K. Navoyan | Searcharxiv

arXiv2018

Factorization of Asplund operators

We give necessary and sufficient conditions for an operator $A:X\to Y$ on a Banach space having a shrinking FDD to factor through a Banach space $Z$ such that the Szlenk index of $Z$ is equal to the Szlenk index of $A$. We also prove that for every ordinal $ξ\in (0, ω_1)\setminus\{ω^η: η<ω_1\text{\ a limit ordinal}\}$, there exists a Banach space $\mathfrak{G}_ξ$ having a shrinking basis and Szlenk index $ω^ξ$ such that for any separable Banach space $X$ and any operator $A:X\to Y$ having Szlenk index less than $ω^ξ$, $A$ factors through a subspace and through a quotient of $\mathfrak{G}_ξ$, and if $X$ has a shrinking FDD, $A$ factors through $\mathfrak{G}_ξ$.

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↗