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$l_{\infty }$ and interpolation between Banach lattices


Authors: Nahum Zobin and Veronica Zobin
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
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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.


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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

DOI: https://doi.org/10.1090/S0002-9939-97-03646-0
Keywords: Banach lattices, interpolation of operators, complex method
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
Article copyright: © Copyright 1997 American Mathematical Society