The complexity of some ordinal determined classes of operators

Abstract

We compute the complexity of the classes of operators ${\mathfrak{G}}_{\xi ,\zeta }\cap \mathcal{L}$ and ${\mathfrak{M}}_{\xi ,\zeta }\cap \mathcal{L}$ in the coding of operators between separable Banach spaces. We also prove the non-existence of universal factoring operators for both $\complement {\mathfrak{G}}_{\xi ,\zeta }$ and $\complement {\mathfrak{M}}_{\xi ,\zeta }$. The latter result is an ordinal extension of a result of Johnson and Girardi.