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- W3146082147 abstract "Let G be a bounded open subset in the complex plane and let H~2(G) denote the Hardy space on G. We call a bounded simply connected domain W perfectly connected if the boundary value function of the inverse of the Riemann map from W onto the unit disk D is almost 1-1 with respect to the Lebesgue measure on D and if the Riemann map belongs to the weak-star closure of the polynomials in H~∞(W). Our main theorem states: in order that for each M∈Lat (M_z), there exist u∈H~∞(G) such that M=∨{uH~2(G)}, it is necessary and sufficient that the following hold: (1) each component of G is a perfectly connected domain; (2) the harmonic measures of the components of G are mutually singular; (3) the set of polynomials is weak-star dense in H~∞(G). Moreover, if G satisfies these conditions, then every M∈Lat (M_z) is of the form uH~2(G), where u∈H~∞(G) and the restriction of u to each of the components of G is either an inner function or zero." @default.
- W3146082147 created "2021-04-13" @default.
- W3146082147 creator A5009807855 @default.
- W3146082147 date "2008-01-01" @default.
- W3146082147 modified "2023-09-24" @default.
- W3146082147 title "Beurling's theorem and invariant subspaces for the shift on Hardy spaces" @default.
- W3146082147 hasPublicationYear "2008" @default.
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