Beurling's theorem and invariant subspaces for the shift on Hardy spaces

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.

Beurling’s theorem and invariant subspaces for the shift on Hardy spaces

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 = ∨{uνH 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. This work was supported By SWUFE’s Key Subjects Construction Items Funds of 211 Project of the 11th Five-Year Plan