Harmonically weighted Dirichlet spaces associated with finitely atomic measures (Q1128242)

Let \(\mu\) be a positive measure on the unit circle and let \(P\mu(z)= \int^{2\pi}_0 {1-| z|^2\over| e^{it}- z|^2} d\mu(t)\) be the Poisson integral of \(\mu\). Associated with \(\mu\) is the harmonically weighted Dirichlet space \(D(\mu)\) of holomorphic functions in \(\mathbb{D}\) for which \(D_\mu(f)= \iint_{\mathbb{D}}| f'|^2 P\mu dA\) is finite, where \(dA\) is the normalized area measure on \(\mathbb{D}\). Note that \(f\in D(\mu)\) implies that \(f\in H^2\), the usual Hardy space. Endowed with the norm \(\| f\|^2_\mu= \| f\|^2_2+ D_\mu(f)\), \(D(\mu)\) becomes a Hilbert space which is contractively contained in \(H^2\). Those spaces arose in the course of investigations of the invariant subspace structure of the shift operator on the classical Dirichlet space [see e.g. \textit{S. Richter} and \textit{C. Sundberg}, Mich. Math. J. 38, No. 3, 355-379 (1991; Zbl 0786.30040) and J. Oper. Theory 28, No. 1, 167-186 (1992; Zbl 0810.46057)]. The paper under review is concerned with the basic structure of the spaces \(D(\mu)\) whenever \(\mu\) is a finite sum of atoms. A description of the wandering vectors for the shift operator on \(D(\mu)\) is given. Recall that a vector is said to be a wandering vector of a given operator if it is orthogonal to its orbit under the positive powers of the operator. As a corollary, the author obtains the result that the unit wandering vectors in \(D(\mu)\) are contractive multipliers of \(D(\mu)\), a result which is known to be true in the classical Dirichlet space [\textit{S. Richter} and \textit{C. Sundberg}, Trans. Am. Math. Soc. 341, No. 2, 863-879 (1994; Zbl 0816.47037)].