ξ-completely continuous operators and ξ-Schur Banach spaces

Abstract

For each ordinal 0⩽ξ⩽ω1, we introduce the notion of a ξ-completely continuous operator and prove that for each ordinal 0<ξ<ω1, the class Vξ of ξ-completely continuous operators is a closed, injective operator ideal which is not surjective, symmetric, or idempotent. We prove that for distinct 0⩽ξ,ζ⩽ω1, the classes of ξ-completely continuous operators and ζ-completely continuous operators are distinct. We also introduce an ordinal rank v for operators such that v(A)=ω1 if and only if A is completely continuous, and otherwise v(A) is the minimum countable ordinal such that A fails to be ξ-completely continuous. We show that there exists an operator A such that v(A)=ξ if and only if 1⩽ξ⩽ω1, and there exists a Banach space X such that v(IX)=ξ if and only if there exists an ordinal γ⩽ω1 such that ξ=ωγ. Finally, prove that for every 0<ξ<ω1, the class {A∈L:v(A)⩾ξ} is Π11-complete in L, the coding of all operators between separable Banach spaces. This is in contrast to the class V∩L, which is Π21-complete in L.