Bounded analytic functions in the Dirichlet space
The Dirichlet space D(G) of a connected open set G in the complex plane is the set of all those analytic functions in G whose first derivative is square integrable with respect to area measure on G. The authors show that each such function can be represented as the quotient of two bounded functions in D(G). This result is used to study the structure of the (closed) subspaces of D(G) that are invariant under the algebra of multiplication operators on the Dirichlet space, in case G is simply connected. It is shown that every nontrivial invariant subspace contains a nontrivial bounded function, and that each two nontrivial invariant subspaces have a nontrivial intersection. When G is the unit disk it is shown that if M is an invariant subspace then zM has codimension one in M. This is in contrast to the situation for the Bergman space (the area square-integrable analytic functions in G) where all these results are known to be false, when G is the unit disk.
scientific article; zbMATH DE number 4021568
scientific article