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Positivity

, Volume 22, Issue 5, pp 1197–1221 | Cite as

Power type asymptotically uniformly smooth and asymptotically uniformly flat norms

  • R. M. Causey
Article

Abstract

We provide a short characterization of p-asymptotic uniform smoothability and asymptotic uniform flatenability of operators and of Banach spaces. We use these characterizations to show that many asymptotic uniform smoothness properties pass to injective tensor products of operators and of Banach spaces. In particular, we prove that the injective tensor product of two asymptotically uniformly smooth Banach spaces is asymptotically uniformly smooth. We prove that for \(1<p<\infty \), the class of p-asymptotically uniformly smoothable operators can be endowed with an ideal norm making this class a Banach ideal. We also prove that the class of asymptotically uniformly flattenable operators can be endowed with an ideal norm making this class a Banach ideal.

Keywords

Szlenk index Operator ideals Ordinal ranks 

Mathematics Subject Classification

Primary 46B03 46B06 Secondary 46B28 47B10 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsMiami UniversityOxfordUSA

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