Erratum to: Ideals of Homogeneous Polynomials and Weakly Compact Approximation Property in Banach Spaces

Abstract

In the proof of the implication (c) =⇒ (a) of Theorem 11 we state that P ∈ PW( E; F ) since P = Q ◦ R with R ∈ W(E; G), Q ∈ P(G; F ) and G Banach. But since PL[W] is not contained in PW (see, e.g., example 27 in G. Botelho: Ideals of polynomials generated by weakly compact operators, Note Mat. 25 (2006), 69–102) this conclusion is not true in general. So the given proof of this implication is not true. The proof of Theorem 11 can be completed as follows: (c) =⇒ (a): By following the pattern of the proof of Lemma 9, having the equality P[W]( E; F ) = PL[W]( E; F ) in mind, one can prove that if P(E; F ) = P[W](E; F ) τc for every Banach space F and for some m ∈ N then, L(E; F ) = W(E; F ) τc for every Banach space F . This shows that E has the WCAP.