Power type $\xi$-asymptotically uniformly smooth norms
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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.References
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Additional Information
- R. M. Causey
- Affiliation: Department of Mathematics, Miami University, Oxford, Ohio 45056
- MR Author ID: 923618
- Email: causeyrm@miamioh.edu
- Received by editor(s): November 1, 2016
- Received by editor(s) in revised form: June 8, 2017
- Published electronically: September 13, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 1509-1546
- MSC (2010): Primary 46B03; Secondary 46B06
- DOI: https://doi.org/10.1090/tran/7336
- MathSciNet review: 3894026