain operators on $\ell ^1$-valued noncommutative symmetric spaces. By duality we may interpolate several well-known noncommutative maximal inequalities. In particular we obtain a version of Doob’s maximal inequality and the dual Doob inequality for noncommutative symmetric spaces. We apply our results to prove the Burkholder-Davis-Gundy and Burkholder-Rosenthal inequalities for noncommutative martingales in these spaces.References
- Turdebek N. Bekjan and Zeqian Chen, Interpolation and $\Phi$-moment inequalities of noncommutative martingales, Probab. Theory Related Fields 152 (2012), no. 1-2, 179–206. MR 2875756, DOI 10.1007/s00440-010-0319-2
- T. Bekjan, Z. Chen, and A. Osȩkowski, Noncommutative moment maximal inequalities. arXiv: 1108.2795
- A. Benedek, A.-P. Calderón, and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356–365. MR 133653, DOI 10.1073/pnas.48.3.356
- Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
- David W. Boyd, Indices of function spaces and their relationship to interpolation, Canadian J. Math. 21 (1969), 1245–1254. MR 412788, DOI 10.4153/CJM-1969-137-x
- A.-P. Calderón, Spaces between $L^{1}$ and $L^{\infty }$ and the theorem of Marcinkiewicz, Studia Math. 26 (1966), 273–299. MR 203444, DOI 10.4064/sm-26-3-301-304
- Sjoerd Dirksen, Ben de Pagter, Denis Potapov, and Fedor Sukochev, Rosenthal inequalities in noncommutative symmetric spaces, J. Funct. Anal. 261 (2011), no. 10, 2890–2925. MR 2832586, DOI 10.1016/j.jfa.2011.07.015
- Sjoerd Dirksen and Éric Ricard, Some remarks on noncommutative Khintchine inequalities, Bull. Lond. Math. Soc. 45 (2013), no. 3, 618–624. MR 3065031, DOI 10.1112/blms/bds107
- Peter G. Dodds, Theresa K.-Y. Dodds, and Ben de Pagter, Noncommutative Banach function spaces, Math. Z. 201 (1989), no. 4, 583–597. MR 1004176, DOI 10.1007/BF01215160
- Peter G. Dodds, Theresa K. Dodds, and Ben de Pagter, Fully symmetric operator spaces, Integral Equations Operator Theory 15 (1992), no. 6, 942–972. MR 1188788, DOI 10.1007/BF01203122
- Peter G. Dodds, Theresa K.-Y. Dodds, and Ben de Pagter, Noncommutative Köthe duality, Trans. Amer. Math. Soc. 339 (1993), no. 2, 717–750. MR 1113694, DOI 10.1090/S0002-9947-1993-1113694-3
- P. Dodds, B. de Pagter, and F. Sukochev, Noncommutative integration. Work in progress.
- Thierry Fack and Hideki Kosaki, Generalized $s$-numbers of $\tau$-measurable operators, Pacific J. Math. 123 (1986), no. 2, 269–300. MR 840845
- Ying Hu, Noncommutative extrapolation theorems and applications, Illinois J. Math. 53 (2009), no. 2, 463–482. MR 2594639
- Yong Jiao, Burkholder’s inequalities in noncommutative Lorentz spaces, Proc. Amer. Math. Soc. 138 (2010), no. 7, 2431–2441. MR 2607873, DOI 10.1090/S0002-9939-10-10267-6
- Yong Jiao, Martingale inequalities in noncommutative symmetric spaces, Arch. Math. (Basel) 98 (2012), no. 1, 87–97. MR 2885535, DOI 10.1007/s00013-011-0343-1
- Marius Junge, Doob’s inequality for non-commutative martingales, J. Reine Angew. Math. 549 (2002), 149–190. MR 1916654, DOI 10.1515/crll.2002.061
- Marius Junge and Quanhua Xu, Noncommutative Burkholder/Rosenthal inequalities, Ann. Probab. 31 (2003), no. 2, 948–995. MR 1964955, DOI 10.1214/aop/1048516542
- Marius Junge and Quanhua Xu, Noncommutative maximal ergodic theorems, J. Amer. Math. Soc. 20 (2007), no. 2, 385–439. MR 2276775, DOI 10.1090/S0894-0347-06-00533-9
- N. J. Kalton, Compact and strictly singular operators on certain function spaces, Arch. Math. (Basel) 43 (1984), no. 1, 66–78. MR 758342, DOI 10.1007/BF01193613
- N. J. Kalton, N. T. Peck, and James W. Roberts, An $F$-space sampler, London Mathematical Society Lecture Note Series, vol. 89, Cambridge University Press, Cambridge, 1984. MR 808777, DOI 10.1017/CBO9780511662447
- N. J. Kalton and F. A. Sukochev, Symmetric norms and spaces of operators, J. Reine Angew. Math. 621 (2008), 81–121. MR 2431251, DOI 10.1515/CRELLE.2008.059
- 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. MR 649411
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367
- Françoise Lust-Piquard, Inégalités de Khintchine dans $C_p\;(1<p<\infty )$, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 7, 289–292 (French, with English summary). MR 859804
- Françoise Lust-Piquard and Gilles Pisier, Noncommutative Khintchine and Paley inequalities, Ark. Mat. 29 (1991), no. 2, 241–260. MR 1150376, DOI 10.1007/BF02384340
- Lech Maligranda, Indices and interpolation, Dissertationes Math. (Rozprawy Mat.) 234 (1985), 49. MR 820076
- J. Marcinkiewicz, Sur l’interpolation d’opérations, C. R. Acad. Sci., Paris, 208:1272–1273, 1939.
- Stephen J. Montgomery-Smith, The Hardy operator and Boyd indices, Interaction between functional analysis, harmonic analysis, and probability (Columbia, MO, 1994) Lecture Notes in Pure and Appl. Math., vol. 175, Dekker, New York, 1996, pp. 359–364. MR 1358172
- Edward Nelson, Notes on non-commutative integration, J. Functional Analysis 15 (1974), 103–116. MR 0355628, DOI 10.1016/0022-1236(74)90014-7
- 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). MR 1648908
- Gilles Pisier and Quanhua Xu, Non-commutative martingale inequalities, Comm. Math. Phys. 189 (1997), no. 3, 667–698. MR 1482934, DOI 10.1007/s002200050224
- E. M. Stein and Guido Weiss, An extension of a theorem of Marcinkiewicz and some of its applications, J. Math. Mech. 8 (1959), 263–284. MR 0107163, DOI 10.1512/iumj.1959.8.58019
- Q. Xu, Operator spaces and noncommutative $L_p$, Nankai University summer school lecture notes, 2007.
- Quan Hua Xu, Analytic functions with values in lattices and symmetric spaces of measurable operators, Math. Proc. Cambridge Philos. Soc. 109 (1991), no. 3, 541–563. MR 1094753, DOI 10.1017/S030500410006998X
- A. Zygmund, On a theorem of Marcinkiewicz concerning interpolation of operations, J. Math. Pures Appl. (9) 35 (1956), 223–248. MR 80887
Additional Information
- Sjoerd Dirksen
- Affiliation: Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
- Address at time of publication: Department of Mathematics, RWTH Aachen University, 52056 Aachen, Germany
- Email: sjoerd.dirksen@hcm.uni-bonn.de, dirksen@mathc.rwth-aachen.de
- Received by editor(s): April 12, 2012
- Received by editor(s) in revised form: March 12, 2013
- Published electronically: December 10, 2014
- Additional Notes: This research was supported by VICI subsidy 639.033.604 of the Netherlands Organisation for Scientific Research (NWO) and the Hausdorff Center for Mathematics
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 4079-4110
- MSC (2010): Primary 46B70, 46L52, 46L53
- DOI: https://doi.org/10.1090/S0002-9947-2014-06185-0
- MathSciNet review: 3324921