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Ryan M Causey

Publications and source records attributed to Ryan M Causey.

6 recordsLinked to original sources
arXiv2017

Power type $ξ$-asymptotically uniformly smooth norms

We extend a precise renorming result of Godefroy, Kalton, and Lancien regarding asymptotically uniformly smooth norms of separable Banach spaces with Szlenk index $ω$. For every ordinal $ξ$, we characterize the operators, and therefore the Banach spaces, which admit a $ξ$-asymptotically uniformly smooth norm with power type modulus and compute for those operators the best possible exponent in terms of the values of $Sz_ξ(\cdot, \varepsilon)$. We also introduce the $ξ$-Szlenk power type and investigate ideal and factorization properties of classes associated with the $ξ$-Szlenk power type.

math.FA↗
arXiv2017

The Szlenk index of injective tensor products and convex hulls

Given any Banach space $X$ and any weak*-compact subset $K$ of $X^*$, we compute the Szlenk index of the weak*-closed, convex hull of $K$ as a function of the Szlenk index of $K$. Also as an application, we compute the Szlenk index of any injective tensor product of two operators. In particular, we compute the Szlenk index of an injective tensor product $\tens$ in terms of $Sz(X)$ and $Sz(Y)$. As another application, we give a complete characterization of those ordinals which occur as the Szlenk index of a Banach space, as well as those ordinals which occur as the Bourgain $\ell_1$ or $c_0$ index of a Banach space.

math.FA↗
arXiv2017

Power type asymptotically uniformly smooth and asymptotically uniformly flat norms

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.

math.FA↗
arXiv2016

A note on the relationship between the Szlenk and $w^*$-dentability indices of arbitrary $w^*$-compact sets

We prove the optimal estimate between the Szlenk and $w^*$-dentability indices of an arbitrary $w^*$-compact subset of the dual of a Banach space. For a given $w^*$-compact, convex subset $K$ of the dual of a Banach space, we introduce a two player game the winning strategies of which determine the Szlenk index of $K$. We give applications to the $w^*$-dentability index of a Banach space and of an operator.

math.FA↗
arXiv2016

Szlenk and $w^*$-dentability indices of $C(K)$

Given any compact, Hausdorff space $K$ and $1<p<\infty$, we compute the Szlenk and $w^*$-dentability indices of the spaces $C(K)$ and $L_p(C(K))$. We show that if $K$ is compact, Hausdorff, scattered, $CB(K)$ is the Cantor-Bendixson index of $K$, and $ξ$ is the minimum ordinal such that $CB(K)\leqslant ω^ξ$, then $Sz(C(K))=ω^ξ$ and $Dz(C(K))=Sz(L_p(C(K)))= ω^{1+ξ}.$

math.FA↗