In the previous chapter we saw that every (closed) invariant subspace of Dμ is cyclic, in other words, that it is generated by a single function in Dμ. In this chapter, we shall take the opposite point of view: starting with f ∈ Dμ, can we identify the invariant subspace that it generates? It is easy to see that g belongs to this invariant subspace if and only if there is a sequence of polynomials (pn) such that ||pnf − g||Dμ → 0, but in practice it is often difficult to determine which g have this property. Therefore we seek other, more explicit descriptions of the invariant subspace generated by f.

In particular, we pose the question: which functions f generate the whole of Dμ? A complete answer to this is not known, even for the special case of the Dirichlet space D. In fact the characterization of those functions cyclic for D is the subject of a conjecture involving the logarithmic capacity of boundary zero sets. We shall present some partial solutions to this conjecture.

Cyclicity inDμ

We begin by formalizing the notions above. Let μ be a finite positive measure on T, and let Dμ be the associated harmonically weighted Dirichlet space. As usual, if μ is normalized Lebesgue measure, then Dμ is just the classical Dirichlet space D, so all the results of this section apply in particular to D.