Remarks on F-spaces of analytic functions


Joel H. Shapiro


Banach spaces of analytic functions (Proceedings of Pelczynski conference, Kent State University 1976), pp 107--124. Lecture Notes in Mathematics, Springer 1977
Abstract: After discussing in a general way certain phenomena associated with the failure of the Hahn-Banach theorem in topological vector spaces which are not locally convex, I show how these phenomena arise in the Hardy spaces H^p of the unit disk, 0 < p < 1. Special attention is paid to closed invariant subspaces of H^p which are weakly dense; the inner function corresponding to such an invariant subspace is called weakly invertible. A sufficient condition for weak invertibility is stated and its proof indicated; (this was later proved necessary by James W. Roberts). Finally---in the only original result of the paper---I show that, contrary to the situation in H^p, every singular inner function is weakly invertible in the Hardy algebra N+ (the functions of Nevanlinna class admitting inner-outer factorizations).
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