This is an announcement for the paper “$xi$-completely continuous operators and $\xi$-Schur Banach spaces” by R.M. Causeyhttps://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1, K. Navoyanhttps://arxiv.org/find/math/1/au:+Navoyan_K/0/1/0/all/0/1.
Abstract: For each ordinal $0\leq\xi\leq\omega_1$, we introduce the notion of a $\xi$-completely continuous operator and prove that for each ordinal $0<\xi<\omega_1$, the class $\mathcal{B}_{\xi}$of $\xi$-completely continuous operators is a closed, injective operator ideal which is not surjective, symmetric, or idempotent. We prove that for distinct $0\leq\xi, \zeta\leq\omega_1$, the classes of $\xi$-completely continuous operators and $\zeta$-completely continuous operators are distinct. We also introduce an ordinal rank $\nu$ for operators such that $\nu(A)=\omega_1$ if and only if $A$ is completely continuous, and otherwise $\nu(A)$ is the minimum countable ordinal such that $A$ fails to be $\xi$-completely continuous. We show that there exists an operator $A$ such that $\nu(A)=\xi$ if and only if $1\leq\xi\leq\omega_1$, and there exists a Banach space $X$ such that $\nu(I_X)=\xi$ if and only if there exists an ordinal $\gamma\leq\omega_1$ such that $\xi=\omega^{\gamma}$. Finally, prove that for every $0<\xi<\omega_1$, the class ${A\in\mathcal{L}:\nu(A)\geq\xi}$ is $\Pi_1^1$-complete in $\mathcal{L}$, the coding of all operators between separable Banach spaces. This is in contrast to the class $\mathcal{B}\cap\mathcal{L}$, which is $\Pi_2^1$-complete in $\mathcal{L}$.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1803.09343