Remarks on Fspaces of analytic
functions

Joel H. Shapiro

Banach spaces of
analytic functions (Proceedings of Pelczynski conference, Kent
State University 1976), pp 107124. Lecture Notes in Mathematics,
Springer 1977


Abstract: After discussing in a general way certain
phenomena associated with the failure of the HahnBanach 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). Finallyin the only original result of
the paperI 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 innerouter factorizations). 
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