Genericity and Universality for Operator Ideals

TL;DR

This paper constructs new universal operators for certain operator ideal complements and uses Descriptive Set Theory to analyze the genericity of classical operator ideals, revealing when universal operators do or do not exist.

Contribution

It introduces new universal operators for some ideals' complements and characterizes the genericity of classical operator ideals using descriptive set theory.

Findings

Many classical operator ideals are shown to be generic.

Conditions are provided for when the complement of an ideal lacks a universal operator.

A new proof is given for the non-existence of a universal operator for completely continuous operators.

Abstract

A bounded linear operator $U$ between Banach spaces is universal for the complement of some operator ideal $\mathfrak{J}$ if it is a member of the complement and it factors through every element of the complement of $\mathfrak{J}$. In the first part of this paper, we produce new universal operators for the complements of several ideals and give examples of ideals whose complements do not admit such operators. In the second part of the paper, we use Descriptive Set Theory to study operator ideals. After restricting attention to operators between separable Banach spaces, we call an operator ideal $\mathfrak{J}$ generic if anytime an operator $A$ has the property that every operator in $\mathfrak{J}$ factors through a restriction of $A$, then every operator between separable Banach spaces factors through a restriction of $A$. We prove that many of classical operator ideals are generic (i.e. strictly singular, weakly compact, Banach-Saks) and give a sufficient condition, based on the complexity of the ideal, for when the complement does not admit a universal operator. Another result is a new proof of a theorem of M. Girardi and W.B. Johnson which states that there is no universal operator for the complement of the ideal of completely continuous operators.