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Proceedings of the American Mathematical Society

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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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$l_\infty$ and interpolation between Banach lattices
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by Nahum Zobin and Veronica Zobin PDF
Proc. Amer. Math. Soc. 125 (1997), 827-833 Request permission

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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Additional 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