Some Banach Space Properties of the Duals of the Disk Algebra and Hoo

摘要

The program of knowing the Banach space properties of the space H∞ of bounded analytic functions on the disk and their duals has been greatly stimulated since the appearance of Pelczynski's notes [Pe2] about Banach spaces of analytic func- tions. Specifically, Pelczynski asked about some classical properties such as weak sequential completeness, the Dunford-pettis property, property (V), or being a Grothendieck space. Most of this program has been realized in a series of papers by Bourgain [B1; B3; B4]. Namely, Bourgain has shown that Hoo and all its duals have the Dunforchpettis property, that all the odd duals of H∞ are wealdy sequen- tially complete and have property (V* ) , and that H∞ is a Grothendieck space and has property (V). Therefore, the following question is open` Are the even duals of Hoo Grothendieck spaces, and do they have property (V)? In this paper, we give a positive answer to this question. In fact, we prove several stronger results which we detall below.