$L_p +L_{\infty }$ and $L_p \cap L_{\infty }$ are not isomorphic for all $1 \leq p < \infty$, $p \neq 2$
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- by Sergey V. Astashkin and Lech Maligranda PDF
- Proc. Amer. Math. Soc. 146 (2018), 2181-2194 Request permission
Abstract:
We prove the result stated in the title. It comes as a consequence of the fact that the space $L_p \cap L_{\infty }$, $1\leq p<\infty$, $p\neq 2$, does not contain a complemented subspace isomorphic to $L_p$. In particular, as a subproduct, we show that $L_p \cap L_{\infty }$ contains a complemented subspace isomorphic to ${\ell }_2$ if and only if $p = 2$.References
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Additional Information
- Sergey V. Astashkin
- Affiliation: Department of Mathematics, Samara National Research University, Moskovskoye shosse 34, 443086, Samara, Russia
- MR Author ID: 197703
- Email: astash56@mail.ru
- Lech Maligranda
- Affiliation: Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden
- MR Author ID: 118770
- Email: lech.maligranda@ltu.se
- Received by editor(s): June 23, 2017
- Received by editor(s) in revised form: August 10, 2017
- Published electronically: February 1, 2018
- Additional Notes: The research of the first author was partially supported by the Ministry of Education and Science of the Russian Federation, project 1.470.2016/1.4, and by the RFBR grant 17-01-00138.
- Communicated by: Thomas Schlumprecht
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 2181-2194
- MSC (2010): Primary 46E30, 46B20, 46B42
- DOI: https://doi.org/10.1090/proc/13928
- MathSciNet review: 3767368