Download PDF by A. Pelczynski: Banach Spaces of Analytic Functions and Absolutely Summing

By A. Pelczynski

This publication surveys effects touching on bases and numerous approximation homes within the classical areas of analytical capabilities. It includes large bibliographical reviews.

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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.

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