Bounded analytic functions in the Dirichlet space

Abstract

In this paper we study the Hilbert space of analytic functions with finite Dirichlet integral in a connected open set C2 in the complex plane. We show that every such function can be represented as a quotient of two bounded analytic functions, each of which has a finite Dirichlet integral. This has several consequences for the structure of invariant subspaces of the algebra of multiplication operators on the Dirichlet space, in case f2 is simply connected. Namely, we show that every nontrivial invariant subspace contains a nontrivial bounded function, that each two nontrivial invariant subspaces have a nontrivial intersection (that is, the algebra is cellular indecomposable), and that each nontrivial invariant subspace has the codimension one property with respect to certain multiplication operators.