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

Abstract

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 $\mathcal{L}$.