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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Power type $ \xi$-asymptotically uniformly smooth norms


Author: R. M. Causey
Journal: Trans. Amer. Math. Soc. 371 (2019), 1509-1546
MSC (2010): Primary 46B03; Secondary 46B06
DOI: https://doi.org/10.1090/tran/7336
Published electronically: September 13, 2018
MathSciNet review: 3894026
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Abstract: We extend a precise renorming result of Godefroy, Kalton, and Lancien regarding asymptotically uniformly smooth norms of separable Banach spaces with Szlenk index $ \omega $. For every ordinal $ \xi $, we characterize the operators, and therefore the Banach spaces, which admit a $ \xi $-asymptotically uniformly smooth norm with power type modulus and compute for those operators the best possible exponent in terms of the values of $ Sz_\xi (\cdot , \varepsilon )$. We also introduce the $ \xi $-Szlenk power type and investigate ideal and factorization properties of classes associated with the $ \xi $-Szlenk power type.


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Additional Information

R. M. Causey
Affiliation: Department of Mathematics, Miami University, Oxford, Ohio 45056
Email: causeyrm@miamioh.edu

DOI: https://doi.org/10.1090/tran/7336
Keywords: Asplund operators, asymptotic uniform smoothness, asymptotic uniform convexity, renorming
Received by editor(s): November 1, 2016
Received by editor(s) in revised form: June 8, 2017
Published electronically: September 13, 2018
Article copyright: © Copyright 2018 American Mathematical Society