On the Dirichlet space for finitely connected regions

Abstract:

This paper is devoted to the study of the Dirichlet space $\operatorname {Dir} (G)$ for finitely connected regions $G$; we are particularly interested in the algebra of bounded multiplication operators on this space. Results in different directions are obtained. One direction deals with the structure of closed subspaces invariant under all bounded multiplication operators. In particular, we show that each such subspace contains a bounded function. For regions with circular boundaries we prove that a finite codimensional closed subspace invariant under multiplication by $z$ must be invariant under all bounded multiplication operators, and furthermore it is of the form $p\operatorname {Dir} (G)$, where $p$ is a polynomial with all its roots lying in $G$. Another direction is to study cyclic and noncyclic vectors for the algebra of all bounded multiplication operators. Typical results are: if $f \in \operatorname {Dir} (G)$ and $f$ is bounded away from zero then $f$ is cyclic; on the other hand, if the zero set of the radial limit function of $f$ on the boundary has positive logarithmic capacity, then $f$ is not cyclic. Also, some other sufficient conditions for a function to be cyclic are given. Lastly, we study transitive operator algebras containing all bounded multiplication operators; we prove that they are dense in the algebra of all bounded operators in the strong operator topology.