xi-completely continuous operators and xi-Schur Banach spaces

摘要

For each ordinal 0 = xi = omega(1), we introduce the notion of a xi-completely continuous operator and prove that for each ordinal 0 xi omega(1), the class D-xi of xi-completely continuous operators is a closed, injective operator ideal which is not surjective, symmetric, or idempotent. We prove that for distinct 0 = xi, zeta = omega(1,) the classes of xi-completely continuous operators and zeta-completely continuous operators are distinct. We also introduce an ordinal rank v for operators such that v(A) = omega(1) if and only if A is completely continuous, and otherwise v(A) is the minimum countable ordinal such that A fails to be xi-completely continuous. We show that there exists an operator A such that v(A) = xi, if and only if 1 = xi = omega(1), and there exists a Banach space X such that v(I-x) = xi if and only if there exists an ordinal gamma = omega(1) such that xi = omega(gamma). Finally, prove that for every 0 xi omega(1), the class {A is an element of L : v(A) = xi} is Pi(1)(1)-complete in L, the coding of all operators between separable Banach spaces. This is in contrast to the class on D boolean AND L, which is Pi(1)(2)-complete in L. (C) 2018 Elsevier Inc. All rights reserved.