Abstract:
We study the closure in the Hardy space or the
disk algebra of algebras
generated by two bounded functions, of which
one is a finite Blaschke product.
We give necessary and sufficient conditions for
density or finite codimension of
such algebras. The conditions are expressed
in terms of the inner part of some
function which is explicitly derived from each
pair of generators. Our results
are based on identifying $z$-invariant subspaces
included in the closure of the
algebra. There are some versions of those
results for the case of the
disk algebra.