Let A be a finitely generated subalgebra of $\Bbb C$[z] and $A_0$ be the collection of all elements of A that are zero at the origin. We consider $H^2$ as a module over A and shall look at the structure of closed A-submodule's of $H^2$.
In the case A=$\Bbb C$[z], the $\Bbb C$[z] module structure of $H^2$ is well known from Beurling's theorem. Any z-invariant subspace M of $H^2$ is in the form M=$g \cdot H^2$, where g is an inner function. For the purpose of application to operators elli ptic in a half cylinder, the vector version of Beurling's theorem was obtained by Peter Lax. It is natural to ask what happens when A is not $\Bbb C$[z] but some finitely generated subalgebra of $\Bbb C$[z]?
Theorem Let A be generated by $p_1$,..,$p_n$ such that
1) deg $p_1$, .. , deg $p_n$ have maximum common divisor = 1
2) $\vert p'_{1}(z)\vert$ + ... +$\vert p'_{n}(z)\vert > 0$.
Let P be the map from $\Bbb C$ to $\Bbb C^n$ by $z\to$($p_1$(z),...,$p_n$(z)) and let $\beta$ = Betti number of the image of P($\Delta$). Then:
\beta \leq sup_{M}\ dim(M \circleddash A(M \circleddash A_{0}M)) \leq m(\beta + r)
where M runs over all finitely generated A-submodules of $H^2$, r is the number of self-tangencies of $P(\Delta)$ and m is a constant depending on the type of self-tangencies.
Corollary (to the proof) Any A-submodule of $H^2$ is finitely generated.