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.

Central to the study of the Dirichlet space is the concept of logarithmic capacity. This assigns to subsets of the unit circle a notion of size which is closely linked to various aspects of functions in D, notably boundary behavior, zeros, multipliers and cyclicity. It will thus be a recurring theme throughout the book.

In this chapter we present a brief but self-contained account of capacity. We do so in an abstract setting, replacing the unit circle by a general compact metric space, and the logarithmic kernel by an arbitrary decreasing function. This entails no extra work, and has the advantage that it covers not only logarithmic capacity but also certain other capacities such as Riesz capacities, which arise naturally in the context of the spaces Dα.

Potentials, energy and capacity

Throughout the chapter, we fix a compact metric space (X, d) and a continuous decreasing function K : (0, ∞) → [0, ∞). The function K is called a kernel (though it has nothing to do with reproducing kernels). We extend the definition of K to 0 by defining K(0) := limt→0+K(t). It may well happen that K(0) = ∞, and in fact this is the case for most interesting kernels, though we do not insist upon it. However, in order to avoid trivialities, we do assume that K ≢ 0.

Definition 2.1.1 Let μ be a finite positive Borel measure on X.

The three classical Hilbert spaces of holomorphic functions in the unit disk are the Hardy, Bergman and Dirichlet spaces. There are several excellent texts covering the Hardy space and the Bergman space. However, to the best of our knowledge, up to now there has been no book devoted to the Dirichlet space. When we began our respective researches into the Dirichlet space, we found ourselves handicapped by the fact that the necessary background information was scattered around the literature, sometimes contained in articles that were difficult to follow. For this reason we began to think about writing an introduction that would be suitable for researchers and graduate students seeking a solid background in the subject. The more we learned about this topic, the more we became convinced that it contains many beautiful ideas that deserve a systematic exposition.

The name Dirichlet space derives from its definition in terms of the so-called Dirichlet integral, arising in Dirichlet's method for solving Laplace's equation (sometimes called the Dirichlet principle). As far as we can determine, the first appearance of the Dirichlet space under that name dates back to two articles of Beurling and Deny in 1958 and 1959, but in fact the notion existed and had been studied at least since Beurling's thesis, which was published in 1933 and written even a little earlier. In the years that followed, Beurling and Carleson laid the foundations of the theory and, after their pioneering work, many other distinguished mathematicians made important contributions.

In this appendix we prove a regularization lemma, Lemma 9.4.5, used in establishing Theorem 9.4.1. The principal tool is the notion of the increasing regularization of a function, and we begin by examining this separately.

Increasing regularization

Definition D.1.1 Given a function u :[0, ∞) → [0, ∞), we define its increasing regularization ũ: [0, ∞) → [0, ∞)by

Clearly ũ is increasing and ũ ≤ u. Also, ũ is maximal with these two properties, in the sense that, if ν is any increasing function with ν ≤ u, then also ν ≤ ũ.

We shall need two results about increasing regularizations. The first is a version of the so-called rising-sun lemma of F. Riesz.

Lemma D.1.2Let u: [0, ∞) → [0, ∞) be a function that is lower semicontinuous and right-continuous. Let ũ be the increasing regularization of u and define U:= {x ∈ [0, ∞): ũ(x) < u(x)}. Then U is open in [0, ∞). Further, if a, b are the endpoints of any component of U, then u(a) ≥ u(b).

Proof Let x ∈ U. Then there exists y > x such that u(y) < u(x). By lower semicontinuity u(y) < u(x′) for all x′ in a neighborhood of x. All such x′ also belong to U. Thus U is open in [0, ∞).

Now let a, b be the endpoints of a component of U.