Spaces of nuclear and compact operators without a complemented copy of [formula omitted

Abstract

This paper deals with spaces of nuclear operators N(X,Y) and spaces of compact operators K(Z,W) containing no complemented copy of C(ωω). For fixed ordinals ω≤α≤β<ω1 and ξ,η<ω1 of the same cardinality, we provide additional conditions on the Banach spaces X, Y, Z and W ensuring that each of the following statements is equivalent to β<αω. (1)N(X⊕C(ξ),Y⊕C(α)) is isomorphic to N(X⊕C(η),Y⊕C(β)).(2)K(Z⊕C(ξ),W⊕C(α)) is isomorphic to K(Z⊕C(η),W⊕C(β)).These results are generalizations of the classical isomorphic classification of spaces C(α), ω≤α<ω1, due to Bessaga and Pełczyński [7], to the setting of the spaces of operators on X⊕C(α) spaces. The generalization (1) covers the case where X is an ℓ1-predual and Y contains no complemented copy of C(ωω). The generalization (2) covers the case Z=ℓp, 1≤p<∞ and W contains no copy of c0.