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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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$l_\infty$ and interpolation between Banach lattices
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by Nahum Zobin and Veronica Zobin
Proc. Amer. Math. Soc. 125 (1997), 827-833
DOI: https://doi.org/10.1090/S0002-9939-97-03646-0

Abstract:

We study the possibility of obtaining the $l_{\infty }$-norm by an interpolation method starting from a couple of Banach lattice norms. We describe all couples of Banach lattice norms in ${\mathbb {R}}^{n}$ such that the $l_{\infty }$-norm is a strict interpolation norm between them. Further we consider the possibility of obtaining the $l_{\infty }$-norm by any method which guarantees interpolation of not only linear operators ( = bilinear forms on ${\mathbb {R}}^{n}\times {\mathbb {R}}^{n})$ but also of all polylinear forms. Here we show that either one of the initial norms has to be proportional to the $l_{\infty }$-norm, or both have to be weighted $l_{\infty }$-norms.
References
  • Yu. A. Brudnyĭ, S. G. Kreĭn, and E. M. Semënov, Interpolation of linear operators, Mathematical analysis, Vol. 24 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1986, pp. 3–163, 272 (Russian). Translated in J. Soviet Math 42 (1988), no. 6, 2009–2112. MR 887950
  • L.Veselova, Duality in Interpolation of Operators, Ph.D. thesis, Kazan State University, Russia (1991), pp. 1–110.
  • Nahum Zobin and Veronica Zobina, A general theory of sufficient collections of norms with a prescribed semigroup of contractions, Nonselfadjoint operators and related topics (Beer Sheva, 1992) Oper. Theory Adv. Appl., vol. 73, Birkhäuser, Basel, 1994, pp. 397–416. MR 1320556
  • V. G. Zobina, Interpolation in spaces with given symmetries and uniqueness of sufficient collections, Soobshch. Akad. Nauk Gruzin. SSR 93 (1979), no. 2, 301–303 (Russian, with English and Georgian summaries). MR 554733
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Bibliographic Information
  • Nahum Zobin
  • Affiliation: Department of Mathematics and Computer Science, University of Miami, Coral Gables, Florida 33124
  • Address at time of publication: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
  • Email: zobin@math.miami.edu, zobin@math.ohio-state.edu
  • Veronica Zobin
  • Affiliation: Department of Mathematics, Technion – I.I.T., Haifa, 32000, Israel
  • Address at time of publication: Department of Mathematics and Computer Science, University of Miami, Coral Gables, Florida 33124
  • Email: zobin@math.miami.edu
  • Received by editor(s): December 1, 1994
  • Received by editor(s) in revised form: September 13, 1995
  • Additional Notes: The research of the first author was partially supported by grants from the Ministry of Absorption, the Ministry of Science and Technology (Israel) and by the Rashi Foundation (France-Israel). The research of the second author was partially supported by a grant from the Ministry of Science, Israel, and “Maagara"—a special project for absorption of new immigrants—at the Department of Mathematics, Technion, Haifa, Israel.
  • Communicated by: Palle E. T. Jorgensen
  • © Copyright 1997 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 125 (1997), 827-833
  • MSC (1991): Primary 46M35
  • DOI: https://doi.org/10.1090/S0002-9939-97-03646-0
  • MathSciNet review: 1353410