There is no strictly singular centralizer on $L_p$
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- by Félix Cabello Sánchez PDF
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Abstract:
We prove that if $\Phi$ is a centralizer on $L_p$, where $0<p<\infty$, then there is a copy of $\ell _2$ inside $L_p$ where $\Phi$ is bounded. If $\Phi$ is symmetric, then it is also bounded on a copy of $\ell _q$, provided $0<p<q<2$. This shows that for a wide class of exact sequences $0\to L_p\to Z\to L_p\to 0$ the quotient map is not strictly singular, which generalizes a recent result of Jesús Suárez.References
- Fernando Albiac and Nigel J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006. MR 2192298
- F. Cabello Sánchez, Nonlinear centralizers in homology, to appear in Math. Ann. DOI 10.1007/S00208-013-0942-1.
- Félix Cabello Sánchez, Jesús M. F. Castillo, and Jesús Suárez, On strictly singular nonlinear centralizers, Nonlinear Anal. 75 (2012), no. 7, 3313–3321. MR 2891170, DOI 10.1016/j.na.2011.12.022
- W. B. Johnson, B. Maurey, G. Schechtman, and L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 19 (1979), no. 217, v+298. MR 527010, DOI 10.1090/memo/0217
- N. J. Kalton, Applications of an inequality for $H_1$-functions, Texas Functional Analysis Seminar 1985–1986 (Austin, TX, 1985–1986) Longhorn Notes, Univ. Texas, Austin, TX, 1986, pp. 41–51. MR 1017041
- Nigel J. Kalton, Nonlinear commutators in interpolation theory, Mem. Amer. Math. Soc. 73 (1988), no. 385, iv+85. MR 938889, DOI 10.1090/memo/0385
- N. J. Kalton, Differentials of complex interpolation processes for Köthe function spaces, Trans. Amer. Math. Soc. 333 (1992), no. 2, 479–529. MR 1081938, DOI 10.1090/S0002-9947-1992-1081938-1
- 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. MR 1999193, DOI 10.1016/S1874-5849(03)80033-5
- N. J. Kalton and N. T. Peck, Twisted sums of sequence spaces and the three space problem, Trans. Amer. Math. Soc. 255 (1979), 1–30. MR 542869, DOI 10.1090/S0002-9947-1979-0542869-X
- Richard Rochberg and Guido Weiss, Derivatives of analytic families of Banach spaces, Ann. of Math. (2) 118 (1983), no. 2, 315–347. MR 717826, DOI 10.2307/2007031
- John V. Ryff, Orbits of $L^{1}$-functions under doubly stochastic transformations, Trans. Amer. Math. Soc. 117 (1965), 92–100. MR 209866, DOI 10.1090/S0002-9947-1965-0209866-5
- Jesús Suárez de la Fuente, The Kalton centralizer on $L_p[0,1]$ is not strictly singular, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3447–3451. MR 3080167, DOI 10.1090/S0002-9939-2013-11599-6
Additional Information
- Félix Cabello Sánchez
- Affiliation: Departamento de Matemáticas, Universidad de Extremadura, 06071-Badajoz, Spain
- Email: fcabello@unex.es
- Received by editor(s): January 31, 2012
- Received by editor(s) in revised form: April 16, 2012, and April 19, 2012
- Published electronically: December 13, 2013
- Additional Notes: This work was supported in part by MTM2010-20190-C02-01 and Junta de Extremadura CR10113 “IV Plan Regional I+D+i, Ayudas a Grupos de Investigación”.
- Communicated by: Thomas Schlumprecht
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 949-955
- MSC (2010): Primary 47B10, 46M18, 46A16
- DOI: https://doi.org/10.1090/S0002-9939-2013-11851-4
- MathSciNet review: 3148529