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- Title
Genericity and Universality for Operator Ideals.
- Authors
Beanland, Kevin; Causey, Ryan M
- 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 whenever 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 classical operator ideals (such as strictly singular, weakly compact, Banach–Saks) are generic 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.
- Subjects
COMPACT operators; BANACH spaces; OPERATOR theory; SET theory; LINEAR operators
- Publication
Quarterly Journal of Mathematics, 2020, Vol 71, Issue 3, p1081
- ISSN
0033-5606
- Publication type
Article
- DOI
10.1093/qmathj/haaa018