Spaces of nuclear and compact operators without a complemented copy of

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 ( ω ω ) . For fixed ordinals ω ≤ α ≤ β < ω 1 and ξ , η < ω 1 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 < α ω . (1) ⊕ C ( ξ ) , Y ⊕ C ( α ) ) is isomorphic to N ( X ⊕ C ( η ) , Y ⊕ C ( β ) ) . ⊕ C ( ξ ) , W ⊕ C ( α ) ) is isomorphic to K ( Z ⊕ C ( η ) , W ⊕ C ( β ) ) . results are generalizations of the classical isomorphic classification of spaces C ( α ) , ω ≤ α < ω 1 , 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 ℓ 1 -predual and Y contains no complemented copy of C ( ω ω ) . The generalization (2) covers the case Z = ℓ p , 1 ≤ p < ∞ and W contains no copy of c 0 .