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**Extra resources for Banach Spaces of Analytic Functions and Absolutely Summing Operators (Regional Conference Series in Mathematics ; No. 30)**

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We show that A, A * and L I /H~ are weakly complete and have the Dunford-Pettis property. We also show that weakly compact operators from A into an arbitrary Banach space are characterized by their behavior on subspaces isomorphic to co' exactly in the same way as the weakly compact operators from a C(S)-space. In the second part of the section, we deal with complemented subspaces of A I and LI/H~. We show, in particular, that LI/H~ does not contain any complemented subspace isomorphic to L I. At the end of the section we discuss v~rious related open problems.

S. sequence in a Banach space X which does not converge to zero in norm, then there is an isomorphic embedding from Co into X which takes the unit vectors of Co onto a subsequence of the sequence (xn)' (iv) '* (i). 1. Hence, by this corollary, the weak closure of W in C(3D)* is weakly compact. Hence T* is weakly compact and therefore T is weakly compact (cf. [D-SI, Chapter VI]). 4. Let W be a bounded subset of A * (resp. L 1 /H~) whose weak closure is not weakly compact. Then there is a sequence (x~) of elements of W which is equiv- alent to the unit vector basis of {1 and spans a complemented subspace of A * (resp.

1 (resp. 1). To prove (b) for L I /H ~, resp. 1 (resp. 1) implies that every weak Cauchy sequence in L I /H~ (resp. in C(aD)*) is the image under a quotient map of a weak Cauchy sequence in L I (resp. in C(aD)*). Now we use the fact that L I (resp. [C(aD)) *) has the Dunford-Pettis property (cf. [D-SI, Chapter VI)). The assertion (b) for A follows from Grothendieck's observation that if X* has the Dunford-Pettis property then X does (cf. [Gr2), [P9]). 2. If E is one of the spaces A, A*, LI /H~ and T: E -+ E a weakly compact operator, then T2 is compact.