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There is no strictly singular centralizer on $ L_p$


Author: Félix Cabello Sánchez
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
Published electronically: December 13, 2013
MathSciNet review: 3148529
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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.


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Additional Information

Félix Cabello Sánchez
Affiliation: Departamento de Matemáticas, Universidad de Extremadura, 06071-Badajoz, Spain
Email: fcabello@unex.es

DOI: https://doi.org/10.1090/S0002-9939-2013-11851-4
Keywords: Strictly singular operator, quasilinear centralizer, Rademacher functions, $q$-stable random variable
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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.