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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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$L_p +L_{\infty }$ 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 $p = 2$.
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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