arXiv subjects

R. M. Causey

Publications and source records attributed to R. M. Causey.

21 records · Page 2Linked to original sources

arXiv2018

Three and a half asymptotic properties

We define and discuss transfinite asymptotic notions of smoothability, type, and equal norm type. We prove distinctness of these notions for a proper class of ordinals and that each class is an ideal. We also extend some results of Godefroy, Kalton, and Lancien to operators and ordinals greater than zero regarding the equivalence of equal norm asymptotic type and uniform renormings with power type smoothness. Finally, we discuss an extension of a non-linear result for quasi-reflexive, asymptotically $p$-smoothable Banach spaces to quasi-reflexive Banach spaces with asymptotic equal norm type $p$.

math.FA↗

arXiv2018

$ξ$-completely continuous operators and $ξ$-Schur Banach spaces

For each ordinal $0\leqslant ξ\leqslant ω_1$, we introduce the notion of a $ξ$-completely continuous operator and prove that for each ordinal $0< ξ< ω_1$, the class $\mathfrak{V}_ξ$ of $ξ$-completely continuous operators is a closed, injective operator ideal which is not surjective, symmetric, or idempotent. We prove that for distinct $0\leqslant ξ, ζ\leqslant ω_1$, the classes of $ξ$-completely continuous operators and $ζ$-completely continuous operators are distinct. We also introduce an ordinal rank $\textsf{v}$ for operators such that $\textsf{v}(A)=ω_1$ if and only if $A$ is completely continuous, and otherwise $\textsf{v}(A)$ is the minimum countable ordinal such that $A$ fails to be $ξ$-completely continuous. We show that there exists an operator $A$ such that $\textsf{v}(A)=ξ$ if and only if $1\leqslant ξ\leqslant ω_1$, and there exists a Banach space $X$ such that $\textsf{v}(I_X)=ξ$ if and only if there exists an ordinal $γ\leqslant ω_1$ such that $ξ=ω^γ$. Finally, prove that for every $0<ξ<ω_1$, the class $\{A\in \mathcal{L}: \textsf{v}(A) \geqslant ξ\}$ is $Π_1^1$-complete in $\mathcal{L}$, the coding of all operators between separable Banach spaces. This is in contrast to the class $\mathfrak{V}\cap \mathcal{L}$, which is $Π_2^1$-complete in $\mathcal{L}$.

math.FA↗

arXiv2017

Power type $ξ$-Asymptotically uniformly smooth and $ξ$-asymptotically uniformly flat norms

For each ordinal $ξ$ and each $1<p<\infty$, we offer a natural, ismorphic characterization of those spaces and operators which admit an equivalent $ξ$-$p$-asymptotically uniformly smooth norm. We also introduce the notion of $ξ$-asymptotically uniformly flat norms and provide an isomorphic characterization of those spaces and operators which admit an equivalent $ξ$-asymptotically uniformly flat norm. Given a compact, Hausdorff space $K$, we prove an optimal renormong theorem regarding the $ξ$-asymptotic smoothness of $C(K)$ in terms of the Cantor-Bendixson index of $K$. We also prove that for all ordinals, both the isomorphic properties and isometric properties we study pass from Banach spaces to their injective tensor products. We study the classes of $ξ$-$p$-asymptotically uniformly smooth, $ξ$-$p$-asymptotically uniformly smoothable, $ξ$-asymptotically uniformly flat, and $ξ$-asymptotically uniformly flattenable operators. We show that these classes are either a Banach ideal or a right Banach ideal when assigned an appropriate ideal norm.

math.FA↗