Power type 𝜉-asymptotically uniformly smooth norms

Causey, R.

doi:10.1090/tran/7336

Power type $\xi$-asymptotically uniformly smooth norms
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by R. M. Causey PDF

Trans. Amer. Math. Soc. 371 (2019), 1509-1546 Request permission

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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