Remarks on F-spaces of analytic
functions
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Joel H. Shapiro
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Banach spaces of
analytic functions (Proceedings of Pelczynski conference, Kent
State University 1976), pp 107--124. Lecture Notes in Mathematics,
Springer 1977
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| 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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