The complexity of some ordinal determined classes of operators

TL;DR

This paper analyzes the complexity of certain classes of operators between separable Banach spaces and proves the non-existence of universal factoring operators for their complements, extending previous ordinal results.

Contribution

It computes the complexity of specific classes of operators and establishes the non-existence of universal operators for their complements, extending known ordinal-based results.

Findings

Computed the complexity of classes rak{G}_{\xi, \xzeta} rak{M}_{\xi, \xzeta} in Banach spaces

Proved non-existence of universal factoring operators for their complements

Extended results of Johnson and Girardi to ordinal cases

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.